Abstract:
The present invention discloses a method for transmitting information in a closed loop multiple antenna system. The method includes that: a transmission apparatus receives information transmitted from a reception apparatus, wherein the information includes codeword index information; the transmission apparatus selects a corresponding codeword from a double-layer-structure-based codebook according to the codeword index information; the transmission apparatus generates a precoding matrix according to the selected codeword; the transmission apparatus performs precoding for data to be transmitted according to the precoding matrix; the transmission apparatus transmits the precoded data to the reception apparatus. The present invention can enable a formed and used codebook to match channel distribution characteristic of a dual-polarized antenna as much as possible, and minimize performance loss brought by codebook quantization, thereby performance of a precoding system can be effectively improved.

Description:
FIELD OF HE INVENTION 
     The present application is a US National Stage of International Application No. PCT/CN2011/000190, filed 31 Jan. 2011, designating the United States, and claiming priority to Chinese Patent Application No. 201010109923.2 filed 8 Feb. 2010. 
     The present invention relates to the field of communications and particularly to a method and device for transmitting/receiving information in a closed-loop multi-antenna system. 
     BACKGROUND OF THE INVENTION 
     At present, a multi-antenna technology at physical layer has become one of crucial technologies in a next-generation wireless communication system. The multi-antenna technology has numerous advantages, for example, the throughput of the system can be improved with a multiplexing gain of multiple antennas, the performance of the system can be improved with a diversity gain of the multiple antennas, receiving terminals can be distinguished from each other with a directivity gain of the antennas to remove interference between the receiving terminals, etc. 
     A Multiple-Input Multiple-Output (MIMO) system uses multiple transmitting and receiving antennas and utilizes joint processing of a signal in the space, time and frequency domains to obtain rate, diversity and array gains. In the MIMO system, if a transmitter can acquire channel state information somehow, a transmission signal can be optimized dependent upon a channel characteristic to improve a reception quality and lower required complexity of a receiver. In a real system, typically a feedback of quantized channel state information is adopted to lower feedback overhead and feedback delay. The channel state information can be quantized in terms of a channel matrix and a statistic thereof or of a pre-processing parameter recommended by the receiver for use. The quantized channel state information is mapped into a set of a limited number of elements, which is referred to as a codebook. 
     In a codebook-based closed-loop MIMO system, a codebook is a key factor influencing the performance of the system. In each of the transmission modes 4, 5 and 6 of the Rel-8 long Term Evolution (LTE), a Precoding Matrix Indicator (PMI) feedback mechanism is adopted, that is, a User Equipment (UE) measures a channel according to a common pilot and reports index information of a codeword that it recommends for use to an eNodeB (eNB). With four common pilot ports, for example, a Householder transformation-based codebook is adopted, and there are sixteen selectable pre-coding matrices for each Rank of the closed-loop spatially multiplexing mode, i.e., the transmission mode 4. General characteristics of an Rel-8 LTE 4-antenna codebook include: 1) a Rank1 codebook is obtained by searching while considering a tradeoff between uncorrelated and correlated channels; 2) a nesting relationship holds between codebooks at the other Ranks and the Rank1 codebook; 3) a 8PSK (Phase Shift Keying) element is adopted for each codebook element to lower calculation complexity; 4) there are constant modulus and unitary characteristics; and 5) complexity of selecting a codeword can be lowered with a characteristic of the Householder transformation. 
     In an MIMO-based wireless communication system, e.g., an Rd-10 LTE-A system, the use of the PMI feedback mechanism can be continued, but the number of configuration combinations of Ranks and the numbers of transmitting antenna ports is far larger than that in the Rel-8 LTE due to an increased number (possibly eight, for example) of transmitting antennas. On one hand, no codebook has been defined for a large number of configuration combinations of Ranks and the number of transmitting antenna ports. On the other hand, the necessity to design a codebook for a spatially correlated channel has become more prominent as compared with the Rel-8 LTE system because the number of antennas constantly increases and the size of a base station is limited so that it is inevitably necessary to shorten the spacing between antennas and/or adopt polarized antennas. In view of the foregoing considerations, a linear codebook for a spatially correlated channel and a designing method thereof are indispensable technology issues in future wireless communication system researches and standardization. Furthermore, researchers consider as much as possible reserving the characteristics of the Rel-8 LTE codebook while designing a codebook in view of the good performance and excellent characteristics of the Rel-8 LTE 4-antenna codebook. 
     The existing Householder transformation method can not be applied directly to designing of an 8-antenna codebook. Generally the following several codebooks can inherit the characteristics (or a part thereof) of the Rel-8 LTE codebook at present. 
     A codebook 1: The Rel-8 codebook is transformed into an 8-antenna array codebook at the Rank1 The codebook 1 has a drawback of failing to satisfy a nesting relationship although the codebook 1 inherits a part of the characteristics of the Rel-8 LTE 4-antenna codebook, i.e., the constant modulus and the use of an 8PSK element. Since the codebook 1 is based upon extending the Rel-8 LTE 4-antenna codebook which results from optimization in view of both spatially correlated and spatially uncorrelated channels and a part of codewords of which is not suitable for the spatially correlated channel, there is a performance loss of the codebook 1 for an array of antennas at a small spacing. 
     A codebook 2: A number 4 7  of 8×8 complex matrices with their elements being 8PSK elements are constructed in complex Hadamard transformation, and then a Rank1 codebook is obtained by searching in the optimizing method with a tradeoff between uncorrelated and correlated channels. The codebook 2 has a drawback of adopting only 8PSK elements although the codebook 2 inherits most of the characteristics of the Rel-8 LTE codebook, i.e., the constant modulus, unitary and nesting characteristics. Since a space consists of a number 4 7  of 8×8 complex matrices with their elements being 8PSK elements is searched for the codebook 2 in a search criterion considering a performance tradeoff between spatially correlated and spatially uncorrelated channels, the performance of the codebook 2 over the spatially correlated channel is degraded. 
     A codebook 3: The Rank codebook is derived by multiplying sixteen different diagonal matrices by a 8-DFT (Discrete Fourier Transform) matrix, and the Rank1 codebook is composed of first columns of the sixteen 8×8 complex matrices. The codebook 3 has such a drawback that although the codebook 3 inherits a part of the characteristics of the Rel-8 LTE codebook, i.e., the constant modulus, unitary and nesting characteristics, complexity of pre-coding calculation will be increased because 16PSK elements are adopted. If there is no significant performance gain over other codebooks, the increased complexity of pre-coding calculation will become an extra burden. 
     A codebook 4: the codebook consists of respective rows (or columns) of one 8-DFT matrix and one 8×8 unitary matrix. The codebook 4 has a drawback of adopting only 8PSK elements although the codebook 4 inherits a part of the characteristics of the Rel-8 LTE codebook, i.e., the constant modulus, unitary and nesting characteristics. Also since the DFT matrix complies with the construction of an array response, both the codebook 3 and the codebook 4 are suitable for a pre-coding MIMO system over a spatially correlated channel. However, the angles of corresponding array responses of respective codewords in the codebook 3 and the codebook 4 are unchanged regardless of a specific distribution characteristic of the channel, so a performance loss of the pre-coding MIMO system may be incurred in a specific scenario. 
     Therefore, it is desired to propose an efficient design of a codebook and a solution to information transmission and reception using the codebook. 
     SUMMARY OF THE INVENTION 
     An object of embodiments of the invention is intended to address at least one of the foregoing drawbacks in the prior an particularly by designing and using a codebook in a two-stage structure and an information transmitting/receiving method using the codebook to improve the performance of a closed-loop multi-antenna system. 
     In order to attain the foregoing object, an aspect of the invention proposes a method for transmitting information in a closed-loop multi-antenna system, which includes the steps of: receiving, by a transmitting device, information including codeword index information transmitted from a receiving device; selecting, by the transmitting device, a corresponding codeword from a codebook in a two-stage structure according to the codeword index information; generating, by the transmitting device, a pre-coding matrix according to the selected codeword; pre-coding, by the transmitting device, data to be transmitted according to the pre-coding matrix; and transmitting, by the transmitting device, the pre-coded data to the receiving device. 
     A further aspect of the invention proposes a method for receiving information in a closed-loop multi-antenna system, which includes the steps of: receiving, by a receiving device, information transmitted from a transmitting device and estimating channel state information according to the information; selecting, by the receiving device, a corresponding codeword from codebook in a two-stage structure according to the channel state information and generating codeword index information of the selected codeword; and transmitting, by the receiving device, the codeword index information of the codeword to the transmitting device. 
     A further aspect of the invention proposes a transmitting device including a receiving module, a codeword determining module, a pre-coding matrix calculating module, a pre-coding module and a transmitting module, wherein: the receiving module is configured to receive information including codeword index information transmitted from a receiving device; the codeword determining module is configured to select a corresponding codeword from a codebook in a two-stage structure according to the codeword index information; the pre-coding matrix calculating module is configured to generate a pre-coding matrix according to the selected codeword; the pre-coding module is configured to pre-code data to be transmitted according to the pre-coding matrix; and the transmitting module is configured to transmit the pre-coded data to the receiving device. 
     A further aspect of the invention proposes a receiving device including a receiving module, a channel estimating module, a codeword selecting module, a codeword index information generating module and a transmitting module, wherein: the receiving module is configured to receive information transmitted from a transmitting device; the channel estimating module is configured to estimate channel state information according to the information; the codeword selecting module is configured to select a corresponding codeword from a codebook in a two-stage structure according to the channel state information; the codeword index information generating module is configured to generate codeword index information of the selected codeword; and the transmitting module is configured to transmit the codeword index information of the codeword to the transmitting device. 
     In the foregoing solutions proposed according to the embodiments of the invention, a formed and used codebook matches as closely as possibly a channel distribution characteristic of a dually polarized antenna to minimize a performance loss due to quantization of the codebook, thereby improving effectively the performance of a pre-coding system. Furthermore, the codebook proposed by the invention has a two-stage structure and can support both a feedback of one PMI and a feedback of two PMIs. 
     Additional aspects and advantages of the invention will be partly presented in the following description, partly become apparent from the following description or be learned from practicing of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing and/or additional aspects and advantages of the invention will become apparent and readily understood from the following description of embodiments in connection with the drawings in which: 
         FIG. 1  is a schematic diagram of the convention of an antenna indexing method of a dually polarized antenna according to an embodiment of the invention; 
         FIG. 2  is a flow chart of a method for transmitting information in a closed-loop multi-antenna system according to an embodiment of the invention; 
         FIG. 3  is a flow chart of a method for receiving information in a closed-loop multi-antenna system according to an embodiment of the invention; 
         FIG. 4  is a schematic structural diagram of a transmitting device according to an embodiment of the invention; and 
         FIG. 5  is a schematic structural diagram of a receiving device according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     Embodiments of the invention will be described in details below; and examples of the embodiments to be described will be illustrated in the drawings throughout identical or similar reference numerals denote identical or similar elements or elements with identical or similar functions. The embodiments to be described below with reference to the drawings are illustrative and merely intended to explain the invention but will not be construed as limiting the invention. 
     Firstly a codeword of the invention will be introduced below, and then a method and device for transmitting/receiving information in a multi-antenna system according to embodiments of the invention will be introduced in details in the form of particular embodiments. 
     Assumed an MIMO system includes N T  transmitting antennas and N R  receiving antennas, and the number of ranks is r, where is an even number and N R ≧1 and r≧1. For the convenience of a description, firstly the convention of an antenna indexing method of a dually polarized antenna will be described. As illustrated in  FIG. 1 , an antenna indexing method is to index firstly antennas sequentially in one polarization direction and then antennas sequentially in the other polarization direction. It shall be noted that a codebook of a dually polarized antenna system and a designing method thereof are independent of an antenna indexing method of dually polarized antennas, and if the convention of the antenna indexing method is changed, corresponding adjusting of indexes of the antennas in the codebook will suffice. 
     In an embodiment of the invention, a codeword c n   r  in a Rank-r codebook in a two-stage structure satisfies the following relationship: 
                       c   n   r     =       ⁢                 [         Φ   k   r     ⁡     (     :     ,   1       )       ⊗       w   m   r     ⁡     (     :     ,   1       )                     Φ   k   r     ⁡     (     :     ,   2       )       ⊗       w   m   r     ⁡     (     :     ,   2       )             …                         Φ   k   r     ⁡     (     :     ,   r       )       ⊗       w   m   r     ⁡     (     :     ,   r       )         ]                       =       ⁢     [               Φ   k   r     ⁡     (     1   ,   1     )       ⁢       w   m   r     ⁡     (     :     ,   1       )                   Φ   k   r     ⁡     (     1   ,   2     )       ⁢       w   m   r     ⁡     (     :     ,   2       )             …             Φ   k   r     ⁡     (     1   ,   r     )       ⁢       w   m   r     ⁡     (     :     ,   r       )                       Φ   k   r     ⁡     (     2   ,   1     )       ⁢       w   m   r     ⁡     (     :     ,   1       )                   Φ   k   r     ⁡     (     2   ,   2     )       ⁢       w   m   r     ⁡     (     :     ,   2       )             …             Φ   k   r     ⁡     (     2   ,   r     )       ⁢       w   m   r     ⁡     (     :     ,   r       )               ]             ,         
where Φ k   r  is a 2×r-dimension matrix, k=0, 1, . . . , K r −1, w m   r  is a N T /2×r-dimension matrix, m=0, 1, . . . , M r −1, X(:,l) represents the l th  column of a matrix X, X(l 1 ,l 2 ) represents an element in the l 1   th  row and in the l 2   th  column of the matrix X, and             represents a direct product of matrices; and a set consists of a number K r  of Φ k   r  is a transformation codebook of the codebook in the two-stage structure, and a set consists of a number M r  of w m   r  codebook of the codebook in the two-stage structure.

     Since the codebook proposed according to the invention has a two-stage structure, both a feedback of one PMI and a feedback of two PMIs can be supported. 
     Specific codebooks will be described below as examples. 
     Codebook 1: 
     In an embodiment of the invention, M 3 =M 4 =8. 
     Basic Codebook: 
     
       
         
               
               
               
             
           
               
                   
               
               
                   
                 Rank3 basic 
                 Rank4 basic 
               
               
                 Index 
                 codebook w m   3   
                 codebook w m   4   
               
               
                   
               
             
             
               
                 0 
                 [v 0  v 6  v 6 ] 
                 [v 0  v 0  v 6  v 6 ] 
               
               
                 1 
                 [v 1  v 7  v 7 ] 
                 [v 1  v 1  v 7  v 7 ] 
               
               
                 2 
                 [v 2  v 0  v 0 ] 
                 [v 2  v 2  v 0  v 0 ] 
               
               
                 3 
                 [v 3  v 1  v 1 ] 
                 [v 3  v 3  v 1  v 1 ] 
               
               
                 4 
                 [v 4  v 2  v 2 ] 
                 [v 4  v 4  v 2  v 2 ] 
               
               
                 5 
                 [v 5  v 3  v 3 ] 
                 [v 5  v 5  v 3  v 3 ] 
               
               
                 6 
                 [v 6  v 4  v 4 ] 
                 [v 6  v 6  v 4  v 4 ] 
               
               
                 7 
                 [v 7  v 5  v 5 ] 
                 [v 7  v 7  v 5  v 5 ] 
               
               
                   
               
             
          
         
       
     
     Transformation Codebook: 
     
       
         
               
               
               
             
           
               
                   
               
               
                   
                 Rank3 transformation 
                 Rank4 transformation 
               
               
                 Index k 
                 codebook Φ k   3   
                 codebook Φ k   4   
               
               
                   
               
             
             
               
                 0 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   2 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
     Codebook 1: 
     
       
         
               
               
               
             
               
               
               
             
           
               
                   
               
               
                 Index 
                 Rank3 codebook 
                 Rank4 codebook 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 2 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 3 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 4 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 5 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 6 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 7 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 8 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 9 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 10 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 11 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 12 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 13 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 14 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 15 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
     Where: 
     
       
         
               
               
             
           
               
                   
               
               
                 Index 
                 v n   
               
               
                   
               
             
             
               
                 0 
                 v 0  = [1 1 1 1] T   
               
               
                 1 
                 v 1  = [1 (1 − j)/{square root over (2)} −j (−1 − j)/{square root over (2)}] T   
               
               
                 2 
                 v 2  = [1 −j −1 j] T   
               
               
                 3 
                 v 3  = [1 (−1 − j)/{square root over (2)} j (1 − j)/{square root over (2)}] T   
               
               
                 4 
                 v 4  = [1 −1 1 −1] T   
               
               
                 5 
                 v 5  = [1 (−1 + j)/{square root over (2)} −j (1 + j)/{square root over (2)}] T   
               
               
                 6 
                 v 6  = [1 j −1 −j] T   
               
               
                 7 
                 v 7  = [1 (1 + j)/{square root over (2)} j (−1 + j)/{square root over (2)}] T   
               
               
                   
               
             
          
         
       
     
     It shall be noted that v n  is a truncated 8-DFT vector or any other truncated 2 K -DFT vector, where 2 K  is larger than or equal to N T /2 and K is a natural number. If a codeword with the unitary characteristic is constructed, mutually orthogonal column vectors of the codeword w m  of the basic codebook, corresponding to mutually non-orthogonal columns of the codeword Φ k  in the transformation codebook, will suffice. For example, first two columns of the codeword 
               Φ   k   3     =     [         1       1       1           1       1         -   1           ]           
with k=0 in the transformation codebook of the codebook 1 are non-orthogonal, and then first two columns of the codeword w m   3  of the basic codebook shall be kept orthogonal; and 1 st  and 3 rd  columns of and 2 nd  and 4 th  columns of the codeword
 
               Φ   k   4     =     [         1       1       1       1           1         -   1         1         -   1           ]           
with k=0 in the transformation codebook are non-orthogonal, and then 1 st  and 3 rd  columns of and also 2 nd  and 4 th  columns of the codeword w m   4  of the basic codebook shall be kept orthogonal.
 
     v n  can be applied to codebooks of the following respective embodiments. 
     Codebook 2: 
     Matrices Φ k  and w m  are defined as follows: 
     
       
         
               
               
               
             
           
               
                   
                   
               
               
                   
                 Index 
                 Φ k   
               
               
                   
                   
               
             
             
               
                   
                 0 
                 
                   
                     
                       
                         
                           
                             Φ 
                             0 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                             ] 
                           
                         
                           
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 1 
                 
                   
                     
                       
                         
                           
                             Φ 
                             1 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                           
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 2 
                 
                   
                     
                       
                         
                           
                             Φ 
                             2 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   j 
                                 
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                                 
                                   j 
                                 
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                               
                             
                             ] 
                           
                         
                           
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 3 
                 
                   
                     
                       
                         
                           
                             Φ 
                             3 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                                 
                                   j 
                                 
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                                 
                                   j 
                                 
                               
                             
                             ] 
                           
                         
                           
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 4 
                 
                   
                     
                       
                         
                           Φ 
                           4 
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     
                                       1 
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     
                                       1 
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 5 
                 
                   
                     
                       
                         
                           
                             Φ 
                             5 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       
                                         1 
                                         + 
                                         j 
                                       
                                       
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       
                                         1 
                                         + 
                                         j 
                                       
                                       
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                           
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 6 
                 
                   
                     
                       
                         
                           
                             Φ 
                             6 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         - 
                                         1 
                                       
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       - 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         - 
                                         1 
                                       
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       - 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                           
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 7 
                 
                   
                     
                       
                         
                           
                             Φ 
                             7 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     
                                       1 
                                       - 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         - 
                                         1 
                                       
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       - 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         - 
                                         1 
                                       
                                       + 
                                       j 
                                     
                                     
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                           
                       
                     
                   
                 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
             
           
               
                   
               
               
                 Index 
                 w m   
               
               
                   
               
             
             
               
                 0 
                 [v 0  v 1  v 2  v 3 ] 
               
               
                 1 
                 [v 1  v 2  v 3  v 4 ] 
               
               
                 2 
                 [v 2  v 3  v 4  v 5 ] 
               
               
                 3 
                 [v 3  v 4  v 5  v 6 ] 
               
               
                 4 
                 [v 4  v 5  v 6  v 7 ] 
               
               
                 5 
                 [v 5  v 6  v 7  v 0 ] 
               
               
                 6 
                 [v 6  v 7  v 0  v 1 ] 
               
               
                 7 
                 [v 7  v 0  v 1  v 2 ] 
               
               
                   
               
             
          
         
       
     
     Basic Codebook: 
     
       
         
               
               
               
               
               
             
           
               
                   
               
               
                   
                 Rank1 
                 Rank2 
                 Rank3 
                 Rank4 
               
               
                   
                 basic 
                 basic 
                 basic 
                 basic 
               
               
                 Index 
                 code- 
                 code- 
                 code- 
                 code- 
               
               
                 m 
                 book w m   1   
                 book w m   2   
                 book w m   3   
                 book w m   4   
               
               
                   
               
             
             
               
                 0 
                 w 0 (:,1) 
                 [w 0 (:,1) w 0 (:,2)] 
                 [w 0 (:,1) w 0 (:,2) 
                 w 0   
               
               
                   
                   
                   
                 w 0 (:,3)] 
               
               
                 1 
                 w 1 (:,1) 
                 [w 1 (:,1) w 1 (:,2)] 
                 [w 1 (:,1) w 1 (:,2) 
                 w 1   
               
               
                   
                   
                   
                 w 1 (:,3)] 
               
               
                 2 
                 w 2 (:,1) 
                 [w 2 (:,1) w 2 (:,2)] 
                 [w 2 (:,1) w 2 (:,2) 
                 w 2   
               
               
                   
                   
                   
                 w 2 (:,3)] 
               
               
                 3 
                 w 3 (:,1) 
                 [w 3 (:,1) w 3 (:,2)] 
                 [w 3 (:,1) w 3 (:,2) 
                 w 3   
               
               
                   
                   
                   
                 w 3 (:,3)] 
               
               
                 4 
                 w 4 (:,1) 
                 [w 4 (:,1) w 4 (:,2)] 
                 [w 4 (:,1) w 4 (:,2) 
                 w 4   
               
               
                   
                   
                   
                 w 4 (:,3)] 
               
               
                 5 
                 w 5 (:,1) 
                 [w 5 (:,1) w 5 (:,2)] 
                 [w 5 (:,1) w 5 (:,2) 
                 w 5   
               
               
                   
                   
                   
                 w 5 (:,3)] 
               
               
                 6 
                 w 6 (:,1) 
                 [w 6 (:,1) w 6 (:,2)] 
                 [w 6 (:,1) w 6 (:,2) 
                 w 6   
               
               
                   
                   
                   
                 w 6 (:,3)] 
               
               
                 7 
                 w 7 (:,1) 
                 [w 7 (:,1) w 7 (:,2)] 
                 [w 7 (:,1) w 7 (:,2) 
                 w 7   
               
               
                   
                   
                   
                 w 7 (:,3)] 
               
               
                   
               
             
          
         
       
     
     Transformation Codebook: 
     
       
         
               
               
               
               
               
             
           
               
                   
               
               
                   
                 Rank1 
                 Rank2 
                 Rank3 
                 Rank4 
               
               
                   
                 transfor- 
                 transfor- 
                 transfor- 
                 transfor- 
               
               
                   
                 mation 
                 mation 
                 mation 
                 mation 
               
               
                 Index 
                 code- 
                 code- 
                 code- 
                 code- 
               
               
                 k 
                 book Φ k   1   
                 book Φ k   2   
                 book Φ k   3   
                 book Φ k   4   
               
               
                   
               
             
             
               
                 0 
                 Φ 0 (:,1) 
                 [Φ 0 (:,1) Φ 0 (:,2)] 
                 [Φ 0 (:,1) Φ 0 (:,2) 
                 Φ 0   
               
               
                   
                   
                   
                 Φ 0 (:,3)] 
               
               
                 1 
                 Φ 1 (:,1) 
                 [Φ 1 (:,1) Φ 1 (:,2)] 
                 [Φ 1 (:,1) Φ 1 (:,2) 
                 Φ 1   
               
               
                   
                   
                   
                 Φ 1 (:,3)] 
               
               
                   
               
             
          
         
       
     
     Codebook 2: 
                                                                   Index   Rank1 codebook   Rank2 codebook   Rank3 codehook   Rank4 codebook                                0             [           v   0               v   0           ]                           [           v   0           v   1               v   0           -     v   1             ]                           [           v   0           v   1           v   2               v   0           -     v   1             v   2           ]                           [           v   0           v   1           v   2           v   3               v   0           -     v   1             v   2           -     v   3             ]                             1             [           v   1               v   1           ]                           [           v   1           v   2               v   1           -     v   2             ]                           [           v   1           v   2           v   3               v   1           -     v   2             v   3           ]                           [           v   1           v   2           v   3           v   4               v   1           -     v   2             v   3           -     v   4             ]                             2             [           v   2               v   2           ]                           [           v   2           v   3               v   2           -     v   3             ]                           [           v   2           v   3           v   4               v   2           -     v   3             v   4           ]                           [           v   2           v   3           v   4           v   5               v   2           -     v   3             v   4           -     v   5             ]                             3             [           v   3               v   3           ]                           [           v   3           v   4               v   3           -     v   4             ]                           [           v   3           v   4           v   5               v   3           -     v   4             v   5           ]                           [           v   3           v   4           v   5           v   6               v   3           -     v   4             v   5           -     v   6             ]                             4             [           v   4               v   4           ]                           [           v   4           v   5               v   4           -     v   5             ]                           [           v   4           v   5           v   6               v   4           -     v   5             v   6           ]                           [           v   4           v   5           v   6           v   7               v   4           -     v   5             v   6           -     v   7             ]                             5             [           v   5               v   5           ]                           [           v   5           v   6               v   5           -     v   6             ]                           [           v   5           v   6           v   7               v   5           -     v   6             v   7           ]                           [           v   5           v   6           v   7           v   0               v   5           -     v   6             v   7           -     v   0             ]                             6             [           v   6               v   6           ]                           [           v   6           v   7               v   6           -     v   7             ]                           [           v   6           v   7           v   0               v   6           -     v   7             v   0           ]                           [           v   6           v   7           v   0           v   1               v   6           -     v   7             v   0           -     v   1             ]                             7             [           v   7               v   7           ]                           [           v   7           v   0               v   7           -     v   0             ]                           [           v   7           v   0           v   1               v   7           -     v   0             v   1           ]                           [           v   7           v   0           v   1           v   2               v   7           -     v   0             v   1           -     v   2             ]                             8             [           v   0               -     v   0             ]                           [           v   0           v   1               -     v   0             v   1           ]                           [           v   0           v   1           v   2               -     v   0             v   1           -     v   2             ]                           [           v   0           v   1           v   2           v   3               -     v   0             v   1           -     v   2             v   3           ]                             9             [           v   1               -     v   1             ]                           [           v   1           v   2               -     v   1             v   2           ]                           [           v   1           v   2           v   3               -     v   1             v   2           -     v   3             ]                           [           v   1           v   2           v   3           v   4               -     v   1             v   2           -     v   3             v   4           ]                             10             [           v   2               -     v   2             ]                           [           v   2           v   3               -     v   2             v   3           ]                           [           v   2           v   3           v   4               -     v   2             v   3           -     v   4             ]                           [           v   2           v   3           v   4           v   5               -     v   2             v   3           -     v   4             v   5           ]                             11             [           v   3               -     v   3             ]                           [           v   3           v   4               -     v   3             v   4           ]                           [           v   3           v   4           v   5               -     v   3             v   4           -     v   5             ]                           [           v   3           v   4           v   5           v   6               -     v   3             v   4           -     v   5             v   6           ]                             12             [           v   4               -     v   4             ]                           [           v   4           v   5               -     v   4             v   5           ]                           [           v   4           v   5           v   6               -     v   4             v   5           -     v   6             ]                           [           v   4           v   5           v   6           v   7               -     v   4             v   5           -     v   6             v   7           ]                             13             [           v   5               -     v   5             ]                           [           v   5           v   6               -     v   5             v   6           ]                           [           v   5           v   6           v   7               -     v   5             v   6           -     v   7             ]                           [           v   5           v   6           v   7           v   0               -     v   5             v   6           -     v   7             v   0           ]                             14             [           v   6               -     v   6             ]                           [           v   6           v   7               -     v   6             v   7           ]                           [           v   6           v   7           v   0               -     v   6             v   7           -     v   0             ]                           [           v   6           v   7           v   0           v   1               -     v   6             v   7           -     v   0             v   1           ]                             15             [           v   7               -     v   7             ]                           [           v   7           v   0               -     v   7             v   0           ]                           [           v   7           v   0           v   1               -     v   7             v   0           -     v   1             ]                           [           v   7           v   0           v   1           v   2               -     v   7             v   0           -     v   1             v   2           ]                                  
Codebook 3:
 
     The codebook 3 is structured substantially the same as the codebook 2. 
     Basic Codebook: 
     
       
         
               
               
               
               
               
             
           
               
                   
               
               
                   
                 Rank1 
                 Rank2 
                 Rank3 
                 Rank4 
               
               
                   
                 basic 
                 basic 
                 basic 
                 basic 
               
               
                 Index 
                 code- 
                 code- 
                 code- 
                 code- 
               
               
                 m 
                 book w m   1   
                 book w m   2   
                 book w m   3   
                 book w m   4   
               
               
                   
               
             
             
               
                 0 
                 w 0 (:,1) 
                 [w 0 (:,1) w 0 (:,2)] 
                 [w 0 (:,1) w 0 (:,2) 
                 w 0   
               
               
                   
                   
                   
                 w 0 (:,3)] 
               
               
                 1 
                 w 1 (:,1) 
                 [w 1 (:,1) w 1 (:,2)] 
                 [w 1 (:,1) w 1 (:,2) 
                 w 1   
               
               
                   
                   
                   
                 w 1 (:,3)] 
               
               
                 2 
                 w 2 (:,1) 
                 [w 2 (:,1) w 2 (:,2)] 
                 [w 2 (:,1) w 2 (:,2) 
                 w 2   
               
               
                   
                   
                   
                 w 2 (:,3)] 
               
               
                 3 
                 w 3 (:,1) 
                 [w 3 (:,1) w 3 (:,2)] 
                 [w 3 (:,1) w 3 (:,2) 
                 w 3   
               
               
                   
                   
                   
                 w 3 (:,3)] 
               
               
                 4 
                 w 4 (:,1) 
                 [w 4 (:,1) w 4 (:,2)] 
                 [w 4 (:,1) w 4 (:,2) 
                 w 4   
               
               
                   
                   
                   
                 w 4 (:,3)] 
               
               
                 5 
                 w 5 (:,1) 
                 [w 5 (:,1) w 5 (:,2)] 
                 [w 5 (:,1) w 5 (:,2) 
                 w 5   
               
               
                   
                   
                   
                 w 5 (:,3)] 
               
               
                 6 
                 w 6 (:,1) 
                 [w 6 (:,1) w 6 (:,2)] 
                 [w 6 (:,1) w 6 (:,2) 
                 w 6   
               
               
                   
                   
                   
                 w 6 (:,3)] 
               
               
                 7 
                 w 7 (:,1) 
                 [w 7 (:,1) w 7 (:,2)] 
                 [w 7 (:,1) w 7 (:,2) 
                 w 7   
               
               
                   
                   
                   
                 w 7 (:,3)] 
               
               
                   
               
             
          
         
       
     
     Transformation Codebook: 
     
       
         
               
               
               
               
               
             
           
               
                   
               
               
                   
                 Rank1 
                 Rank2 
                 Rank3 
                 Rank4 
               
               
                   
                 transfor- 
                 transfor- 
                 transfor- 
                 transfor- 
               
               
                   
                 mation 
                 mation 
                 mation 
                 mation 
               
               
                 Index 
                 code- 
                 code- 
                 code- 
                 code- 
               
               
                 k 
                 book Φ k   1   
                 book Φ k   2   
                 book Φ k   3   
                 book Φ k   4   
               
               
                   
               
             
             
               
                 0 
                 Φ 0 (:,1) 
                 [Φ 0 (:,1) Φ 0 (:,2)] 
                 [Φ 0 (:,1) Φ 0 (:,2) 
                 Φ 0   
               
               
                   
                   
                   
                 Φ 0 (:,3)] 
               
               
                 1 
                 Φ 1 (:,1) 
                 [Φ 1 (:,1) Φ 1 (:,2)] 
                 [Φ 1 (:,1) Φ 1 (:,2) 
                 Φ 1   
               
               
                   
                   
                   
                 Φ 1 (:,3)] 
               
               
                 2 
                 Φ 2 (:,1) 
                 [Φ 2 (:,1) Φ 2 (:,2)] 
                 [Φ 2 (:,1) Φ 2 (:,2) 
                 Φ 2   
               
               
                   
                   
                   
                 Φ 2 (:,3)] 
               
               
                 3 
                 Φ 3 (:,1) 
                 [Φ 3 (:,1) Φ 3 (:,2)] 
                 [Φ 3 (:,1) Φ 3 (:,2) 
                 Φ 3   
               
               
                   
                   
                   
                 Φ 3 (:,3)] 
               
               
                   
               
             
          
         
       
     
     Codebook 3: 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
               
               
                   
                 Rank1 
                 Rank2 
                   
                   
               
               
                 Index 
                 codebook 
                 codebook 
                 Rank3 codebook 
                 Rank4 codebook 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 2 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 3 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 4 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 5 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 6 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 7 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 8 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 9 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 10 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 11 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 12 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 13 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 14 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 15 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 16 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 17 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 18 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 19 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 20 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 21 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 22 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 23 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 24 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 25 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 26 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 27 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 28 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 29 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 30 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 31 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
     In other embodiments of the invention, codebooks resulting from column-exchange of the foregoing codebooks can also be applied in the invention. For example: 
     Codebook 4 (Resulting from Column-Exchange of the Codebook 1): 
                                                   Index   Rank3 codebook   Rank4 codebook                                0             [           v   0             2     ⁢     v   6           0             v   0         0           2     ⁢     v   6             ]                           [           v   0         0         v   6         0           0         v   0         0         v   6           ]                             1             [           v   1             2     ⁢     v   7           0             v   1         0           2     ⁢     v   7             ]                           [           v   1         0         v   7         0           0         v   1         0         v   7           ]                             2             [           v   2             2     ⁢     v   0           0             v   2         0           2     ⁢     v   0             ]                           [           v   2         0         v   0         0           0         v   2         0         v   0           ]                             3             [           v   3             2     ⁢     v   1           0             v   3         0           2     ⁢     v   1             ]                           [           v   3         0         v   1         0           0         v   3         0         v   1           ]                             4             [           v   4             2     ⁢     v   2           0             v   4         0           2     ⁢     v   2             ]                           [           v   4         0         v   2         0           0         v   4         0         v   2           ]                             5             [           v   5             2     ⁢     v   3           0             v   5         0           2     ⁢     v   3             ]                           [           v   5         0         v   3         0           0         v   5         0         v   3           ]                             6             [           v   6             2     ⁢     v   4           0             v   6         0           2     ⁢     v   4             ]                           [           v   6         0         v   4         0           0         v   6         0         v   4           ]                             7             [           v   7             2     ⁢     v   5           0             v   7         0           2     ⁢     v   5             ]                           [           v   7         0         v   5         0           0         v   7         0         v   5           ]                             8             [           v   0           v   6           v   6               v   0           v   6           -     v   6             ]                           [           v   6           v   0           v   6           v   0               v   6           v   0           -     v   6             -     v   0             ]                             9             [           v   1           v   7           v   7               v   1           v   7           -     v   7             ]                           [           v   7           v   1           v   7           v   1               v   7           v   1           -     v   7             -     v   1             ]                             10             [           v   2           v   0           v   0               v   2           v   0           -     v   0             ]                           [           v   0           v   2           v   0           v   2               v   0           v   2           -     v   0             -     v   2             ]                             11             [           v   3           v   1           v   1               v   3           v   1           -     v   1             ]                           [           v   1           v   3           v   1           v   3               v   1           v   3           -     v   1             -     v   3             ]                             12             [           v   4           v   2           v   2               v   4           v   2           -     v   2             ]                           [           v   2           v   4           v   2           v   4               v   2           v   4           -     v   2             -     v   4             ]                             13             [           v   5           v   3           v   3               v   5           v   3           -     v   3             ]                           [           v   3           v   5           v   3           v   5               v   3           v   5           -     v   3             -     v   5             ]                             14             [           v   6           v   4           v   4               v   6           v   4           -     v   4             ]                           [           v   4           v   6           v   4           v   6               v   4           v   6           -     v   4             -     v   6             ]                             15             [           v   7           v   5           v   5               v   7           v   5           -     v   5             ]                           [           v   5           v   7           v   5           v   7               v   5           v   7           -     v   5             -     v   7             ]                                  
Codebook 5 (Resulting from Column-Exchange of the Codebook 2):
 
                                                                       Rank1   Rank2   Rank3   Rank4        Index   codebook   codebook   codebook   codebook                                0             [           v   0               v   0           ]                           [           v   0           v   1               v   0           -     v   1             ]                           [           v   0           v   1           v   2               v   0           -     v   1             v   2           ]                           [           v   0           v   1           v   2           v   3               v   0           -     v   1             v   2           -     v   3             ]                             1             [           v   1               v   1           ]                           [           v   1           v   2               v   1           -     v   2             ]                           [           v   1           v   2           v   3               v   1           -     v   2             v   3           ]                           [           v   1           v   2           v   3           v   4               v   1           -     v   2             v   3           -     v   4             ]                             2             [           v   2               v   2           ]                           [           v   2           v   3               v   2           -     v   3             ]                           [           v   2           v   3           v   4               v   2           -     v   3             v   4           ]                           [           v   2           v   3           v   4           v   5               v   2           -     v   3             v   4           -     v   5             ]                             3             [           v   3               v   3           ]                           [           v   3           v   4               v   3           -     v   4             ]                           [           v   3           v   4           v   5               v   3           -     v   4             v   5           ]                           [           v   3           v   4           v   5           v   6               v   3           -     v   4             v   5           -     v   6             ]                             4             [           v   4               v   4           ]                           [           v   4           v   5               v   4           -     v   5             ]                           [           v   4           v   5           v   6               v   4           -     v   5             v   6           ]                           [           v   4           v   5           v   6           v   7               v   4           -     v   5             v   6           -     v   7             ]                             5             [           v   5               v   5           ]                           [           v   5           v   6               v   5           -     v   6             ]                           [           v   5           v   6           v   7               v   5           -     v   6             v   7           ]                           [           v   5           v   6           v   7           v   0               v   5           -     v   6             v   7           -     v   0             ]                             6             [           v   6               v   6           ]                           [           v   6           v   7               v   6           -     v   7             ]                           [           v   6           v   7           v   0               v   6           -     v   7             v   0           ]                           [           v   6           v   7           v   0           v   1               v   6           -     v   7             v   0           -     v   1             ]                             7             [           v   7               v   7           ]                           [           v   7           v   0               v   7           -     v   0             ]                           [           v   7           v   0           v   1               v   7           -     v   0             v   1           ]                           [           v   7           v   0           v   1           v   2               v   7           -     v   0             v   1           -     v   2             ]                             8             [           v   0               -     v   0             ]                           [           v   1           v   0               v   1           -     v   0             ]                           [           v   0           v   1           v   2               -     v   0             v   1           -     v   2             ]                           [           v   0           v   1           v   2           v   3               -     v   0             v   1           -     v   2             v   3           ]                             9             [           v   1               -     v   1             ]                           [           v   2           v   1               v   2           -     v   1             ]                           [           v   1           v   2           v   3               -     v   1             v   2           -     v   3             ]                           [           v   1           v   2           v   3           v   4               -     v   1             v   2           -     v   3             v   4           ]                             10             [           v   2               -     v   2             ]                           [           v   3           v   2               v   3           -     v   2             ]                           [           v   2           v   3           v   4               -     v   2             v   3           -     v   4             ]                           [           v   2           v   3           v   4           v   5               -     v   2             v   3           -     v   4             v   5           ]                             11             [           v   3               -     v   3             ]                           [           v   4           v   3               v   4           -     v   3             ]                           [           v   3           v   4           v   5               -     v   3             v   4           -     v   5             ]                           [           v   3           v   4           v   5           v   6               -     v   3             v   4           -     v   5             v   6           ]                             12             [           v   4               -     v   4             ]                           [           v   5           v   4               v   5           -     v   4             ]                           [           v   4           v   5           v   6               -     v   4             v   5           -     v   6             ]                           [           v   4           v   5           v   6           v   7               -     v   4             v   5           -     v   6             v   7           ]                             13             [           v   5               -     v   5             ]                           [           v   6           v   5               v   6           -     v   5             ]                           [           v   5           v   6           v   7               -     v   5             v   6           -     v   7             ]                           [           v   5           v   6           v   7           v   0               -     v   5             v   6           -     v   7             v   0           ]                             14             [           v   6               -     v   6             ]                           [           v   7           v   6               v   7           -     v   6             ]                           [           v   6           v   7           v   0               -     v   6             v   7           -     v   0             ]                           [           v   6           v   7           v   0           v   1               -     v   6             v   7           -     v   0             v   1           ]                             15             [           v   7               -     v   7             ]                           [           v   0           v   7               v   0           -     v   7             ]                           [           v   7           v   0           v   1               -     v   7             v   0           -     v   1             ]                           [           v   7           v   0           v   1           v   2               -     v   7             v   0           -     v   1             v   2           ]                                  
Codebook 6 (Resulting from Column-Exchange of the Codebook 3):
 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
               
               
                   
                 Rank1 
                 Rank2 
                 Rank3 
                 Rank4 
               
               
                 Index 
                 codebook 
                 codebook 
                 codebook 
                 codebook 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 0 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 2 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 3 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 4 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 5 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 6 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 7 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 8 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 9 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 10 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 11  
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     3 
                                   
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 12 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     4 
                                   
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 13 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     5 
                                   
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 14 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     6 
                                   
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     0 
                                   
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 15 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     v 
                                     7 
                                   
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     v 
                                     1 
                                   
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 16 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 17 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 18 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 19 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 20 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 21 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   5 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     6 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 22 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   6 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     7 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   0 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 23 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   7 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   7 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 24 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   0 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     0 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 25 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   1 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     1 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   4 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 26 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   2 
                                 
                               
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
                                   v 
                                   4 
                                 
                               
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     2 
                                   
                                 
                               
                               
                                 
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                                   3 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   5 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 27 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
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                                   v 
                                   4 
                                 
                               
                               
                                 
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                                   3 
                                 
                               
                             
                             
                               
                                 
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                                   4 
                                 
                               
                               
                                 
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                                     3 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
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                                   v 
                                   3 
                                 
                               
                               
                                 
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                                     3 
                                   
                                 
                               
                               
                                 
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                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   3 
                                 
                               
                               
                                 
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                                   4 
                                 
                               
                               
                                 
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                                     3 
                                   
                                 
                               
                               
                                 
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                                   4 
                                 
                               
                               
                                 
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                                     5 
                                   
                                 
                               
                               
                                 
                                   jv 
                                   6 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 28 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     jv 
                                     4 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
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                                   5 
                                 
                               
                               
                                 
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                                     4 
                                   
                                 
                               
                             
                           
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                                   4 
                                 
                               
                               
                                 
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                                     4 
                                   
                                 
                               
                               
                                 
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                                   5 
                                 
                               
                               
                                 
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                                     6 
                                   
                                 
                               
                             
                           
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                                   7 
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
               
               
                   
               
               
                 29 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   5 
                                 
                               
                             
                             
                               
                                 
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                                     5 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
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                                   6 
                                 
                               
                               
                                 
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                                   6 
                                 
                               
                               
                                 
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                                     5 
                                   
                                 
                               
                             
                           
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                                     5 
                                   
                                 
                               
                               
                                 
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                                   6 
                                 
                               
                               
                                 
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                                     7 
                                   
                                 
                               
                             
                           
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                 30 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   v 
                                   6 
                                 
                               
                             
                             
                               
                                 
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                                     6 
                                   
                                 
                               
                             
                           
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                                   7 
                                 
                               
                               
                                 
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                                     6 
                                   
                                 
                               
                             
                           
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                 31 
                 
                   
                     
                       
                         
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                                   7 
                                 
                               
                             
                             
                               
                                 
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                                   1 
                                 
                               
                               
                                 
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                                   2 
                                 
                               
                             
                             
                               
                                 
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                                   0 
                                 
                               
                               
                                 
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                                   2 
                                 
                               
                             
                           
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     As illustrated in  FIG. 2 , which is a flow chart of a method for transmitting information in a multi-antenna system according to an embodiment of the invention, the method includes the following steps. 
     S 201 : A transmitting device receives information including codeword index information transmitted from a receiving device and extracts the codeword index information therefrom. Specifically, in order to know the channel condition, the transmitting device receives a Channel Quality Indicator (CQI), a selected Rank, a selected codeword and other information fed back from a remote receiver. Such information of the CQI, the selected Rank and the selected codeword will be used in a subsequent transmission process. 
     In the embodiment of the invention, a codeword c n   r  codebook satisfies the following relationship:
 
 c   n   r =[Φ k   r (:,1)             w   m   r (:,1) Φ k   r (:,2)           w   m   r (:,2) . . . Φ k   r (:, r )           w   m   r (:, r )],
 
where Φ k   r  is a 2×r-dimension matrix, k=0, 1, . . . , K r −1, w m   r  is a N T /2×r-dimension matrix, m=0, 1, . . . , M r −1, X(:,l) represents the l th  column of a matrix X, and           represents a direct product of matrices; and a set consists of a number K r  of Φ k   r  is a transformation codebook of the codebook in the two-stage structure, and set consists of a number M r  of w m   r  is a basic codebook of the codebook in the two-stage structure. Therefore, the fed-back codeword index information can be the codeword index n in the codebook, the codeword indexes k and m in the transformation codebook and the basic codebook, or another combination of k and m. In the codebook 3, for example, if the fed-back codeword index information is nε{0, . . . , 31}, a corresponding codeword can be retrieved directly from the codebook; and if the fed-back codeword index information is kε{0, . . . , 3} and mε{0, . . . , 7}, a corresponding codeword can also be retrieved from the codebook; and also since the codeword in this codebook satisfies the following characteristic:

                         c   n   r     =       ⁢     [             c   l   r     ⁡     (       1   :       N   T     /   2       ,   :     )                   f   ⁡     (   l   )       ⁢       c   l   r     ⁡     (             N   T     /   2     +   1     :     N   T       ,   :     )               ]       ,       where   ⁢           ⁢     f   ⁡     (   l   )         =     {           1         l   ≤   7             j         l   &gt;   7           ⁢           ⁢   and                       c   l   r     =       ⁢     [               Φ   k   r     ⁡     (     :     ,   1       )       ⊗       w   m   r     ⁡     (     :     ,   1       )                   Φ   k   r     ⁡     (     :     ,   2       )       ⊗       w   m   r     ⁡     (     :     ,   2       )             …             Φ   k   r     ⁡     (     :     ,   r       )       ⊗       w   m   r     ⁡     (     :     ,   r       )               ]             ,         
where X(l 1 ,l 2 ,:) represents a matrix consists of the l 1   th  to l 2   th  row vectors in the matrix X, therefore, if the fed-back codeword index information is lε{0, . . . , 15} and f(l), a corresponding codeword can also be retrieved from the codebook.
 
     S 202 : The transmitting device selects a corresponding codeword from a codebook in a two-stage structure according to the codeword index information, and all of the foregoing codebooks 1 to 5 can be applied in this embodiment. 
     S 203 : The transmitting device performs a transformation process (e.g., a zero-forcing process, etc.) based upon the codeword to obtain a pre-coding matrix. Of course, the corresponding codeword can alternatively be taken directly as a pre-coding matrix in another embodiment. 
     S 204 : The transmitting device pre-codes data to be transmitted according to the pre-coding matrix. 
     S 205 : The transmitting device transmits the pre-coded data to the receiving device. 
     As illustrated in  FIG. 3 , which is a flow chart of a method for receiving information in a multi-antenna system according to an embodiment of the invention, the method includes the following steps. 
     S 301 : A receiving device receives information transmitted from a transmitting device. 
     S 302 : The receiving device estimates channel state information according to the received information. 
     S 303 : The receiving device selects a corresponding codeword from a codebook in a two-stage structure according to the channel state information, and all of the foregoing codebooks 1 to 5 can be applied in this embodiment. 
     S 304 : The receiving device generates codeword index information of the selected codeword. Like the foregoing embodiment, since the codeword satisfies the following relationship:
 
 c   l   r =[Φ k   r (:,1)             w   m   r (:,1) Φ k   r (:,2)           w   m   r (:,2) . . . Φ k   r (:, r )           w   m   r (:, r )],
 
where Φ k   r  is a 2×r-dimension matrix, k=0, 1, . . . , K r −1, w m   r  is a N T /2×r-dimension matrix, m=0, 1, . . . , M r −1, X(:,l) represents the l th  column of a matrix X, and           represents a direct product of matrices; and a set consists of a number K r  of Φ k   r  is a transformation codebook of the codebook in the two-stage structure, and a set consists of a number M r  of w m   r  is a basic codebook of the codebook in the two-stage structure, therefore the generated codeword index information can be the codeword index n in the codebook, the codeword indexes k and m in the transformation codebook and the basic codebook, or another combination of k and m. In the codebook 3, for example, the fed-back codeword index information can be nε{0, . . . , 31}; or the fed-back codeword index information can be kε{0, . . . , 3} and mε{0, . . . , 7}, and a transmitting device can retrieve a corresponding codeword from the codebook according to either the codeword index n or the combination of the codeword indexes k and m; and also since the codeword in this codebook satisfies the following characteristic:

                         c   n   r     =       ⁢     [             c   l   r     ⁡     (       1   :       N   T     /   2       ,   :     )                   f   ⁡     (   l   )       ⁢       c   l   r     ⁡     (             N   T     /   2     +   1     :     N   T       ,   :     )               ]       ,       where   ⁢           ⁢     f   ⁡     (   l   )         =     {           1         l   ≤   7             j         l   &gt;   7           ⁢           ⁢   and                       c   l   r     =       ⁢           [               Φ   k   r     ⁡     (     :     ,   1       )       ⊗       w   m   r     ⁡     (     :     ,   1       )                   Φ   k   r     ⁡     (     :     ,   2       )       ⊗       w   m   r     ⁡     (     :     ,   2       )             …                           Φ   k   r     ⁡     (     :     ,   r       )       ⊗       w   m   r     ⁡     (     :     ,   r       )         ]                   ,         
where X(l 1 :l 2 ,:) represents a matrix consists of the l 1   th  to l 2   th  row vectors in the matrix X, therefore if the fed-back codeword index information is lε{0, . . . , 15} and f(l), the transmitting device can also retrieve a corresponding codeword from the codebook.
 
     S 305 : The receiving device transmits information including the codeword index information to the transmitting device. 
     Reference is made to  FIG. 4  illustrating a schematic structural diagram of a transmitting device according to an embodiment of the invention, which includes a receiving module  110 , a codebook storing module  120 , a codeword determining module  130 , a pre-coding matrix calculating module  140 , a pre-coding module  150  and a transmitting module  160 . The receiving module  110  is configured to receive information including codeword index information transmitted from a receiving device. The codebook storing module  120  is configured to store a codebook in a two-stage structure. The codeword determining module  130  is configured to select a corresponding codeword from the codebook in the two-stage structure stored in the codebook storing module  120  according to the codeword index information. The pre-coding matrix calculating module  140  is configured to generate a pre-coding matrix according to the selected codeword, for example, perform a transformation process (e.g., zero-forcing process, etc.) based upon the codeword to obtain a pre-coding matrix or take the corresponding codeword directly as a pre-coding matrix. The pre-coding module  150  is configured to pre-code data to be transmitted according to the pre-coding matrix. The transmitting module  160  is configured to transmit the pre-coded data to the receiving device. 
     Reference is made to  FIG. 5  illustrating a schematic structural diagram of a receiving device according to an embodiment of the invention, which includes a receiving module  210 , a channel estimating module  220 , a codebook storing module  230 , a codeword selecting module  240 , a codeword index information generating module  250  and a transmitting module  260 . The receiving module  210  is configured to receive information transmitted from a transmitting device. The channel estimating module  220  is configured to estimate channel state information according to the information received by the receiving module  210 . The codebook storing module  230  is configured to store a codebook in a two-stage structure. The codeword selecting module  240  is configured to select a corresponding codeword from the codebook in the two-stage structure according to the e channel state information. The codeword index information generating module  250  is configured to generate codeword index information of the selected codeword. The transmitting module  260  is configured to transmit information including the codeword index information to the transmitting device. 
     In the foregoing solutions proposed according to the embodiments of the invention, a formed and used codebook matches as closely as possibly a channel distribution characteristic of a dually polarized antenna to minimize a performance loss due to quantization of the codebook, thereby improving effectively the performance of a pre-coding system. Furthermore, the codebook proposed by the invention has a two-stage structure and can support both a feedback of one PMI and a feedback of two PMIs. 
     Those ordinarily skilled in the art can appreciate that all or a part of the steps in the methods according to the foregoing embodiments of the invention can be perfumed in program instructing relevant hardware, which may be stored in a computer readable storage medium and which upon execution can perform one or combination of the steps in the methods according to the embodiments. 
     Furthermore, the respective functional elements in the respective embodiments of the invention can be integrated in a processing module or can physically exist separately or two or inure of the elements can be integrated in a module. The integrated module can be embodied in the form of hardware or in the form of a software functional module. If the integrated module is embodied in the form of a software functional module and sold or used as a separate product, it can be stored in a computer readable storage medium. 
     The storage medium mentioned above can be a read only memory, a magnetic disk, an optical disk, etc. 
     The foregoing description is merely illustrative of the preferred embodiments of the invention, and it shall be pointed out that those ordinarily skilled in the art can further make several adaptations and modifications without departing from the principle of the invention and these adaptations and modifications shall also be construed as coming into the scope of the invention.