Abstract:
A code vector comprising 2N data elements is generated in response to N input data elements. The first N data elements of the code vector represent a replica of the N input data elements, and the second N data elements of the code vector comprise N parity data elements. The combination of the first N data elements and the second N data elements forms a Kerdock code vector. The Kerdock code vector is decoded to recover the N input data elements by selecting one of a possible 2 N  Kerdock code vectors as a best estimate of the Kerdock code vector. The selected Kerdock vector comprises 2N data elements, and only the first N data elements of the selected Kerdock code vector are used as the N input data elements.

Description:
RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 60/324,096 filed on Sep. 24, 2001. 
    
    
     TECHNICAL FIELD OF THE INVENTION 
     The present invention relates to the coding and decoding of data, such as map data that indicate the distribution of first and second data segments in a transmitted data field. 
     BACKGROUND OF THE INVENTION 
     The ATSC digital television standard presently provides for the transmission of successive data fields each comprising 313 segments extending over a 24.2 ms time interval. FIG. 1 discloses an exemplary format for a data field according to this standard. The first segment of each field is a field sync segment. The field sync segment is composed of four two-level segment sync symbols and space for 828 other two-level symbols. A portion of this space is used for a field sync, and another portion of this field is reserved. Each of the remaining segments of each field comprises four two-level segment sync symbols and 828 n-level data symbols where n is currently eight, although n could be other integers such as two, four, sixteen, etc. 
     As indicated by U.S. patent application Ser. No. 09/804,262 filed on Mar. 13, 2001, there is presently some interest in extending the ATSC digital television standard to allow a field to contain a mix of more robustly coded data (referred to herein as E-VSB data) and the data currently provided for in the standard (referred to herein as VSB data). Preferably, the data mix is employed on a segment-by-segment basis such that some segments of a field are used to transmit VSB data exclusively and the remaining segments of the field are used to transmit E-VSB segments exclusively. However, it is possible that all data segments of a field could contain either E-VSB data segments exclusively or VSB data segments exclusively. Moreover, it is also possible that the E-VSB data contained in some segments of a field may be coded with one robust coding rate and that the E-VSB data in other segments of the field may be coded at other robust coding rates. 
     As disclosed in the above mentioned &#39;262 application, a map that indicates which segments contain the more robust (E-VSB) data and which segments contain standard VSB data is preferably provided by the transmitter to the receiver so that the receiver can properly decode and otherwise process the received VSB and E-VSB data. Assuming that a field contains E-VSB data at different coding rates, the map in that case must also designate the coding rates that apply to the differently coded E-VSB data segments. 
     The &#39;262 application describes one mapping system. Co-pending U.S. patent application Ser. No. (7024) filed on describes another mapping system that reliably identifies which segments contain first data (such as VSB data) and which segments contain second data (such as E-VSB data). The present invention is directed to the coding and decoding of data, such as the map data that identifies which segments contain the first data and which segments contain the second data. 
     SUMMARY OF THE INVENTION 
     In accordance with one aspect of the present invention, a method of generating a code vector comprises the following: providing N input data elements; and, generating a code vector comprising 2N data elements in response to the N input data elements, the first N data elements of the code vector representing a replica of the N input data elements, and the second N data elements of the code vector comprising N parity data elements. The combination of the first N data elements and the second N data elements form a Kerdock code vector. 
     In accordance with another aspect of the present invention, a method is provided to decode a received Kerdock code vector in order to produce an output data signal. The received Kerdock code vector comprises 2N data elements. The method comprises the following: selecting one of a possible 2 N  Kerdock code vectors as a best estimate of the received Kerdock code vector, wherein the selected Kerdock vector comprises 2N data elements; and, using only the first N data elements of the selected Kerdock code vector as the output data signal. 
     In accordance with yet another aspect of the present invention, a method of decoding a systematic code vector to produce a decoded data signal comprises the following: receiving the systematic code vector; re-arranging the data elements in the received systematic code vector to form a non-systematic code vector; and, decoding the non-systematic code vector using a non-systematic decoder to produce the decoded data signal. 
     In accordance with still another aspect of the present invention, a method of decoding a code vector comprises the following: receiving an N element map data unit encoded in the form of a 2N element systematic Kerdock code vector, wherein the map data unit represents a distribution of first and second data segments in a received field; decoding the 2N element systematic Kerdock code vector by selecting one of a possible 2 N  Kerdock code vectors as a best estimate of the 2N element systematic Kerdock code vector, wherein the selected Kerdock code vector comprises 2N elements; and, using the first N elements of the selected Kerdock code vector as the N element map data unit. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     These and other features and advantages of the present invention will become more apparent from a detailed consideration of the invention when taken in conjunction with the drawings in which: 
     FIG. 1 is a diagram illustrating an exemplary format of a field as defined in the ATSC digital television standard; 
     FIG. 2 is a schematic diagram of a map insertion system that inserts a map into fields to be transmitted; 
     FIG. 3 shows an example of a convolutional interleaver that can be used in the map insertion system shown in FIG. 2; 
     FIG. 4 is a schematic diagram of a portion of a receiver involved in de-formatting a received field based upon a received map; 
     FIG. 5 shows an example of a convolutional de-interleaver that can be used in the receiver portion shown in FIG. 4; 
     FIG. 6 shows a first embodiment of a Kerdock encoder that can be used in the map insertion system shown in FIG. 2; 
     FIG. 7 shows a second embodiment of a Kerdock encoder that can be used in the map insertion system shown in FIG. 2; 
     FIG. 8 shows a first embodiment of a Kerdock decoder that can be used in the receiver portion shown in FIG. 4; 
     FIG. 9 shows a second embodiment of a Kerdock decoder that can be used in the receiver portion shown in FIG. 4; 
     FIG. 10 shows a third embodiment of a Kerdock decoder that can be used in the receiver portion shown in FIG. 4; and, 
     FIGS. 11A and 11B are flow charts illustrating the operation of the receiver portion shown in FIG.  4 . 
    
    
     DETAILED DESCRIPTION 
     In order to indicate which segments of a field contain VSB data and which segments of a field contain E-VSB data, a twelve bit map data unit (mdu) is defined for each data field. Accordingly, the map data unit is capable of designating one of 4096 possible combinations of VSB and E-VSB data segments for a respective field. The map data unit for an odd ATSC transmitted field may be denoted as {A 0  B 0  C 0 }, and the map data unit for the next succeeding even ATSC transmitted field may be denoted as {A e  B e  C e }, where A 0 , B 0 , C 0 , A e , B e , and C e  each comprises four bits and is referred to herein as a map data sub-unit. Thus, each map data unit comprises twelve bits and two map data units for successive odd and even fields comprise twenty-four bits. 
     As shown in FIG. 2, a controller  10  generates a sequence of map data units for application to a Kerdock encoder  12 . Kerdock encoders that may be used for the Kerdock encoder  12  are disclosed below. Eight bits at a time are supplied to the Kerdock encoder  12 . Thus, the first eight bits supplied to the Kerdock encoder  12  correspond to map data sub-units A 0  B 0 , the next eight bits supplied to the Kerdock encoder  12  correspond to map data sub-units C 0  A e , and the next eight bits supplied to the Kerdock encoder  12  correspond to map data sub-units B e  C e . The map data units for succeeding fields are applied to the Kerdock encoder  12  in a like fashion. 
     For each eight bit input, the Kerdock encoder  12  produces a sixteen bit code word or vector that consists of the eight input bits and eight parity bits P x . Accordingly, for input map data sub-units A 0  B 0 , the output of the Kerdock encoder  12  is a code word or vector {A 0  B 0  P 1 }; for map data sub-units C 0  A e , the output of the Kerdock encoder  12  is {C 0  A e  P 2 }; and, for map data sub-units B e  C e , the output of the Kerdock encoder  12  is {B e  C e P 3 }. Thus, three map data sub-units covering successive odd and even fields n and n+1 are thereby encoded into three sixteen bit output vectors containing forty-eight bits in all. 
     The code vectors that are produced by the Kerdock encoder  12  are processed by a convolutional interleaver  14  in order to provide protection from burst noise. Convolutional interleavers and de-interleavers are described in the ATSC digital television standard. An interleaver that may be used for the convolutional interleaver  14  is disclosed in U.S. Pat. No. 5,572,532. The convolutional interleaver  14  is preferably characterized by the parameters N=48, B=16 and M=3, where N is M times the block size (16 data elements) corresponding to three encoded map vectors produced by the Kerdock encoder  12 , B is the interleave depth, and M is the delay unit size of the interleaver. Thus, the convolutional interleaver  14  delays the individual bits of the forty-eight bits of each block of three code vectors by 0, 3, 6, . . . , 45 bits at the output of the convolutional interleaver  14 . 
     The convolutional interleaver  14  is preferably synchronized to the ATSC field sync signal that is generated by the controller  10  so that the successive delays on the input bits are reset at the end of each field. Accordingly, each field begins with zero delay. As will be explained in further detail hereinafter, each set of forty-eight interleaved bits of the blocks of three code vectors are duplicated for transmission as two level symbols in the reserved portion of two consecutive field sync segments. It will be appreciated that this Kerdock coding and duplication results in an effective coding rate of ¼ because the map bits are doubled in number by the Kerdock encoder  12  and are doubled in number again by the duplication, so that twenty-four bits representing two map data units are coded into ninety-six bits in two field sync segments. 
     It will also be appreciated that, considering the corresponding de-interleaver in the receiver, a latency interval L must be accounted for when associating the map data units with the corresponding fields. The latency interval of the interleaver/de-interleaver combination is given by the expression L=N×(B−1). In the specific example of the convolutional interleaver  14  given above, N=48 and B=16. Therefore, the latency interval of the interleaver/de-interleaver combination according to this example is L=48×15=720 bits or 15 (720/48) fields. If two additional fields are allowed for processing time, the system may be characterized by the following relationships: 
     
       
         Coded mdu for field  n: A   0   B   0   P   1   C   0   A   E   P   2   B   E   C   E   P   3   
       
     
     
       
         Coded mdu for field  n +1 : A   0   B   0   P   1   C   0   A   E   P   2   B   E   C   E   P 3 
       
     
     where mdu A 0 B 0 C 0  identifies the mix of VSB and E-VSB segments for field n+2+L and where mdu A E B E C E  identifies the mix of VSB and E-VSB segments for field n+3+L. 
     A VSB data source  16  provides VSB data and an E-VSB data source  18  provides E-VSB data. One result of the Kerdock encoding applied by the Kerdock encoder  12  is that the mdus are more robustly encoded than are the VSB data and the E-VSB data. The controller  10  controls the VSB data source  16  and the E-VSB data source  18  so as to control the mix of VSB and E-VSB data segments in a particular field. Because of the system latency interval, the map data unit, which notifies the receiver of this mix and which is encoded by the Kerdock encoder  12 , is transmitted beginning in a field that is transmitted 17 or 18 fields earlier than the field containing that mix and ends in a field that is transmitted 1 or 2 fields earlier that the field containing that mix. That is, the map data unit supplied by the controller  10  to the Kerdock encoder  12  during formatting of the current field corresponds to VSB and/or E-VSB data to be transmitted 17 or 18 fields later. However, because of the interleaving performed by the convolutional interleaver  38 , this map data unit is spread over 15 fields. 
     The data segments supplied by the VSB data source  16  and the E-VSB data source  18 , together with the encoded and interleaved map data unit bits from the convolutional interleaver  14 , are applied to a data field formatter  20 . The data field formatter  20  is synchronized to the field sync signal from the controller  10  and formats the transmitted field so that the forty-eight encoded and interleaved map data unit bits are inserted into the reserved portion of two successive field sync segments. The VSB data source  16  and the E-VSB data source  18  are controlled by the controller  10  so that the VSB and E-VSB data segments supplied by the VSB data source  16  and the E-VSB data source  18  to the data field formatter  20  correspond to a map data unit transmitted beginning n+2+L or n+3+L fields prior thereto. The data field formatter  20  is synchronized so that these VSB and E-VSB data segments are appropriately multiplexed throughout the current field in accordance with that previously transmitted map data unit. 
     Finally, the formatted fields are successively applied to a standard ATSC modulator and transmitter  22  for transmission. 
     An example of the convolutional interleaver  14  is shown in FIG.  3  and includes sixteen paths coupled between an input  24  and an output  26  by corresponding synchronized switching functions  28  and  29 . As shown in FIG. 2, the input  24  is coupled to the Kerdock encoder  12  and the output  26  is coupled to the data field formatter  20 . The switching functions  28  and  29  synchronously step through the sixteen paths on a data element-by-data element basis so that one data element received on the input  24  is coupled through the first path to the output  26 , so that the next data element received on the input  24  is coupled through the second path to the output  26 , and so on. 
     The first path of the convolutional interleaver  14  imposes no delay on the data elements passing therethrough, the second path of the convolutional interleaver  14  imposes a three element delay on the data elements passing therethrough, the third path of the convolutional interleaver  14  imposes a six element delay on the data elements passing therethrough, . . . , and the sixteenth path of the convolutional interleaver  14  imposes a forty-five element delay on the data elements passing therethrough. 
     As shown in FIG. 4, the signal transmitted by the ATSC modulator and transmitter  22  is received by a receiver comprising a tuner  30 . The IF output of the tuner  30  is demodulated by an ATSC demodulator  32  in order to provide an analog baseband output representing the transmitted symbols. This analog signal is sampled by an A/D converter  34  under control of a digital processor  38  to convert the demodulated symbols into corresponding multibit digital values. The encoded and interleaved map data unit symbols, which are duplicated in successive fields as discussed above, are applied to a convolutional de-interleaver  40 . The remaining symbols are directly applied to the digital processor  38 , which converts these remaining symbols to corresponding bits, arranged in data bytes, for application to a segment de-formatter  42 . The segment de-formatter  42  receives a de-interleaved and decoded map data unit from a Kerdock decoder  44 . The segment de-formatter  42  responds to this de-interleaved and decoded map data unit by passing the VSB segments in the field to a VSB processor  46  and by passing the E-VSB segments in the field to an E-VSB processor  48 . The VSB processor  46  and the E-VSB processor  48  decode and otherwise process the respective VSB data and E-VSB data from the segment de-formatter  42 . 
     As an example, the VSB processor  46  may perform Reed-Solomon decoding and, in the case where the VSB data has been trellis encoded in the transmitter, the VSB processor  46  may also perform Viterbi decoding. The E-VSB processor  48 , for example, may perform the same decoding as the VSB processor  46  and, in addition, perform the additional decoding corresponding to the additional coding that was performed in the transmitter in order to add robustness to the data. Moreover, the VSB processor  46  and the E-VSB processor  48  may perform de-interleaving and de-randomization. 
     The interleaved map data unit symbols from the A/D converter  34  are applied to the convolutional de-interleaver  40  which de-interleaves the map data unit symbols in inverse fashion relative to the convolutional interleaver  14  in order to provide the vectors produced by the Kerdock encoder  12 . The de-interleaved vectors corresponding to a map data unit and to its corresponding duplicate map data unit are averaged on a bit-by-bit basis by an averaging circuit  50  in order to improve the reliability of the map data units. The de-interleaved and averaged vectors are decoded by the Kerdock decoder  44  in order to recover the map data units that control the segment de-formatter  42 . Since the mdus were encoded more robustly than either the VSB data or the E-VSB data, the mdus will be recovered in the receiver with less errors than the data. 
     As explained previously, the latency interval of the interleave/de-interleave process is accommodated in the system because the map data units provided by the controller  10  define the mix of VSB and E-VSB data segments that are to be transmitted L fields later in time. Exemplary embodiments of the Kerdock decoder  44  and the convolutional de-interleaver  40  are disclosed in the previously referenced U.S. Pat. Nos. 6,226,318 B1 and 5,572,532 respectively. 
     As discussed below in more detail, the Kerdock decoder  44  may be arranged to provide an estimation of the reliability of the decoding process. In terms of the map data unit specifically, the Kerdock decoder  44  may be arranged to provide an estimation of the reliability of the decoding of the map data unit. If this reliability indicates that the decoded map data unit is not reliable, the immediately previous map data unit that was reliably decoded is used to de-format the field instead of the currently decoded map data unit. This operation is justified by assuming that the mix between VSB data and E-VSB data changes from field to field at a relatively slow rate so that the substitute map data unit will likely define the appropriate segment mix. 
     An example of the convolutional de-interleaver  40  is shown in FIG.  5  and includes sixteen paths coupled between an input  60  and an output  62  by corresponding synchronized switching functions  64  and  66 . As shown in FIG. 4, the input  60  is coupled to the A/D converter  34  and the output  62  is coupled to the averaging circuit  50 . The switching functions  64  and  66  synchronously step through the sixteen paths on a data element-by-data element basis so that one data element received on the input  60  is coupled through the first path to the output  62 , so that the next data element received on the input  60  is coupled through the second path to the output  62 , and so on. 
     The first path of the convolutional de-interleaver  40  imposes a forty-five element delay on the data elements passing therethrough, the second path of the convolutional interleaver  14  imposes a forty-two delay on the data elements passing therethrough, . . . , the fourteenth path of the convolutional interleaver  14  imposes a six element delay on the data elements passing therethrough, the fifteenth path of the convolutional interleaver  14  imposes a three element delay on the data elements passing therethrough, and the sixteen path of the convolutional interleaver  14  imposes no delay on the data elements passing therethrough. 
     A systematic Kerdock encoder  70  is shown in FIG.  6  and may be used for the Kerdock encoder  12 . The systematic Kerdock encoder  70  accepts an input having N data elements, such as the map data sub-units A 0  and B 0  having a total of eight bits, and outputs a corresponding code word having 2N data elements, such as the code word {A 0  B 0  P 1 } having a total of sixteen bits, by appending N parity bits, such as the eight parity bits P 1 , to the end of the N input data elements, such as the map data sub-units A 0  and B 0  having eight bits. The N parity data elements are read out of a look-up table  72  based on the N input data elements. 
     In the case where each map data sub-unit is four bits so that two map data sub-units are eight bits, the look-up table  72  stores 256 sets of parity bits where each set contains eight bits. Appendix A shows exemplary data for the look-up table  72 . Each bit in this exemplary data has one of only two values, 1 or −1. In describing the relationship between the input bits and the sets of parity bits stored in the look-up table, it is useful to think of a bit having a value of −1 as a bit having a value of 0. The eight bits that are provided as an input to the systematic Kerdock encoder  70  are used as an address into the look-up table  72 . 
     The data stored in the look-up table  72  are arranged so that, when a set of eight parity bits is read out according to eight input bits and is appended to the eight input bits, a Kerdock code word is formed. A Kerdock code word has a minimum distance of six from any other Kerdock code word. Distance is a measure of how many corresponding bits differ between two code words. 
     The relationship between the input bits and the bits stored in the look-up table  72  fosters the creation of the Kerdock code words that are output by the systematic Kerdock encoder  70 . This relationship is as follows: the input bits having a value of −1 −1 −1 −1 −1 −1 −1 −1 (i.e., the address 0) are used to address the first row of Appendix A; the input bits having a value of −1 −1 −1 −1 −1 −1 −1 1 (i.e., the address 1) are used to address the second row of Appendix A; the input bits having a value of −1 −1 −1 −1 −1 −1 1 −1 (i.e., the address 2) are used to address the third row of Appendix A; the input bits having a value of −1 −1 −1 −1 −1 −1 1 1 (i.e., the address 3) are used to address the fourth row of Appendix A; and so on. 
     As an example, when the input −1 −1 −1 −1 −1 −1 −1 −1 is received by the systematic Kerdock encoder  70 , the first row of Appendix A is read out from the look-up table  72  and is appended to this input to form the Kerdock code word −1 −1 −1 −1 −1 −1 −1 −1 −1 1 −1 1 1 −1 1 −1. As another example, when the input −1 −1 −1 −1 −1 −1 −1 1 is received by the systematic Kerdock encoder  70 , the second row of Appendix A is read out from the look-up table  72  and is appended to this input to form the Kerdock code word −1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 −1 1 1 −1 −1. It is noted that these two Kerdock code words have a distance of six from each other because the eighth, ninth, tenth, twelfth, fourteenth, and fifteenth bits are different between the two Kerdock code words. 
     Alternatively, a systematic Kerdock encoder  74  is shown in FIG.  7  and may be used for the Kerdock encoder  12 . The systematic Kerdock encoder  74  accepts an input having N data elements, such as the map data sub-units A 0  and B 0  having a total of eight bits, and outputs a corresponding code word having 2N data elements, such as the code word {A 0  B 0  P 1 } having a total of sixteen bits, by reading out the 2N data element code word from a look-up table  76 . 
     In the case where each map data sub-unit is four bits so that two map data sub-units are eight bits, the look-up table  76  stores 256 code words where each code word contains sixteen bits. Appendix B shows exemplary data for the look-up table  76 . As in the case of Appendix A, each bit in this exemplary data has one of only two values, 1 or −1. The eight bits that are provided as an input to the systematic Kerdock encoder  74  are used as an address into the look-up table  76  and correspond to the first eight bits of a row in the data shown in Appendix B. The row of Appendix B that is addressed by a set of eight input bits is the row in which the first eight bits match the eight input bits. Each code word stored in the look-up table  76  is a Kerdock code word because each code word stored in the look-up table  76  has a minimum distance of six from any other Kerdock code word stored in the look-up table  76 . 
     As an example, when the input −1 −1 −1 −1 −1 −1 −1 −1 is received by the systematic Kerdock encoder  74 , a row  78  of Appendix A is read out from the look-up table  76 . The row  78  contains the following bits: −1 −1 −1 −1 −1 −1 1 −1 1 −1 1 −1 1 1 −1 1 −1. As another example, when the input −1 −1 −1 −1 −1−1 −1 1 is received by the systematic Kerdock encoder  74 , a row  80  of Appendix A is read out from the look-up table  76 . The row  80  contains the following bits: −1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 −1 1 1 −1 −1. It is noted that these two Kerdock code words have a distance of six from each other because the eighth, ninth, tenth, twelfth, fourteenth, and fifteenth bits are different between the two Kerdock code words. 
     A systematic Kerdock decoder  82  is shown in FIG.  8  and may be used for the Kerdock decoder  44 . The systematic Kerdock decoder  82  accepts an input having 2N data elements, such as eight bits corresponding to the map data sub-units A 0  and B 0  and 8 bits corresponding to the parity bits, and outputs a vector of N data elements, such as the map data sub-units A 0  and B 0  having eight bits. 
     More specifically, a correlator  84  correlates the 2N input data elements with each of 256 Kerdock code words stored in a look-up table  86 , where each Kerdock code word comprises sixteen data elements such as bits. Appendix B shows exemplary data for the look-up table  86 . The correlation implemented by the correlator  84 , for example, may be a cross product of the input 2N data elements and each of the Kerdock code words stored in the look-up table  86 . 
     Thus, the first data element of the 2N input data elements is multiplied by the first data element of a first Kerdock code word stored in the look-up table  86  to form a first product, the second data element of the 2N input data elements is multiplied by the second data element of the first Kerdock code word stored in the look-up table  86  to form a second product, . . . , and the sixteenth data element of the 2N input data elements is multiplied by the sixteenth data element of the first Kerdock code word stored in the look-up table  86  to form a sixteenth product. The resulting sixteen products are added to form a first correlation between the 2N input data elements and the first Kerdock code word stored in the look-up table  86 . This process is repeated for each of the other 255 Kerdock code words stored in the look-up table  86 . 
     An identifier  88  identifies the Kerdock code word from the look-up table  86  that produced the largest correlation and outputs the first eight data elements of this Kerdock code word as the eight data elements making up two map data sub-units of a map data unit to be applied to the segment de-formatter  42 . The identifier  88  may also form the difference between the largest correlation and the next largest correlation as a reliability factor that indicates the reliability with which the 2N input data elements have been decoded. 
     Alternatively, a systematic Kerdock decoder  90  is shown in FIG.  9  and may be used for the Kerdock decoder  44 . The systematic Kerdock decoder  90  accepts an input having 2N data elements, such as the map data sub-units A 0  and B 0  having eight bits and the corresponding eight parity bits P 1 , and outputs a vector having N data elements, such as the map data sub-units A 0  and B 0  having eight bits. 
     More specifically, a correlator  92  correlates the 2N input data elements with each of 256 Kerdock code words that are supplied to the correlator  92  from a sequence generator  94  and a look-up table  96 . Each Kerdock code word supplied to the correlator  92  from the sequence generator  94  and the look-up table  96  comprises sixteen data elements such as bits. The correlation implemented by the correlator  92 , for example, may be the same correlation as implemented by the correlator  84 . 
     The first eight bits of the first Kerdock code word supplied to the correlator  92  comprises a first sequence of eight bits generated by the sequence generator  94 . For example, this first sequence may be −1 −1 −1 −1 −1 −1 −1 −1 (i.e., 0). The second eight bits of the first Kerdock code word supplied to the correlator  92  comprise eight bits read out of the look-up table  96  based on an address corresponding to the eight bits generated by the sequence generator  94 . These two sets of eight bits are appended together and are supplied to the correlator  92 . 
     Appendix A shows exemplary data for the look-up table  96 . The relationship between the input bits from the sequence generator  94  and the bits stored in the look-up table  96  may be the same as that used by the systematic Kerdock encoder  70 . Accordingly, the input bits having a value of −1 −1 −1 −1 −1 −1 −1 −1 (i.e., the address 0) are used to address the first row of Appendix A, the input bits having a value of −1 −1 −1 −1 −1 −1 −1 1 (i.e., the address 1) are used to address the second row of Appendix A, and so on. 
     The correlator  92  produces a first correlation based upon the input 2N bits and the first Kerdock code word produced by the sequence generator  94  and the look-up table  96 . 
     The first eight bits of the second Kerdock code word supplied to the correlator  92  comprises a second sequence of eight bits generated by the sequence generator  94 . For example, this second sequence may be −1 −1 −1 −1 −1 −1 −1 1 (i.e., 1). The second eight bits of the second Kerdock code word supplied to the correlator  92  comprise eight bits read out of the look-up table  96  based on an address corresponding to the eight bits generated by the sequence generator  94 . These two sets of eight bits are appended together and are supplied to the correlator  92 . 
     The correlator  92  produces a second correlation based upon the input 2N bits and the second Kerdock code word produced by the sequence generator  94  and the look-up table  96 , and so on. 
     An identifier  98  identifies the Kerdock code word from the sequence generator  94  and the look-up table  96  that produced the largest correlation and outputs the first eight data elements of this Kerdock code word as the eight data elements making up two map data sub-units of a map data unit to be applied to the segment de-formatter  42 . The identifier  98  may also form the difference between the largest correlation and the next largest correlation as a reliability factor indicating the reliability with which the 2N input data elements have been decoded. 
     As a further alternative, a systematic decoder  100  shown in FIG. 10 may be used for the Kerdock decoder  44 . The systematic decoder  100  is a modified form of the non-systematic decoder disclosed in U.S. Pat. No. 6,226,318 B1. The systematic decoder  100  includes two column rotators  102  and  104 , and eight vector multipliers  106 ,  108 ,  110 ,  112 ,  114 ,  116 ,  118 , and  120 . The modification involves the addition of the two column rotators to the non-systematic decoder disclosed in U.S. Pat. No. 6,226,318 B1. 
     Appendix C shows the coset leaders that are applied to first inputs of the multipliers  106 ,  108 ,  110 ,  112 ,  114 ,  116 ,  118 , and  120 . Accordingly, the coset leader in the first row of Appendix C is applied to the first input of the multiplier  106 , the coset leader in the second row of Appendix C is applied to the first input of the multiplier  108 , the coset leader in the third row of Appendix C is applied to the first input of the multiplier  110 , . . . , and the coset leader in the eighth row of Appendix C is applied to the first input of the multiplier  120 . 
     The input sixteen data elements to be decoded are re-arranged (such as rotated) according to the first column of the following table and this rotated input is applied to each of the second inputs of the multipliers  106 ,  108 ,  110 , and  112 . The input to be decoded is rotated according to the second column of the following table and this rotated input is applied to each of the second inputs of the multipliers  114 ,  116 ,  118 , and  120 . 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 
               
               
                   
                   
               
             
             
               
                   
                 4″1 
                 4″1 
               
               
                   
                 15″2 
                 15″2 
               
               
                   
                 14″3 
                 14″3 
               
               
                   
                 9″4 
                 9″4 
               
               
                   
                 5″5 
                 6″5 
               
               
                   
                 12″6 
                 11″6 
               
               
                   
                 7″7 
                 8″7 
               
               
                   
                 2″8 
                 1″8 
               
               
                   
                 3″9 
                 3″9 
               
               
                   
                 16″10 
                 16″10 
               
               
                   
                 13″11 
                 13″11 
               
               
                   
                 10″12 
                 10″12 
               
               
                   
                 6″13 
                 5″13 
               
               
                   
                 11″14 
                 12″14 
               
               
                   
                 8″15 
                 7″15 
               
               
                   
                 1″16 
                 2″16 
               
               
                   
                   
               
             
          
         
       
     
     According to the first column of the above table, the fourth input data element is moved to the first data element position of the output to be supplied to the second inputs of the multipliers  106 ,  108 ,  110 , and  112 , the fifteenth input data element is moved to the second data element position of the output to be supplied to the second inputs of the multipliers  106 ,  108 ,  110 , and  112 , the fourteenth input data element is moved to the third data element position of the output to be supplied to the second inputs of the multipliers  106 ,  108 ,  110 , and  112 , . . . , and the first input data element is moved to the sixteenth data element position of the output to be supplied to the second inputs of the multipliers  106 ,  108 ,  110 , and  112 . Accordingly, the sixteen data element input is rotated by the column rotator  102  to form a sixteen data element output to be supplied to the second inputs of the multipliers  106 ,  108 ,  110 , and  112 . 
     Similarly, the second column of the above table shows the rotation imposed by the column rotator  104  on the sixteen input data elements to form a sixteen data element output to be supplied to the second inputs of the multipliers  114 ,  116 ,  118 , and  120 . The column rotators  102  and  104  in effect convert a systematic code vector to a non-systematic code vector. 
     The outputs of the multipliers  106 ,  108 ,  110 ,  112 ,  114 ,  116 ,  118 , and  120  are processed by a corresponding one of 16×16 Hadamard transforms  119   1 - 119   8  to produce corresponding spectra that are analyzed by a spectral analyzer  122 . The spectral analyzer  122  determines which spectra contains the largest coefficient and decodes the largest coefficient to produce the corresponding Kerdock code word. The first eight bits of this Kerdock code word are supplied by the spectral analyzer  122  as the eight data elements making up two map data sub-units of a map data unit to be applied to the segment de-formatter  42 . The spectral analyzer  122  may also form the difference between the largest coefficient and the next largest coefficient as a reliability factor indicating the reliability with which the 2N input data elements have been decoded. 
     The receiving arrangement shown in FIG. 4 performs the functions illustrated by the flow chart of FIGS. 11A and 11B. A block  200  receives a field, and a block  202  parses the field in order to recover the map data unit symbols. A block  204  de-interleaves the map data unit symbols, and a block  206  stores the de-interleaved map data unit symbols in a memory. When a full map data unit has been de-interleaved as determined by a block  208 , a block  210  determines whether this map data unit corresponds to an odd field or an even field. If the block  210  determines that this map data unit corresponds to an odd field, a block  212  simply stores the map data unit awaiting de-interleaving and decoding of the duplicate of this data map unit because data map units transmitted in odd fields are duplicated in even fields. After the non-duplicate map data unit is stored by the block  212 , flow returns to the block  200 . 
     If the block  210  determines that this map data unit corresponds to an even field, the recovered map data unit is a duplicate of the map data unit previously de-interleaved and decoded. Accordingly, a block  214  averages the current map data unit and the previous map data unit. A block  216  decodes the map data unit average, and a block  218  computes a reliability factor for the map data unit average. A block  220  stores the average map and the corresponding reliability factor. 
     A block  222  determines whether the reliability factor of a decoded map data unit corresponding to the field received at the block  200  indicates that the decoded map data unit has been reliably decoded. If the reliability factor indicates reliable decoding, a block  224  de-formats the field corresponding to the appropriate map data unit and sends the VSB data and/or the E-VSB data to the VSB processor  46  and/or the E-VSB processor  48 , as appropriate, in accordance with the de-formatting. 
     On the other hand, if the reliability factor indicates that the decoding was not reliable as determined at the block  222 , a block  226  retrieves the immediately previous map data unit that was reliably decoded, and a block  228  de-formats the field in accordance with the retrieved immediately previous map data unit and sends the VSB data and/or the E-VSB data to the VSB processor  46  and/or the E-VSB processor  48 , as appropriate, in accordance with the de-formatting. 
     
       
         
               
             
               
               
             
           
               
                   
               
               
                 APPENDIX A 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 { −1, 1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, −1, 1, },  
               
               
                   
                 { −1, −1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 {−1, 1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, −1, 1, }, 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
           
               
                   
               
               
                 APPENDIX B 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 { 1, −1, −1, −1, 1, 1, 1, 1, 1, 1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, 1, 1, −1, −1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, −1, −1, −1, −1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, 1, −1, −1, 1, 1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, −1, −1, 1, 1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, −1, −1, −1, −1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, 1, 1, −1, −1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, 1, −1, 1, −1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, 1, −1, −1, 1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, −1, 1, −1, 1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, −1, 1, 1, −1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, −1, 1, 1, −1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, −1, 1, −1, 1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, 1, −1, −1, 1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, 1, −1, 1, −1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, 1, 1, −1, 1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, 1, 1, 1, −1, −1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, −1, −1, 1, −1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, −1, −1, −1, 1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, −1, −1, −1, −1, 1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, −1, −1, 1, −1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, 1, 1, 1, −1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, −1, 1, 1, −1, 1, 1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, 1, −1, −1, −1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, 1, −1, 1, 1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, −1, 1, 1, 1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, −1, 1, −1, −1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, −1, 1, −1, −1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, −1, 1, 1, 1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, 1, −1, 1, 1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, 1, −1, −1, −1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, −1, 1, 1, −1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, −1, 1, −1, 1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, 1, 1, −1, −1, 1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, 1, 1, −1, 1, −1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, −1, 1, −1, 1, −1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, 1, −1, −1, 1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, −1, −1, 1, −1, 1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, −1, −1, 1, 1, −1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, −1, −1, 1, 1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, −1, −1, −1, −1, −1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, −1, 1, 1, −1, −1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, −1, 1, 1, 1, 1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, 1, 1, 1, 1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, 1, 1, −1, −1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, 1, −1, −1, −1, −1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, 1, −1, −1, 1, 1, 1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, −1, 1, 1, 1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, 1, −1, 1, −1, −1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, 1, 1, −1, −1, −1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, 1, 1, −1, 1, 1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, 1, −1, 1, 1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, 1, −1, −1, −1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, −1, −1, 1, −1, −1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, −1, −1, 1, 1, 1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, −1, −1, 1, −1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, −1, −1, −1, 1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, −1, 1, 1, −1, 1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, −1, 1, 1, 1, −1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, 1, 1, 1, −1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, 1, 1, −1, 1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, 1, −1, −1, −1, 1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, 1, −1, −1, 1, −1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, 1, −1, −1, −1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, 1, −1, 1, 1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, −1, 1, 1, 1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, −1, 1, −1, −1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, −1, 1, −1, −1, −1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, −1, 1, 1, 1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, 1, −1, 1, 1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, 1, −1, −1, −1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, −1, −1, −1, 1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, −1, −1, 1, −1, −1, −1, 1, 1, −1, 1, } 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, 1, 1, 1, −1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, 1, 1, −1, 1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, 1, 1, −1, 1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, 1, 1, 1, −1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, −1, −1, 1, −1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, −1, −1, −1, 1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, 1, −1, 1, −1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, 1, −1, −1, 1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, −1, 1, −1, 1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, −1, 1, 1, −1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, −1, 1, 1, −1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, −1, 1, −1, 1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, 1, −1, −1, 1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, 1, −1, 1, −1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, −1, −1, 1, 1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, −1, −1, −1, −1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, 1, 1, −1, −1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, 1, 1, 1, 1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, 1, 1, 1, 1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, 1, 1, −1, −1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, −1, −1, −1, −1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, −1, −1, 1, 1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, 1, 1, −1, 1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, 1, 1, 1, −1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, −1, 1, −1, −1, 1, −1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, −1, 1, −1, −1, −1, 1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, −1, −1, −1, 1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, −1, −1, 1, −1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, −1, 1, −1, 1, 1, 1, −1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, −1, 1, −1, 1, 1, −1, 1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, −1, 1, −1, −1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, −1, 1, 1, 1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, 1, 1, 1, −1, 1, 1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, 1, 1, 1, −1, −1, −1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, 1, −1, −1, −1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, 1, −1, 1, 1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, −1, −1, −1, −1, 1, 1, 1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, −1, −1, −1, −1, 1, −1, −1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, 1, 1, −1, −1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, 1, 1, 1, 1, −1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, −1, 1, −1, −1, 1, 1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, −1, 1, −1, −1, −1, −1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, −1, −1, −1, −1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, −1, −1, 1, 1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, −1, 1, −1, 1, 1, 1, 1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, −1, 1, −1, 1, 1, −1, −1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, −1, 1, −1, 1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, −1, 1, 1, −1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, 1, 1, 1, −1, 1, −1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, 1, 1, 1, −1, −1, 1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, 1, −1, −1, 1, −1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, 1, −1, 1, −1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, −1, −1, −1, −1, 1, 1, −1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, −1, −1, −1, −1, 1, −1, 1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, −1, −1, −1, −1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, −1, −1, 1, 1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, 1, 1, 1, 1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, 1, 1, −1, −1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, 1, 1, −1, −1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, 1, 1, 1, 1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, −1, −1, 1, 1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, −1, −1, −1, −1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, −1, 1, −1, 1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, −1, 1, 1, −1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, 1, −1, 1, −1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, 1, −1, −1, 1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, 1, −1, −1, 1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, 1, −1, 1, −1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, −1, 1, 1, −1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, −1, 1, −1, 1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, −1, −1, 1, −1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, −1, −1, −1, 1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, 1, 1, −1, 1, −1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, 1, 1, 1, −1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, 1, 1, 1, −1, 1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, 1, 1, −1, 1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, −1, −1, −1, 1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, −1, −1, 1, −1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, −1, 1, 1, 1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, −1, 1, −1, −1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, 1, −1, −1, −1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, 1, −1, 1, 1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, −1, 1, −1, 1, 1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, 1, −1, −1, −1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, −1, 1, −1, −1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, −1, −1, 1, 1, 1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, 1, −1, −1, 1, 1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, 1, −1, 1, −1, −1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, −1, −1, 1, 1, −1, 1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, −1, −1, 1, −1, 1, −1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, −1, 1, −1, 1, −1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, −1, 1, 1, −1, 1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, 1, 1, −1, 1, −1, −1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, 1, 1, −1, −1, 1, 1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, 1, 1, −1, −1, −1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, 1, 1, 1, 1, 1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, 1, −1, −1, 1, 1, −1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, 1, −1, −1, −1, −1, 1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, −1, −1, −1, −1, −1, 1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, −1, −1, 1, 1, −1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, −1, 1, 1, 1, 1, 1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, −1, 1, 1, −1, −1, −1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, 1, −1, −1, −1, 1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, 1, −1, 1, 1, −1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, −1, −1, 1, 1, 1, 1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, −1, −1, 1, −1, −1, −1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, −1, 1, −1, −1, −1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, −1, 1, 1, 1, 1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, 1, 1, −1, 1, 1, −1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, 1, 1, −1, −1, −1, 1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, 1, 1, −1, 1, −1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, 1, 1, 1, −1, 1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, 1, −1, −1, 1, −1, −1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, 1, −1, −1, −1, 1, 1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, −1, −1, −1, 1, 1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, −1, −1, 1, −1, −1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, −1, 1, 1, 1, −1, 1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, −1, 1, 1, −1, 1, −1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, −1, 1, 1, 1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, −1, 1, −1, −1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, 1, −1, −1, −1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, 1, −1, 1, 1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, 1, −1, 1, 1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, 1, −1, −1, −1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, −1, 1, −1, −1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, −1, 1, 1, 1, −1, 1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, 1, 1, 1, −1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, 1, 1, −1, 1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, −1, −1, −1, 1, −1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, −1, −1, 1, −1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, −1, −1, 1, −1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, −1, −1, −1, 1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, 1, 1, −1, 1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, 1, 1, 1, −1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, −1, 1, −1, 1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, −1, 1, 1, −1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, 1, −1, 1, −1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, 1, −1, −1, 1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, 1, 1, −1, −1, 1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, 1, −1, 1, −1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, −1, 1, 1, −1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, 1, −1, 1, −1, 1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, 1, 1, −1, −1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, 1, 1, 1, 1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, −1, −1, 1, 1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, −1, −1, −1, −1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, −1, −1, −1, −1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, −1, −1, 1, 1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, 1, 1, 1, 1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, 1, 1, −1, −1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, −1, −1, 1, −1, 1, −1, 1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, −1, −1, −1, 1, −1, 1, 1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, 1, −1, 1, 1, −1, 1, 1, −1, −1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, 1, −1, 1, 1, 1, −1, −1, 1, −1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, 1, 1, 1, 1, −1, −1, 1, 1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, 1, 1, −1, 1, 1, −1, 1, 1, 1, 1, }, 
               
               
                   
                 { 1, −1, 1, 1, −1, 1, −1, −1, −1, 1, −1, 1, −1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, 1, 1, −1, 1, −1, −1, 1, −1, 1, −1, −1, −1, 1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, 1, −1, 1, 1, 1, 1, −1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, 1, −1, −1, −1, −1, −1, −1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, −1, −1, −1, 1, −1, −1, 1, 1, 1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, −1, −1, −1, 1, 1, 1, −1, −1, 1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, −1, 1, 1, 1, −1, −1, −1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, −1, 1, −1, −1, 1, 1, −1, 1, 1, −1, }, 
               
               
                   
                 { 1, 1, −1, 1, 1, 1, 1, −1, −1, −1, −1, −1, 1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, −1, 1, 1, 1, 1, −1, 1, 1, 1, 1, 1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, −1, −1, 1, 1, −1, −1, −1, −1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, −1, −1, −1, −1, 1, 1, −1, −1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, 1, −1, 1, 1, −1, −1, −1, −1, 1, 1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, 1, −1, 1, 1, 1, 1, 1, 1, 1, 1, 1, −1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, 1, 1, 1, 1, 1, 1, −1, −1, −1, 1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, 1, 1, −1, −1, −1, −1, −1, −1, 1, −1, }, 
               
               
                   
                 { 1, 1, 1, 1, −1, 1, −1, −1, −1, −1, 1, 1, 1, 1, −1, 1, }, 
               
               
                   
                 { −1, −1, 1, 1, −1, 1, −1, −1, 1, 1, −1, −1, 1, 1, 1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, 1, −1, 1, −1, −1, 1, 1, −1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, 1, −1, −1, 1, 1, −1, 1, −1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, −1, −1, −1, 1, −1, 1, −1, 1, −1, 1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, −1, −1, −1, 1, 1, −1, 1, −1, −1, 1, 1, 1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, −1, 1, 1, −1, 1, −1, 1, −1, −1, −1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, −1, 1, −1, 1, −1, 1, 1, −1, 1, 1, }, 
               
               
                   
                 { 1, −1, −1, 1, 1, 1, 1, −1, −1, 1, 1, −1, −1, 1, −1, −1, }, 
               
               
                   
                 { −1, 1, −1, 1, 1, 1, 1, −1, 1, −1, −1, 1, −1, 1, 1, 1, } 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
           
               
                   
               
               
                 APPENDIX C 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 −1 1 1 1 1 1 1 −1 −1 1 −1 1 1 1 1 1 
               
               
                   
                 −1 1 −1 −1 1 −1 1 1 −1 −1 1 1 1 1 1 1 
               
               
                   
                 −1 1 −1 1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 1 
               
               
                   
                 −1 1 1 1 1 1 −1 1 −1 −1 1 1 1 −1 1 −1 
               
               
                   
                 −1 1 1 −1 1 1 −1 1 −1 1 −1 −1 −1 −1 1 1 
               
               
                   
                 −1 1 −1 1 1 −1 −1 −1 −1 −1 1 −1 −1 −1 1 1 
               
               
                   
                 −1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 −1 1 1 1 
               
               
                   
                 −1 1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1 1 −1 
               
               
                   
                   
               
             
          
         
       
     
     Certain modifications of the present invention have been discussed above. Other modifications will occur to those practicing in the art of the present invention. For example, the blocks that have been described above in relation to the various drawing figures described herein may be hardware blocks, software modules, logic arrays, etc. Moreover, the arrangements shown in these drawing figures may be implemented as separate blocks as shown, or the blocks may be combined or divided as desired without departing from the scope of the invention. 
     Moreover, as disclosed above, the convolutional interleaver  14  and de-interleaver  40  are preferably characterized by the parameters N=48, B=16 and M=3. However, the convolutional interleaver  14  and de-interleaver  40  may be characterized by the other values for the parameters N, B, and M. 
     Accordingly, the description of the present invention is to be construed as illustrative only and is for the purpose of teaching those skilled in the art the best mode of carrying out the invention. The details may be varied substantially without departing from the spirit of the invention, and the exclusive use of all modifications which are within the scope of the appended claims is reserved.