Abstract:
Disclosed is an apparatus for encoding k consecutive inputs indicating a TFCI (Transport Format Combination Indicator) of each of successively transmitted frames into a sequence of m symbols in an NB-TDD (Narrowband-Time Division Duplex) mobile communication system. An encoder encodes the k input bits into a sequence of at least 2 n  symbols where 2 n &gt;m, using an extended Reed-Muller code from a Kasami sequence. A puncturer performs puncturing on the sequence of 2 n  symbols from the encoder so as to output a sequence of m symbols.

Description:
PRIORITY 
   This application is a divisional of application Ser. No. 09/879,688, filed Jun. 12, 2001, and this application claims priority to an application entitled “Apparatus and Method for Encoding and Decoding TFCI in a Mobile Communication System” filed in the Korean Industrial Property Office on Jun. 12, 2000 and assigned Ser. No. 2000-33107, the contents of each of which are incorporated herein by reference. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates generally to an apparatus and method for a TFCI (Transport Format Combination Indicator) code generator in a CDMA mobile communication system, and in particular, to an apparatus and method for encoding a TFCI in an NB-TDD (Narrowband-Time Division Duplex) mobile communication system. 
   2. Description of the Related Art 
   In general, a CDMA mobile communication system (or an IMT-2000 system) transmits data frames of various services such as a voice service, an image service and a data service all together, using a single physical channel. Such service frames are transmitted at either a fixed data rate or a variable data rate. As for the different services transmitted at a fixed data rate, it is not necessary to inform a receiver of a spreading rate of the respective service frames. However, regarding the services transmitted at a variable data rate, it is necessary to inform the receiver of a spreading rate of the respective service frames. In the IMT-2000 system, the data rate is in inverse proportion to the data spreading rate. 
   When the respective services use different frame transfer rates, a TFCI is used to indicate a combination of the currently transmitted services. The TFCI secures correct reception of the respective services. 
     FIG. 1  illustrates an example in which an NB-TDD communication system uses the TFCI. Herein, the NB-TDD system employs 8PSK (8-ary Phase Shift Keying) modulation for high-speed transmission, and the TFCI bits are encoded to a code of length 48 before transmission. As shown in  FIG. 1 , one frame is divided into two sub-frames of a sub-frame# 1  and a sub-frame# 2 . Each sub-frame is comprised of 7 time slots TS# 0 –TS# 6 . Among the 7 time slots, the odd-numbered time slots TS# 0 , TS# 2 , TS# 4  and TS# 6  are used for an uplink transmitted from a mobile station to a base station, while the even-numbered time slots TS# 1 , TS# 3  and TS# 5  are used for a downlink transmitted from a base station to a mobile station. Each time slot has a structure in which data symbols, a first part of TFCI, a midamble signal, SS symbols, TPC symbols, a second part of TFCI, data symbols and GP are sequentially time-multiplexed. 
     FIG. 2  illustrates a structure of a transmitter for transmitting a frame in a conventional NB-TDD communication system. Referring to  FIG. 2 , a TFCI encoder  200  encodes an input TFCI and outputs a TFCI symbols. A first multiplexer (MUX)  210  multiplexes the TFCI symbols output from the TFCI encoder  200  and other signals. Here, the “other signals” refer to the data symbol, the SS symbol and the TCP symbol included in each slot of  FIG. 1 . That is, the first multiplexer  210  multiplexes the TFCI symbol and the other signals except for the midamble signal of  FIG. 1 . A channel spreader  220  channel-spreads the output of the first multiplexer  210  by multiplying it by a given orthogonal code. A scrambler  230  scrambles the output of the channel spreader  220  by multiplying it by a scrambling code. A second multiplexer  240  multiplexes the output of the scrambler  230  and the midamble signal as shown in  FIG. 1 . Here, the first multiplexer  210  and the second multiplexer  240  generate the frame structure of  FIG. 1 , under the control of a controller (not shown). 
     FIG. 3  illustrates a structure of a receiver in the conventional NB-TDD communication system. Referring to  FIG. 3 , a first demultiplexer  340  demultiplexes an input frame signal under the control of a controller (not shown), and outputs a midamble signal and other signals. Here, the “other signals” include the TFCI symbol, the data symbol, the SS symbol and the TCP symbol. A descrambler  330  descrambles the other signals output from the demultiplexer  340  by multiplying them by a scrambling code. A channel despreader  320  channel-despreads the output of the descrambler  330  by multiplying it by an orthogonal code. A second demultiplexer  310  demultiplexes the signals output from the channel despreader  320  into the TFCI symbol and other signals, under the control of the controller. Here, the “other signals” include the data symbol, the SS symbol, and the TCP symbol. A TFCI decoder  300  decodes the TFCI symbol output from the second demultiplexer  310  and outputs TFCI bits. 
   The TFCI is comprised of 1 to 2 bits to indicate 1 to 4 combinations of the services, comprised of 3 to 5 bits to indicate 8 to 32 combinations of the services, or comprised of 6 to 10 bits to indicate 64 to 1024 combinations of the services. Since the TFCI is information indispensable when the receiver analyzes the respective service frames, a transmission error of the TFCI may prevent the receiver from correctly analyzing the respective service frames. Therefore, the TFCI is encoded using an error correcting code so that even though a transmission error occurs on the TFCI, the receiver can correct the error. 
     FIG. 4  illustrates a scheme for encoding the TFCI using an error correcting code according to the prior art. Referring to  FIG. 4 , an extended Reed-Muller encoder  400  encodes an input 10-bit TFCI and outputs a 32-symbol TFCI codeword. A repeater  410  outputs intact even-numbered symbols of the TFCI codeword output from the extended Reed-Muller encoder  400  and repeats odd-numbered symbols, thereby outputting a total of 48 coded symbols. In  FIG. 4 , a less-than-10-bit TFCI is constructed to have a 10-bit format by padding a value of 0 from the MSB (Most Significant Bit), i.e., from the leftmost bit. The (32,10) extended Reed-Muller encoder  400  is disclosed in detail in Korean patent application No. 1999-27932, the contents of which are hereby incorporated by reference. 
   In the (32,10) extended Reed-Muller encoder  400 , a minimum distance between codes is 12. After repetition, an input code is converted to a (48,10) code having a minimum distance of 16. In general, an error correction capability of binary linear codes is determined depending on the minimum distance between the binary linear codes. The minimum distance (dmin) between the binary linear codes to become optimal codes is disclosed in a paper entitled “An Updated Table of Minimum-Distance Bounds for Binary Linear Codes” (A. E. Brouwer and Tom Verhoeff, IEEE Transactions on information Theory, VOL 39, NO. 2, MARCH 1993). 
   The paper discloses that the minimum distance required for the binary linear codes used to obtain a 48-bit output from a 10-bit input is 19 to 20. However, since the encoder  400  has a minimum distance of 16, the error correction encoding scheme of  FIG. 4  does not have optimal codes, causing an increase in TFCI error probability in the same channel environment. Because of the TFCI error, the receiver may misjudge a rate of the data frame and decode the data frame at the misjudged rate, thereby increasing a frame error rate (FER). Therefore, it is important to minimize a frame error rate of the error correction encoder for encoding the TFCI. 
   SUMMARY OF THE INVENTION 
   It is, therefore, an object of the present invention to provide a (48,10) encoding and decoding apparatus and method for encoding a TFCI. 
   It is another object of the present invention to provide an apparatus and method for encoding a TFCI in an NB-TDD CDMA mobile communication system. 
   It is further another object of the present invention to provide an apparatus and method for decoding a TFCI in an NB-TDD CDMA mobile communication system. 
   To achieve the above and other objects, there is provided an apparatus for encoding k consecutive inputs indicating a TFCI of each of successively transmitted frames into a sequence of m symbols in an NB-TDD mobile communication system. An encoder encodes the k input bits into a sequence of at least 2 n  symbols where 2 n &gt;m, using an extended Reed-Muller code from a Kasami sequence. A puncturer performs puncturing on the sequence of 2 n  symbols from the encoder so as to output a sequence of m symbols. 
   Preferably, the encoder comprises: a 1-bit generator for generating a sequence of same symbols; a basis orthogonal sequence generator for generating a plurality of basis orthogonal sequences; a basis mask sequence generator for generating a plurality of basis mask sequences; and an operator for receiving the TFCI including a first information part indicating conversion to a biorthogonal sequence, a second information part indicating conversion to an orthogonal sequence and a third information part indicating conversion to a mask sequence. The operator is also for generating the sequence of 2 n  symbols by combining an orthogonal sequence selected from the basis orthogonal sequences by the second information part, a biorthogonal sequence constructed by a combination of the selected orthogonal sequence and the same symbols selected by the first information part, and a mask sequence selected by the third information part. 
   Preferably, the operator comprises a first multiplier for multiplying the same symbols by the first information part; a plurality of second multipliers for multiplying the basis orthogonal sequences by TFCI bits constituting the second information part; a plurality of third multipliers for multiplying the basis mask sequences by TFCI bits constituting the third information part; and an adder for generating the sequence of 2 n  symbols by adding outputs of the first to third multipliers. 
   To achieve the above and other objects, there is provided an method for encoding 10 consecutive input bits indicating a TFCI of each of successively transmitted frames into a sequence of 48 coded symbols in an NB-TDD mobile communication system, comprising: creating first sequences having a length 48 punctured orthogonal sequences; creating second sequences having a length 48 punctured mask sequences; multiplying the first sequences with each associated TFCI bit and the second sequences with each associated TFCI bit; and adding the each resulting sequences calculated by the multiplication and outputting the sequence of 48 symbols wherein the punctured orthogonal sequences and the punctured mask sequences are sequences generated by puncturing following positions out of length 64 Walsh codes and length 64 masks; {0, 4, 8,13,16,20,27,31,34,38,41,44,50,54,57,61} 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The above and other objects, features and advantages of the present invention will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings in which: 
       FIG. 1  is a diagram illustrating a frame format used in a conventional NB-TDD CDMA communication system; 
       FIG. 2  is a diagram illustrating a structure of a transmitter for transmitting a frame in the conventional NB-TDD communication system; 
       FIG. 3  is a diagram illustrating a structure of a receiver for the conventional NB-TDD communication system; 
       FIG. 4  is a diagram illustrating a scheme for encoding a TFCI using an error correcting code according to the prior art; 
       FIG. 5  is a diagram illustrating a scheme for encoding a linear error correcting code; 
       FIG. 6  is a flow chart illustrating a procedure for creating a mask function using a Kasami sequence family; 
       FIG. 7A  is a diagram illustrating an apparatus for encoding a TFCI according to a first embodiment of the present invention; 
       FIG. 7B  is a diagram illustrating an apparatus for encoding a TFCI according to a second embodiment of the present invention; 
       FIG. 8  is a flow chart illustrating an operation performed by the encoder of  FIG. 7A ; 
       FIG. 9  is a diagram illustrating an apparatus for decoding a TFCI according to an embodiment of the present invention; 
       FIG. 10  is a flow chart illustrating an operation performed by the comparator shown in  FIG. 9 ; 
       FIG. 11  is a diagram illustrating a structure of 1024 codes output from a (64,10) encoder according to an embodiment of the present invention; and 
       FIG. 12  is a flow chart illustrating an operation performed by the encoder of  FIG. 7B . 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   A preferred embodiment of the present invention will be described herein below with reference to the accompanying drawings. In the following description, well-known functions or constructions are not described in detail since they would obscure the invention in unnecessary detail. 
   A CDMA mobile communication system according an embodiment of the present invention uses extended Reed-Muller codes to create optimal codes when encoding a TFCI. 
   Commonly, a measure, i.e., a parameter indicating performance of a linear error correcting code, includes distribution of a Hamming distance of a codeword of an error correcting code. The Hamming distance refers to the number of non-zero symbols in the respective codewords. That is, for a codeword ‘0111’, the number of 1&#39;s included in this codeword, i.e., the Hamming distance is 3. The least value among the Hamming distance values of several codewords is called a “minimum distance (dmin)”. The linear error correcting code has superior error correcting performance (or capability), as the minimum distance is increased more and more. 
   The extended Reed-Muller code can be derived from a sequence determined by the sum (or XOR) of a specific sequence and an m-sequence. In order to use a sequence family (or group) including the sum of the sequences as its elements, the sequence family must have a large minimum distance. Such specific sequence family includes a Kasami sequence family, a Gold sequence family and a Kerdock code family. Such specific sequences have a minimum distance of (2 2m −2 m )/2 for the full length L=2 2m , and a minimum distance of 2 2m −2 m  for the full distance L=2 2m+1 . That is, the minimum distance is 28 for the full length 64. 
   Now, a description will be made regarding a method for creating an extended error correcting code which is a linear error correcting code having high performance, using the above stated sequence families. 
   According to a coding theory, there exists a column permutation function for creating a Walsh code by cyclic-shifting the m-sequence. The m-sequence becomes a Walsh code when the sequences comprised of the specific sequence and the m-sequence are subjected to column permutation using the column permutation function. The minimum distance by the sum (XOR) of the specific sequence and the Walsh code satisfies the optimal code property. Herein, a sequence obtained by column-permuting the specific sequence will be referred to as a “mask function (or mask sequence).”  FIG. 5  illustrates a scheme for encoding the linear error correcting code. As illustrated, the present invention provides a TFCI encoding scheme for making a complete coded symbol (or TFCI codeword) by adding a first coded symbol (or mask function) created by a first TFCI bit and a second coded symbol (or orthogonal code) created by a second TFCI bit. 
   Referring to  FIG. 5 , TFCI bits to be transmitted are divided into a first TFCI bit and a second TFCI bit and then, provided to a mask function generator  502  and a Walsh code generator  504 , respectively. The mask function generator  504  outputs a given mask sequence by encoding the first TFCI bit, and the Walsh code generator  504  outputs a given orthogonal sequence by encoding the second TFCI bit. An adder  510  then adds (XORs) the mask sequence from the mask function generator  502  and the orthogonal sequence from the orthogonal code generator  504 , and outputs a complete TFCI codeword (or TFCI coded symbol). The mask function generator  502  may have mask sequences associated with every set of the first TFCI bits in the form of a coding table. The orthogonal code generator  504  may also have orthogonal sequences associated with every set of the second TFCI bits in the form of a coding table. 
   Now, a description will be made of a method for creating the mask functions (or mask sequences) in the case where a (2 n ,n+k) code is created using the Kasami sequence. Here, the “(2 n ,n+k) code” refers to a code for outputting a TFCI codeword (or coded symbol) comprised of 2 n  symbols by receiving (n+k) TFCI bits (input information bits). Actually, it is known that the Kasami sequence is represented by the sum of different m-sequences. Therefore, in order to create the (2 n ,n+k) code, a Kasami sequence of length 2 n −1 must be created first. The Kasami sequence is equivalent to the sum of an m-sequence created by a generator polynomial fl(x) and a sequence obtained by repeating 2 (n/2) +1 times a sequence of length 2 (n/2) −1 determined by decimating the m-sequence in a unit of 2 (n/2) +1. In addition, if the generator polynomial is determined, the respective m-sequences m(t), i.e., m 1 (t) and m 2 (t) can be calculated using a trace function in accordance with Equation (1) below. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             m 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                         
                           Tr 
                           ⁡ 
                           
                             ( 
                             
                               A 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 α 
                                 t 
                               
                             
                             ) 
                           
                         
                       
                       , 
                       
                         t 
                         = 
                         0 
                       
                       , 
                       1 
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       30 
                     
                   
                 
                 
                   
                     
                       where 
                       , 
                       
                           
                       
                       ⁢ 
                       
                         
                           Tr 
                           ⁡ 
                           
                             ( 
                             α 
                             ) 
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               n 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             α 
                             
                               2 
                               k 
                             
                           
                         
                       
                       , 
                       
                         α 
                         ∈ 
                         
                           GF 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               n 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
   
   In Equation (1), A indicates a value determined according to an initial value of the m-sequence, α indicates a root of the generator polynomial, and n indicates the degree of the generator polynomial. 
     FIG. 6  illustrates a procedure for creating the mask function in the case where the (2 n ,n+k) code (i.e., a code for outputting a 2 n -bit coded symbol by receiving (n+k) input information bits) is created using the Kasami sequence among the above-mentioned sequences. It is known that the Kasami sequence is represented by the sum of the different m-sequences. Therefore, in order to create the (2 n ,n+k) code, a Kasami sequence of length 2 n −1 must be created first. The Kasami sequence, as described above, is created by the sum of an m-sequence created by a generator polynomial fl(x) and a sequence obtained by repeating 2 (n/2) +1 times a sequence of length 2 (n/2) −1 determined by decimating the m-sequence in a unit of 2 (n/2) +1. 
   Referring to  FIG. 6 , in step  610 , an m-sequence m 1 (t) created by the generator polynomial fl(x) and a sequence m 2 (t) obtained by repeating 2 (n/2) +1 times a sequence of length 2 (n/2) −1 determined by decimating the m-sequence m 2 (t) in a unit of 2 (n/2) +1 are calculated in accordance with Equation (1). In step  620 , a column permutation function σ(t) for converting the m-sequence m 1 (t) into a Walsh code shown in Equation (2) below is calculated. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         σ 
                         : 
                         
                           { 
                           
                             0 
                             , 
                             1 
                             , 
                             2 
                             , 
                             … 
                             ⁢ 
                             
                                 
                             
                             , 
                             
                               
                                 2 
                                 n 
                               
                               - 
                               2 
                             
                           
                           } 
                         
                       
                       -&gt; 
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             
                               2 
                               n 
                             
                             - 
                             1 
                           
                         
                         } 
                       
                     
                   
                 
                 
                   
                     
                       
                         σ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             n 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               m 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⁢ 
                           
                             2 
                             
                               n 
                               - 
                               1 
                               - 
                               i 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
   
   In step  630 , 7 sequence families obtained by cyclic-shifting the m-sequence m 2 (t) 0 to 6 times are subjected to column permutation using σ −1 (t)+2, where σ −1 (t) is an inverse function of the column permutation function σ(t) for converting the sequence m 1 (t) to the Walsh code. Further, ‘0’ is added to the head of every sequence created by the column permutation so as to make the sequences have a length 2 n , thereby creating 2 n −1 sequence families di(t) of length 2 n , where i=0, . . . ,2 n −1 and t=1, . . . ,2 n . The sequence families created in step  630  can be represented by Equation (3) below. 
   
     
       
         
           
             
               
                 
                   
                     
                       { 
                       
                         
                           
                             
                               
                                 d 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             | 
                             t 
                           
                           = 
                           1 
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           2 
                           ⁢ 
                           n 
                         
                         , 
                         
                             
                         
                         ⁢ 
                         
                           i 
                           = 
                           0 
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           
                             2 
                             
                               n 
                               2 
                             
                           
                           - 
                           2 
                         
                       
                       } 
                     
                   
                 
                 
                   
                     
                       
                         
                           d 
                           i 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       = 
                       
                         ( 
                         
                           
                             
                               
                                 0 
                                 , 
                               
                             
                             
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   t 
                                 
                                 = 
                                 1 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     m 
                                     d 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       + 
                                       i 
                                       - 
                                       2 
                                     
                                     ) 
                                   
                                 
                                 , 
                               
                             
                             
                               
                                 
                                   
                                     if 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   = 
                                   1 
                                 
                                 , 
                                 2 
                                 , 
                                 3 
                                 , 
                                 … 
                                 ⁢ 
                                 
                                     
                                 
                                 , 
                                 
                                   2 
                                   n 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
   
   The calculated sequence families di(t) are mask functions which can be used as 7 masks. 
   One of the properties of the calculated sequence families di(t) is that a mask created by adding two different masks out of the above masks becomes another mask out of the 2 (n/2) −1 masks. To generalize further, all of the 2 (n/2) −1 masks including a mask of all 0&#39;s can be represented by a predefined sum of n masks out of the 2 (n/2) −1 masks. The n masks are defined as basis sequences (or basis sequences). 
   The total number of codewords required in creating the (2 n ,n+k) code is 2 n+k  which is the number of possible sets of the input information bits. Here, the number of biorthogonal sequences indicating 2 n  orthogonal sequences (or Walsh sequences) and their complements is 2 n ×2=2 n −1, and the number of non-zero masks required to create the (2 n ,n+k) code is (2 n+k /2 n+1 )−1=2 k−1 −1. In addition, all of the 2 k−1 −1 masks can also be represented by a predefined sum of the (k−1) masks on the basis of the property similar to that described above. 
   Next, a method for selecting the (k−1) masks will be described. In step  630 , a sequence family is created by cyclic-shifting the m 2 (t) 0 to 2 (n/2)−1  times. An m-sequence created by cyclic-shifting the m 2 (t) i times can be expressed as Tr(α i ·α t ) using Equation (1). That is, a sequence family created by cyclic-shifting the m 2 (t) 0 to 6 times include the sequences created according to initial values A=1,α, . . . , α 2     n     −2 . At this moment, (k−1) linearly independent basis elements are searched from the Galois elements 1,α, . . . , α 2     n     −2 . The sequences corresponding to the output sequences of the trace function taking the (k−1) basis elements as initial values become basis mask sequences. In this process, the linearly independent condition is represented by Equation (4) below.
 
α 1 , . . . ,α k−1 : linearly independent              c   1 α 1   +c   2 α 2   + . . . +c   k−1 α k−1 ≠0, ∀ c   1   ,c   2   , ,. . . , c   k−1   Equation (4)

   A method for creating the generalized mask function will be described with reference to  FIG. 6 , for the case where a (64,10) code is created using the Kasami sequence family. Actually, it is well known that the Kasami sequence is represented by the sum of the different m-sequences. Therefore, in order to create the (64,10) code, a Kasami sequence of length 63 must be created first. The Kasami sequence is comprised of an m-sequence created by a generator polynomial x 6 +x+1 and a sequence created by repeating 2 (n/2) +1 times a sequence of length 2 (n/2) −1 determined by decimating the m-sequence in a unit of 2 (n/2) +1. Here, if the generator polynomial is determined, each m-sequence m(t) can be calculated using the trace finction as shown in Equation (5) below. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             m 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                         
                           Tr 
                           ⁡ 
                           
                             ( 
                             
                               A 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 α 
                                 t 
                               
                             
                             ) 
                           
                         
                       
                       , 
                       
                         t 
                         = 
                         0 
                       
                       , 
                       1 
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       63 
                     
                   
                 
                 
                   
                     
                       where 
                       , 
                       
                           
                       
                       ⁢ 
                       
                         
                           Tr 
                           ⁡ 
                           
                             ( 
                             α 
                             ) 
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             4 
                           
                           ⁢ 
                           
                             α 
                             
                               2 
                               n 
                             
                           
                         
                       
                       , 
                       
                         α 
                         ∈ 
                         
                           GF 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               5 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 Equation 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
   
   In Equation (5), A indicates a value determined according to an initial value of the m-sequence and a indicates a root of the generator polynomial. In addition, n=6 because the generator polynomial is of the 6 th  degree. 
     FIG. 6  illustrates a procedure for creating the mask function in the case where the (64,10) code (i.e., a code for outputting a 64-bit coded symbol by receiving 10 input information bits) is created using a Kasami sequence family among the above-stated sequence families. Referring to  FIG. 6 , in step  610 , an m-sequence m 1 (t) created by the generator polynomial x 6 +x+1 and a sequence m 2 (t) obtained by repeating 2 (n/2) +1 times a sequence of length 2 (n/2) −1 determined by decimating the m-sequence m 2 (t) in a unit of 2 (n/2) +1 are calculated in accordance with Equation (5). In step  620 , a column permutation function σ(t) for converting the m-sequence m 1 (t) into a Walsh code shown in Equation (6) below is calculated. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         σ 
                         : 
                         
                           { 
                           
                             0 
                             , 
                             1 
                             , 
                             2 
                             , 
                             … 
                             ⁢ 
                             
                                 
                             
                             , 
                             63 
                           
                           } 
                         
                       
                       -&gt; 
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           64 
                         
                         } 
                       
                     
                   
                 
                 
                   
                     
                       
                         σ 
                         ⁡ 
                         
                           ( 
                           t 
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   In step  630 , 7 sequence families obtained by cyclic-shifting the m-sequence m 2 (t) 0 to 6 times are subjected to column permutation using σ −1 (t)+2, where σ −1 (t) is an inverse function of the column permutation function σ(t) for converting the sequence m 1 (t) to the Walsh code. Further, ‘0’ is added to the head of every sequence created by the column permutation so as to make the sequences have a length 64, thereby creating 7 sequence families di(t) of length 64, where i=0, . . . ,6 and t=1, . . . ,64. The sequence families created in step  630  can be represented by Equation (7) below. 
   
     
       
         
           
             
               
                 
                   
                     
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   The sequence families di(t) calculated by Equation (7) are mask functions which can be used as 7 mask sequences. 
   One of the properties of the calculated sequence families di(t) is that a mask created by adding two different masks out of the above masks becomes another mask out of the 7 masks. To generalize further, all of the 7 masks can be represented by a predefined sum of 3 masks out of the 7 masks. As mentioned above, all of the mask sequences which can be represented by the predefined sum of the masks, are defined as basis sequences. 
   The total number of codewords required in creating the (64,10) code is 2 10 =1024, which is the number of possible sets of the input information bits. Here, the number of biorthogonal codewords of length 64 is 64×2=128, and the number of masks required to create the (64,10) code is (1024/128)−1=7. In addition, all of the 7 masks can also be represented by a predefined sum of the 3 masks on the basis of the property similar to that described above. Therefore, a method for selecting the 3 masks is required. The method for selecting the 3 masks will be described below. In step  630 , a sequence family is created by cyclic-shifting the m 2 (t) 0 to 6 times. An m-sequence created by cyclic-shifting the m 2 (t) i times can be expressed as Tr(α i ·α t ) using Equation (5). That is, a sequence family created by cyclic-shifting the m 2 (t) 0 to 6 times include the sequences created according to initial values A=1,α, . . . ,α 6 . At this moment, 3 linearly independent basis elements are searched from the Galois elements 1,α, . . . ,α 6 . It is possible to create all the 7 masks by the predefined sum of the 3 masks by selecting the sequences taking the 3 basis elements as initial values. In this process, the linearly independent condition is represented by Equation (8) below.
 
α,β,γ,δ: linearly independent              c   1   α+c   2   β+c   3   γ+c   4 Δ≠0,  ∀c   1   ,c   2   ,c   3   , c   4    Equation (8)

   Actually, 1, α and α 2  in the Galois field GF(2 3 ) are basis polynomials well known as the above 4 linear independent elements. Therefore, the following 3 mask functions M 1 , M 2  and M 4  are calculated by substituting the basis polynomials into Equation (5) 
   M 1 =0011010101101111101000110000011011110110010100111001111111000101 
   M 2 =0100011111010001111011010111101101111011000100101101000110111000 
   M 4 =0001100011100111110101001101010010111101101111010111000110001110 
   Now, a detailed description will be made regarding an apparatus and method for encoding and decoding a TFCI in a NB-TDD CDMA mobile communication system according to an embodiment of the present invention. In the embodiments of the present invention, the encoder and the decoder use the basis mask sequences calculated in the above method. Specifically, a method for creating the basis mask sequences will be described below. 
   First Embodiment 
     FIG. 7A  illustrates an apparatus for encoding a TFCI in a NB-TDD CDMA mobile communication system according to a first embodiment of the present invention. Referring to  FIG. 7A , 10 input information bits a 0 –a 9  are provided to their associated multipliers  740 – 749 , respectively. A basis Walsh code generator  710  generates basis Walsh codes having a predetermined length. Here, the “basis Walsh codes” refer to predetermined Walsh codes, by a predetermined sum of which all of desired Walsh codes can be created. For example, for a Walsh code of length 64, the basis Walsh codes include a 1 st  Walsh code W 1 , a 2 nd  Walsh code W 2 , a 4 th  Walsh code W 4 , an 8 th  Walsh code W 8 , a 16 th  Walsh code W 16  and a 32 nd  Walsh code W 32 . A 1-bit generator  700  continuously generates a predetermined code bit. That is, as the invention is applied to the biorthogonal sequences, the 1-bit generator  700  generates a bit required for using orthogonal sequences as biorthogonal codes. For example, the 1-bit generator  700  constantly generates a bit having a value ‘1’, thereby to invert the Walsh codes generated from the basis Walsh code generator  710 . 
   The Walsh code generator  710  simultaneously outputs Walsh codes W 1 , W 2 , W 4 , W 8 , W 16  and W 32  of length 64. The multiplier  740  multiplies the 1 st  Walsh code W 1  (=0101010101010101010101010101010101010101010101010101010101010101) from the Walsh code generator  710  by the first input information bit a 0 . The multiplier  741  multiplies the 2 nd  Walsh code W 2  (=0011001100110011001100110011001100110011001100110011001100110011) from the Walsh code generator  710  by the second input information bit a 1 . The multiplier  742  multiplies the 4 th  Walsh code W 4  (=0000111100001111000011110000111100001111000011110000111100001111) from the Walsh code generator  710  by the third input information bit a 2 . The multiplier  743  multiplies the 8 th  Walsh code W 8  (=00000000 11111111000000001111111100000000111111110000000011111111) from the Walsh code generator  710  by the fourth input information bit a 3 . The multiplier  744  multiplies the 16 th  Walsh code W 16  (=0000000000000000111111111111111100000000000000001111111111111111) from the Walsh code generator  710  by the fifth input information bit a 4 . The multiplier  745  multiplies the 32 nd  Walsh code W 32  (=00000000000000000000000000000000111111111111111111 11111111111111) from the Walsh code generator  710  by the sixth input information bit a 5 . That is, the multipliers  740 – 745  multiply the input basis Walsh codes W 1 , W 2 , W 4 , W 8 , W 16  and W 32  by their associated input information bits a 0 –a 5  in a symbol unit. Meanwhile, the multiplier  746  multiplies the symbols of all 1&#39;s output from the 1-bit generator  700  by the seventh input information bit a 6 . 
   A mask generator  720  generates mask sequences having a predetermined length. The method for generating the mask sequences will not be described, since it has already been described above. For example, when the (64,10) code is generated using the Kasami sequence, the basis mask sequences include a 1 st  mask sequence M 1 , a 2 nd  mask sequence M 2  and a 4 th  mask sequence M 4 . The mask generator  720  simultaneously outputs the mask functions M 1 , M 2  and M 4  of length 64. The multiplier  747  multiplies the 1 st  mask function M 1  (=0011010101101111101000110000011011110110010100111001111111000101) from the mask generator  720  by the eighth input information bit a 7 . The multiplier  748  multiplies the 2 nd  mask function M 2  (=0100011111010001111011010111101101111011000100101101000110111000) from the mask generator  720  by the ninth input information bit a 8 . The multiplier  749  multiplies the 4 th  mask function M 4  (0001100011100111110101001101010010111101101111010111000110001110) from the mask generator  720  by the tenth input information bit a 9 . The multipliers  747 – 749  multiply the input basis mask sequences M 1 , M 2  and M 4  by the associated input information bits a 7 –a 9  in a symbol unit. 
   An adder  760  adds (or XORS) the symbols output from the multipliers  740 – 749  in a symbol unit, and then, outputs 64 coded symbols. A symbol puncturer  770  punctures the 64 symbols output from the adder  760  according to a predetermined rule and outputs 48 symbols. That is, the (48,10) encoder punctures 16 symbols from the 64 symbols created by the (64,10) code. The minimum distance of the (48,10) encoder varies depending on the positions of the 16 punctured symbols. Combinations of the 16 punctured positions, providing superior performance, are shown below. When using the following combinations of the punctured positions, the (48,10) encoder has the minimum distance of 18 and provides superior weight distribution. 
   {0, 4, 8,13,16,20,27,31,34,38,41,44,50,54,57,61} 
   {0, 4, 8,13,16,21,25,28,32,37,43,44,49,52,56,62} 
   {0, 4, 8,13,16,21,25,31,32,37,43,44,49,52,56,61} 
   {0, 4, 8,13,18,21,25,30,35,36,40,46,50,53,57,62} 
   {0, 4, 8,13,18,21,25,30,35,37,40,47,50,53,57,62} 
   {0, 4, 8,13,19,22,27,30,33,36,41,44,49,55,58,61} 
   {0, 4, 8,13,19,22,27,30,33,36,41,44,50,52,56,63} 
   {0, 4, 8,13,19,22,27,30,33,36,41,44,50,52,58,61} 
   {0, 4, 8,13,16,20,27,31,34,38,41,44,50,54,57,61} 
     FIG. 8  illustrates a control flow for encoding a TFCI in an NB-TDD CDMA mobile communication system according to the first embodiment of the present invention. Referring to  FIG. 8 , in step  800 , a sequence of 10 input information bits a 0 –a 9  is input and then, parameters code[ ] and j are initialized to ‘0’. The parameter code[ ] indicates the 64 coded symbols finally output from the encoder and the parameter j is used to count the 64 symbols constituting one codeword. 
   Thereafter, it is determined in step  810  whether the first information bit a 0  is ‘1’. If the first information bit a 0  is ‘1’, the 1 st  Walsh code W 1  (=010101010101010101010101010101010101010101010101010101010101) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the first information bit a 0  is not ‘1’, the control flow skips to step  812 . After step  810 , it is determined in step  812  whether the second information bit a 1  is ‘1’. If the second information bit a 1  is ‘1’, the 2 nd  Walsh code W 2  (=0011001100110011001100110011001100110011001100110011001100110011) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the second information bit a 1  is not ‘1’, the control flow skips to step  814 . After step  812 , it is determined in step  814  whether the third information bit a 2  is ‘1’. If the third information bit a 2  is ‘1’, the 4 th  Walsh code W 4 (=0000111100001111000011110000111100001111000011110000111100001111) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the third information bit a 2  is not ‘1’, the control flow skips to step  816 . After step  814 , it is determined in step  816  whether the fourth information bit a 3  is ‘1’. If the fourth information bit a 3  is ‘1’, the 8 th  Walsh code W 8  (=0000000011111111000000001111111100000000111111110000000011111111) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the fourth information bit a 3  is not ‘1’, the control flow skips to step  818 . After step  816 , it is determined in step  818  whether the fifth information bit a 4  is ‘1’. If the fifth information bit a 4  is ‘1’, the 16 th  Walsh code W 16 (=000000000000000011111111111111110000000000000000111111111111111) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the fifth information bit a 4  is not ‘1’, the control flow skips to step  820 . After step  818 , it is determined in step  820  whether the sixth information bit a 5  is ‘1’. If the sixth information bit a 5  is ‘1’, the 32 nd  Walsh code W 32 (=0000000000000000000000000000000011111111111111111111111111111111) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the sixth information bit a 5  is not ‘1’, the control flow jumps to step  822 . 
   After step  820 , it is determined in step  822  whether the seventh information bit a 6  is ‘1’. If the seventh information bit a 6  is ‘1’, a sequence of all 1&#39;s is XORed with the coded symbol sequence parameter code [ ] of length 64. Otherwise, if the seventh information bit a 6  is not ‘1’, the control flow jumps to step  824 . That is, in step  822 , the Walsh code created in the preceding steps is XORed by 1 thereby to create a biorthogonal code. More specifically, if the seventh information bit a 6  is ‘1’, the parameter j is initialized to ‘0’ and a j th  parameter code[j] is XORed with ‘1’. Further, it is determined whether the parameter j is 63, in order to determine whether the parameter j is the last symbol of the codeword. If the parameter j is not 63, this process is repeated after increasing the parameter j by 1. In other words, in step  822 , when the seventh information bit a 6  is ‘1’, a length-64 sequence of all 1&#39;s is XORed with a coded symbol sequence of length 64. Therefore, after repeating this process 64 times, the control flow proceeds to step  824  from the step for determining whether the parameter j is 63. 
   After step  822 , it is determined in step  824  whether the eighth information bit a 7  is ‘1’. If the eighth information bit a 7  is ‘1’, the first mask function M 1  (=001101010110111110100011000001101111011001010011100111111) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the eighth information bit a 7  is not ‘1’, the control flow skips to step  826 . After step  824 , it is determined in step  826  whether the ninth information bit a 8  is ‘1’. If the ninth information bit a 8  is ‘1’, the second mask function M 2  (=0100011111010001111011010111101101111011000100101101000110111000) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the ninth information bit a 8  is not ‘1’, the control flow skips to step  828 . After step  826 , it is determined in step  828  whether the tenth information bit a 9  is ‘1’. If the tenth information bit a 9  is ‘1’, the fourth mask function M 4  (=000110001110011111010100110101001011110110 1111010111000110001110) is XORed with the coded symbol sequence parameter code[ ] of length 64. Otherwise, if the tenth information bit a 9  is not ‘1’, the control flow skips to step  830 . In step  830 , only the sequences corresponding to information bits  1 &#39;s out of the 10 sequences W 1 , W 2 , W 4 , W 8 , W 16 , W 32 ,  1 , M 1 , M 2  and M 4  of length 64 associated respectively with the 10 input information bits a 0 –a 9  are all XORed to output a value of the coded symbol sequence parameter code[ ]. 
   The (64,10) encoder operating in the method of  FIG. 8  creates 64 Walsh codes of length 64, 64 inverted Walsh codes determined by inverting the 64 Walsh codes, and a total of 896 codes determined by the combination of a total of 7 mask sequences calculated by the combination of a total of 128 orthogonal codes and 3 mask functions. Therefore, the total number of codewords is 1024. In addition, a (64,9) encoder creates 64 Walsh codes of length 64, Walsh codes calculated by adding all 1&#39;s to (or multiplying −1 by, in case of a real number) symbols of every Walsh code among the 1024 codewords, and codes determined by combining a total of 4 mask functions calculated by the combination of a total of 128 orthogonal codes and 2 mask functions among the 3 mask functions, and a (64,8) encoder creates 64 Walsh codes of length 64, Walsh codes calculated by adding all 1&#39;s to (or multiplying −1 by, in case of a real number) symbols of every Walsh code among the 1024 codewords, and codes determined by combining a total of 2 mask functions calculated by the combination of a total of 128 biorthogonal codes and 1 mask function among the 3 mask functions. The (64,9) encoder and the (64,8) encoder both have a minimum distance of 28. The (64,9) encoder can be realized using only two of the 3 mask functions output from the mask function generator  720  of  FIG. 7A , while the (64,8) encoder can be realized using only one of the 3 mask functions output from the mask function generator  720 . As stated above, the encoder can adaptively perform encoding according to the number of input information bits, and can also have superior performance by increasing the minimum distance determining the performance of the encoder, as high as possible. 
   The (64,10) encoder uses, as codewords, 64 Walsh codes of length 64, 64 inverted Walsh codes calculated by inverting the 64 Walsh codes, and 896 sequences calculated by combining a total of 128 biorthogonal codes with 7 masks functions of length 64, the structure of which is illustrated in  FIG. 11 . 
     FIG. 9  illustrates an apparatus for decoding a TFCI according to an embodiment of the present invention. Referring to  FIG. 9 , the decoder inserts ‘0’ in the positions, punctured by the encoder, of a received signal corresponding to the TFCI symbol of length of 48, having a value of +1/−1, thereby to create a received signal r(t) of length 64. The received signal r(t) is provided to 7 multipliers  901 – 907  and a correlation calculator  920 . The received signal r(t) is a signal encoded by a predetermined Walsh code and a predetermined mask sequence in the encoder of the transmitter. A mask generator  910  creates possible mask functions M 1 –M 7  which can be created by 3 basis masks, and provides the generated mask functions to multipliers  901 – 907 , respectively. The multiplier  901  multiplies the received signal r(t) by the mask function M 1  output from the mask generator  910 , and provides its output to a correlation calculator  921 . The multiplier  902  multiplies the received signal r(t) by the mask function M 2  output from the mask generator  910 , and provides its output to a correlation calculator  922 . The multiplier  907  multiplies the received signal r(t) by the mask function M 7  output from the mask generator  910 , and provides its output to a correlation calculator  927 . That is, the multipliers  901 – 907  multiply the received signal r(t) by their associated mask functions M 1 –M 7  from the mask generator  910 , and provide their outputs to the associated correlation calculators  921 – 927 , respectively. By doing so, the received signal r(t) and the signals calculated by multiplying the received signal r(t) by the possible 7 mask functions, i.e., a total of 8 signals are provided to the 8 correlation calculators  920 – 927 , respectively. If the transmitter has encoded the TFCI using a predetermined mask function, any one of the outputs from the multipliers  901 – 907  will be a mask function-removed signal. Then, the correlation calculators  920 – 927  calculate 128 correlation values by correlating the received signal r(t) and the outputs of the multipliers  901 – 907  with 64 Walsh codes of length 64 and 64 inverted Walsh codes calculated by inverting the 64 Walsh codes, i.e., a total of 128 bi-Walsh (or biorthogonal) codes. The largest one of the calculated correlation values, an index of then-correlated Walsh code and an index of the correlation calculator are provided to a correlation comparator  940 . The 128 Walsh codes have already been defined above. The correlation calculator  920  calculates 128 correlation values by correlating the received signal r(t) with 128 bi-Waish codes of length 64. Further, the correlation calculator  920  provides the correlation comparator  940  with the largest one of the calculated correlation values, an index of then-calculated Walsh code and an index ‘0’ of the correlation calculator  920 . Here, the index of the correlation calculator is equivalent to an index of the mask function indicating which mask function is multiplied by the received signal for the signal input to the correlation calculator. However, the mask index ‘0’ means that no mask is multiplied by the received signal. Further, the correlation calculator  921  also calculates 128 correlation values by correlating the received signal r(t) multiplied by the mask function M 1  by the multiplier  901  with 128 bi-Walsh codes of length 64. Further, the correlation calculator  921  provides the correlation comparator  940  with the largest one of the calculated correlation values, an index of then-calculated Walsh code and an index ‘1’ of the correlation calculator  921 . The correlation calculator  922  calculates 128 correlation values by correlating the received signal r(t) multiplied by the mask function M 2  by the multiplier  902  with 128 bi-Walsh codes of length 64. Further, the correlation calculator  922  provides the correlation comparator  940  with the largest one of the 128 calculated correlation values, an index of then-calculated Walsh code and an index ‘2’ of the correlation calculator  922 . The correlation calculator  927  calculates 128 correlation values by correlating the received signal r(t) multiplied by the mask function M 7  by the multiplier  907  with 128 bi-Walsh codes of length 64. Further, the correlation calculator  927  provides the correlation comparator  940  with the largest one of the calculated correlation values, an index of then-calculated Walsh code and an index ‘7’ of the correlation calculator  927 . 
   The correlation comparator  940  then compares the 8 largest correlation values provided from the correlation calculators  920 – 927 , and determines the largest one of them. After determining the largest correlation value, the correlation comparator  940  outputs TFCI information bits transmitted from the transmitter according to the index of the Walsh code provided from the correlation calculator associated with the determined correlation value and an index (or mask index) of the same correlation calculator. That is, the correlation comparator  940  determines a decoded signal of the received signal using the index of the Walsh code and the index of the mask function. 
     FIG. 10  illustrates a procedure for determining a Walsh code index and a mask function index for the largest correlation value by comparing the 8 correlation values in the correlation comparator  940  according to the first embodiment of the present invention, and outputting the TFCI information bits accordingly. Referring to  FIG. 10 , in step  1000 , a frequency indicating index parameter i is initialized to 1, and a maximum value, a Walsh code index and a mask index are all initialized to ‘0’. In step  1010 , the correlation value, the Walsh code index for the correlation value and the mask index, output from the first correlation calculator  920 , are stored as a first maximum value, a first Walsh code index and a first mask sequence index, respectively. Thereafter, in step  1020 , the first maximum value is compared with a previously stored maximum value. If the first maximum value is larger than the previously stored maximum value, the procedure goes to step  1030 . Otherwise, if the first maximum value is smaller than or equal to the previously stored maximum value, the procedure proceeds to step  1040 . In step  1030 , the first maximum value is designated as the maximum value, and the first Walsh code index and the first mask index are designated as the Walsh code index and the mask index, respectively. In step  1040 , a value set for the index parameter i is compared with the number ‘8’ of the correlation calculators, in order to determine whether comparison has been completely performed on all of the 8 correlation values. If the frequency indicating index i is not equal to the number ‘8’ of the correlation calculators in step  1040 , the correlation comparator  940  increases the frequency indicating index i by 1 in step  1060  and thereafter, returns to step  1010  to repeat the above-described process using the i th  maximum value, the i th  Walsh code index and the i th  mask index, output from the increased i th  correlation calculator. After the above process is repeatedly performed on the 8 th  maximum value, the 8 th  Walsh code index and the 8 th  mask index, the frequency indicating index i becomes 8. Then, the procedure goes to step  1050 . In step  1050 , the correlation comparator  940  outputs decoded bits (TFCI information bits) associated with the Walsh code index and the mask index. The Walsh code index and the mask index corresponding to the decoded bits are the Walsh code index and the mask index corresponding to the largest one of the 8 correlation values provided from the 8 correlation calculators. 
   In the first embodiment, the (48,10) encoder creates 48 symbols by puncturing 16 symbols after creating 64 codes. In the second embodiment below, however, unlike  FIG. 7A , the encoder outputs 48 symbols after puncturing 16 symbols according to a predetermined puncturing pattern in the Walsh code generator, the 1-bit generator and the mask generator. 
   Second Embodiment 
   The encoding apparatus according to the second embodiment of the present invention is similar in structure to the encoder described with reference to the first embodiment. However, the only difference is that the sequences output from the 1-bit generator, the Walsh code generator and the mask generator are the sequences of length 48, to which a puncturing pattern is previously applied. For example, the sequences output from the Walsh code generator, the 1-bit generator and the mask generator according to the first embodiment, from which 0 th , 4 th , 8 th , 13 th , 16 h , 20 h , 27 th , 31 st , 34 th , 38 th , 41 st , 44 th , 50 th , 54 th , 57 th  and 61 st  terms are punctured, are used in the second embodiment. 
     FIG. 7B  illustrates an apparatus for encoding a TFCI in an NB-TDD CDMA mobile communication system according to the second embodiment of the present invention. Referring to  FIG. 7B , 10 input information bits a 0 –a 9  are provided to their associated multipliers  7400 ,  7410 ,  7420 ,  7430 ,  7440 ,  7450 ,  7460 ,  7470 ,  7480  and  7490 , respectively. A basis Walsh code generator  7100  simultaneously generates Walsh codes W 1 ′, W 2 ′, W 4 ′, W 8 ′, W 16 ′ and W 32 ′ of length 48, calculated by puncturing the basis Walsh codes according to a predetermined puncturing rule as described above. Here, the “basis Walsh codes” refer to predetermined Walsh codes, by a predetermined sum of which all of desired Walsh codes can be created. For example, for a Walsh code of length 64, the basis Walsh codes include a 1 st  Walsh code W 1 , a 2 nd  Walsh code W 2 , a 4 th  Walsh code W 4 , an 8 th  Walsh code W 8 , a 16 th  Walsh code W16 and a 32 nd  Walsh code W 32 . A 1-bit generator  7000  continuously generates a predetermined code bit. The multiplier  7400  multiplies the Walsh code W 1 ′ (=101101101001101101010010011011001101011011001001) punctured according to a predetermined puncturing rule by the Walsh code generator  7100  by the input information bit a 0 . The multiplier  7410  multiplies the punctured Walsh code W 2 ′ (=011011011011011011001001001001011011001001011011) from the Walsh code generator  7100  by the input information bit a 1 . The multiplier  7420  multiplies the punctured Walsh code W 4 ′ (=000111000111000111000111000111000111000111000111) from the Walsh code generator  7100  by the input information bit a 2 . The multiplier  7430  multiplies the punctured Walsh code W 8 ′ (=000000111111000000111111000000111111000000111111) from the Walsh code generator  7100  by the input information bit a 3 . The multiplier  7440  multiplies the punctured Walsh code W 16 ′ (=0000000000001111111111000000000000111111111111) from the Walsh code generator  7100  by the input information bit a 4 . The multiplier  7450  multiplies the punctured Walsh code W 32 ′ (=000000000000000000000000111111111111111111111111) from the Walsh code generator  7100  by the input information bit multiplier  7460  multiplies the symbols of all 1&#39;s output from the 1-bit generator  7000  by the input information bit a 6 . 
   A mask generator  7200  simultaneously outputs punctured basis mask functions M 1 ′, M 2 ′ and M 4 ′ of length 48, determined by puncturing the basis masks according to a predetermined puncturing pattern. The method for creating the mask functions will not be described, since it has already been described above. The multiplier  7470  multiplies the punctured mask function M 1 ′ (=011101110111010011000011111010001011101111100001) from the mask generator  7200  by the input information bit a 7 . The multiplier  7480  multiplies the punctured mask function M 2 ′ (=100111101001110101011101011101001010111001111100) from the mask generator  7200  by the input information bit a 8 . The multiplier  7490  multiplies the punctured mask function M 4 ′ (=001000110011101100110010101111111101011001100110) from the mask generator  7200  by the input information bit a 9 . That is, the multipliers  7470 – 7490  multiply the input basis mask sequences M 1 ′, M 2 ′ and M 4 ′ by the associated input information bits a 7 –a 9  in a symbol unit. An adder  7600  then adds (or XORs) the symbols output from the multipliers  7400 – 7490  in a symbol unit, and outputs  48  coded symbols (TFCI symbols). 
     FIG. 12  illustrates a control flow for encoding a TFCI in an NB-TDD CDMA mobile communication system according to the second embodiment of the present invention. Referring to  FIG. 12 , in step  1200 , a sequence of 10 input information bits a 0 –a 9  is input and then, parameters code[ ] and j are initialized to ‘0’. Here, the coded symbol sequence parameter code[ ] indicates the 48 coded symbols finally output from the encoder and the parameter j is used to count the 48 coded symbols constituting one codeword. 
   Thereafter, it is determined in step  1210  whether the first information bit a 0  is ‘1’. If the first information bit a 0  is ‘1’, the punctured basis Walsh code W 1 ′ (=101101101001101101010010011011001101011011001001) is XORed with the coded symbol sequence parameter code[ ]. Otherwise, if the first information bit a 0  is not ‘1’, the control flow skips to step  1212 . Specifically, if the information bit a 0  is ‘1’, the parameter j is initialized to ‘0’ and a j th  symbol of the first punctured Walsh code W 1 ′ is XORed with a j th  position code[j] of the coded symbol sequence parameter. Here, since j=0, the 0 th  symbol of the first Walsh code is XORed with the 0 th  position of the coded symbol sequence parameter. Further, it is determined whether the parameter j is 47, in order to determine whether the parameter j indicates the last coded symbol. If the parameter j is not equal to 47, the parameter j is increased by 1 and then the above process is repeated. Otherwise, if the parameter j is equal to 47, the control flow proceeds to step  1212 . That is, after completion of XORing on the 48 coded symbols, the control flow proceeds to the next step. 
   After step  1210 , it is determined in step  1212  whether the second information bit a 1  is ‘1’. If the second information bit a 1  is ‘1’, the punctured basis Walsh code W 2 ′ (=011011011011011011001001001001011011001001011011) is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the second information bit a 1  is not ‘1’, the control flow skips to step  1214 . After step  1212 , it is determined in step  1214  whether the third information bit a 2  is ‘1’. If the third information bit a 2  is ‘1’, the punctured basis Walsh code W 4 ′ (000111000111000111000111000111000111000111000111) is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the third information bit a 2  is not ‘1’ , the control flow skips to step  1216 . After step  1214 , it is determined in step  1216  whether the fourth information bit a 3  is ‘1’. If the fourth information bit a 3  is ‘1’, the punctured basis Walsh code W 8 ′ (=000000111111000000111111000000111111000000111111) is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the fourth information bit a 3  is not ‘1’, the control flow skips to step  1218 . After step  1216 , it is determined in step  1218  whether the fifth information bit a 4  is ‘1’. If the fifth information bit a 4  is ‘1’, the punctured basis Walsh code W 16 ′ (=000000000000111111111111000000000000111111111111) is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the fifth information bit a 4  is not ‘1’, the control flow skips to step  1220 . After step 1218, it is determined in step  1220  whether the sixth information bit a 5  is ‘1’. If the sixth information bit a 5  is ‘1’, the punctured basis Walsh code W 32 ′ (=000000000000000000000000111111111111111111111111) is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the sixth information bit a 5  is not ‘1’, the control flow jumps to step  1222 . 
   After step  1220 , it is determined in step  1222  whether the seventh information bit a 6  is ‘1’. If the seventh information bit a 6  is ‘1’, a length-48 sequence of all 1&#39;s is XORed with the coded symbol sequence parameter code [ ]. Otherwise, if the seventh information bit a 6  is not ‘1’, the control flow jumps to step  1224 . That is, in step  1222 , the symbols of the Walsh code created in the preceding steps are inverted to create a bi-Walsh code corresponding to the Walsh code, thereby to create 128 bi-Walsh codes of length 48. 
   After step  1222 , it is determined in step  1224  whether the eighth information bit a 7  is ‘1’. If the eighth information bit a 7  is ‘1’, the basis mask function M 1 ′ (=011101110111010011000011111010001011101111100001) punctured according to a predetermined puncturing rule is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the eighth information bit a 7  is not ‘1’, the control flow skips to step  1226 . After step  1224 , it is determined in step  1226  whether the ninth information bit a 8  is ‘1’. If the ninth information bit a 8  is ‘1’, the punctured basis mask function M 2 ′ (=100111101001110101011101011101001010111001111100) is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the ninth information bit a 8  is not ‘1’, the control flow skips to step  1228 . After step  1226 , it is determined in step  1228  whether the tenth information bit a 9  is ‘1’. If the tenth information bit a 9  is ‘1’, the punctured basis mask function M 4 ′ (=001000110011101100110010101111111101011001100110) is XORed with the coded symbol sequence parameter code[ ] of length 48. Otherwise, if the tenth information bit a 9  is not ‘1’, the control flow is ended. After the process of  FIG. 12 , the coded symbols determined by XORing only the sequences corresponding to information bits  1 &#39;s out of the 10 sequences W 1 ′, W 2 ′, W 4 ′, W 8 ′, W 16 ′, W 32 ′,  1 , M 1 ′, M 2 ′ and M 4 ′ of length 32 associated respectively with the 10 input information bits a 0 –a 9  are stored in the parameter code[ ]. 
   The (48,10) encoder creates 1024 codewords by puncturing, for example, 0 th , 4 th , 8 th , 13 th , 16 th , 20 th , 27 th , 31 st , 34 th , 38 th , 41 st , 44 th , 50 th , 54 th , 57 th  and 61 st  symbols from all of the codewords (Walsh codes or mask functions) of length 64 described in the first embodiment. Therefore, the total number of the codewords is 1024. In addition, a (48,9) encoder creates 64 Walsh codes of length 64 determined by puncturing 0 th , 4 th , 8 th , 13 th , 16 th , 20 th , 27 th ,31 st , 34 th , 38 th , 41 st , 44 th , 50 th , 54 th , 57 th  and 61 st  symbols from the 64 Walsh codes of length 64, codes calculated by adding all 1&#39;s to (or multiplying −1 by, in case of a real number) symbols of all the punctured Walsh codes among the 1024 codewords, and codes determined by combining a total of 4 mask functions calculated by the combination of a total of 128 codes and 2 mask functions among the 3 punctured mask functions, and a (48,8) encoder creates 64 Walsh codes of length 48, codes calculated by adding all 1&#39;s to (or multiplying −1 by, in case of a real number) symbols of every punctured Walsh code among the 1024 codewords, and codes determined by combining a total of 2 mask functions calculated by the combination of a total of 128 codes and 1 mask function among the 3 punctured mask functions. The (48,9) encoder and the (48,8) encoder both have a minimum distance of 18. 
   The (48,9) encoder can be realized using only two of the 3 mask functions output from the mask function generator of  FIG. 7B , while the (48,8) encoder can be realized using only one of the 3 mask functions output from the mask function generator of  FIG. 7B . In addition, a (48,7) encoder can be realized using none of the 3 mask functions output from the mask function generator of  FIG. 7B . As stated above, the encoder can adaptively perform encoding according to the number of input information bits, and can also have superior performance by increasing the minimum distance determining the performance of the encoder, as high as possible. 
   Next, a description of a decoder according to the second embodiment of the present invention will be made with reference to  FIG. 9 . 
   Referring to  FIG. 9 , a received signal r(t) corresponding to a TFCI symbol of length 48 having a value of +1/−1 is commonly input to 7 multipliers  901 – 907 . The received signal r(t) is a signal encoded by a given punctured Walsh code and a given punctured mask sequence in the encoder ( FIG. 7B ) of the transmitter. A mask generator  910  creates every possible mask function which can be created by the 3 basis masks, i.e., mask functions M 1 ′–M 7 ′ of length 48 punctured according to a given puncturing rule, and provides the generated mask functions to multipliers  901 – 907 , respectively. The multiplier  901  multiplies the received signal r(t) of length 48 by the mask function M 1 ′ output from the mask generator  910 , and provides its output to a correlation calculator  921 . The multiplier  902  multiplies the received signal r(t) by the mask function M 2 ′ output from the mask generator  910 , and provides its output to a correlation calculator  922 . The multiplier  907  multiplies the received signal r(t) by the mask function M 7 ′ output from the mask generator  910 , and provides its output to a correlation calculator  927 . That is, the multipliers  901 – 907  multiply the received signal r(t) by their associated mask functions M 1 ′–M 7 ′ from the mask generator  910 , and provide their outputs to the associated correlation calculators  921 – 927 , respectively. By doing so, the received signal r(t) and the signals calculated by multiplying the received signal r(t) by the possible 7 mask functions, i.e., a total of 8 signals, are provided to the 8 correlation calculators  920 – 907 , respectively. If the transmitter has encoded the TFCI bits using a predetermined mask function, any one of the outputs from the multipliers  901 – 907  will be a mask function-removed signal. Then, the correlation calculators  920 – 927  calculate 128 correlation values by correlating the received signal r(t) and the outputs of the multipliers  901 – 907  with 128 bi-Walsh codes of length 48. The largest one of the calculated correlation values, an index of then-correlated Walsh code and an index of the correlation calculator are provided to a correlation comparator  940 . Here, the index of the correlation calculator is equivalent to an index of the mask function indicating which mask function is multiplied by the received signal, for the signal input to the correlation calculator. However, the mask index ‘0’ means that no mask is multiplied by the received signal. The correlation calculator  920  calculates correlation values by correlating the received signal r(t) with 128 biorthogonal codes of length 48. Further, the correlation calculator  920  provides the correlation comparator  940  with the largest one of the calculated correlation values, an index of then correlated Walsh code and an index ‘0’ of the correlation calculator  920 . At the same time, the correlation calculator  921  also calculates 128 correlation values by correlating the received signal r(t) multiplied by the mask function M 1 ′ by the multiplier  901  with 128 bi-Walsh codes of length 48. Further, the correlation calculator  921  provides the correlation comparator  940  with the largest one of the calculated correlation values, an index of then-calculated Walsh code and an index ‘1’ of the correlation calculator  921 . The correlation calculator  922  calculates 128 correlation values by correlating the received signal r(t) multiplied by the mask function M 2 ′ by the multiplier  902  with 128 bi-Walsh codes of length 48. Further, the correlation calculator  922  provides the correlation comparator  940  with the largest one of the 128 calculated correlation values, an index of then-calculated Walsh code and an index ‘2’ of the correlation calculator  922 . The correlation calculator  927  calculates 128 correlation values by correlating the received signal r(t) multiplied by the mask function M 7 ′ by the multiplier  907  with 128 bi-Walsh codes of length 48. Further, the correlation calculator  927  provides the correlation comparator  940  with the largest one of the calculated correlation values, an index of then-calculated Walsh code and an index ‘7’ of the correlation calculator  927 . 
   The correlation comparator  940  then compares the 8 largest correlation values provided from the correlation calculators  920 – 927 , and determines the largest one of them. After determining the largest correlation value, the correlation comparator  940  outputs TFCI information bits transmitted from the transmitter according to the index of the Walsh code provided from the correlation calculator associated with the determined correlation value and an index (or an index of a mask function multiplied by the received signal r(t)) of the same correlation calculator. 
   The correlation comparator according to the second embodiment has the same operation as that of the correlation comparator according to the first embodiment. An operation of the correlation comparator according to the second embodiment will be described below with reference to  FIG. 10 . 
   Referring to  FIG. 10 , in step  1000 , a frequency indicating index i is initialized to 1, and a maximum value, a Walsh code index and a mask index are all initialized to ‘0’. In step  1010 , the correlation value, the Walsh code index for the correlation value and the mask index, output from the first correlation calculator  920 , are stored as a first maximum value, a first Walsh code index and a first mask sequence index, respectively. Thereafter, in step  1020 , the first maximum value is compared with a previously stored maximum value. If the first maximum value is larger than the previously stored maximum value, the procedure goes to step  1030 . Otherwise, if the first maximum value is smaller than or equal to the previously stored maximum value, the procedure proceeds to step  1040 . In step  1030 , the first maximum value is designated as the maximum value, and the first Walsh code index and the first mask index are designated as the Walsh code index and the mask index, respectively. In step  1040 , a count value set for the index parameter i is compared with the number ‘8’ of the correlation calculators, in order to determine whether comparison has been completely performed on all of the 8 correlation values. If the frequency indicating index i is not equal to the number ‘8’ of the correlation calculators in step  1040 , the correlation comparator  940  increases the frequency indicating index i by 1 in step  1060  and thereafter, returns to step  1010  to repeat the above-described process using the i th  maximum value, the i th  Walsh code index and the i th  mask index, output from the increased i th  correlation calculator. After the above process is repeatedly performed on the 8 th  maximum value, the 8 th  Walsh code index and the 8 th  mask index, the frequency indicating index i becomes 8. Then, the procedure goes to step  1050 . In step  1050 , the correlation comparator  940  outputs decoded bits (TFCI bits) associated with the Walsh code index and the mask index. The Walsh code index and the mask index corresponding to the decoded bits are the Walsh code index and the mask index corresponding to the largest one of the 8 correlation values provided from the 8 correlation calculators. 
   As described above, the novel NB-TDD CDMA mobile communication system according to the present invention can efficiently encode and decode the TFCI, so as to increase error correcting capability. 
   While the invention has been shown and described with reference to a certain preferred embodiment thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.