Patent Publication Number: US-2023147023-A1

Title: Transmitting apparatus and interleaving method thereof

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This is continuation of U.S. application Ser. No. 17/329,870 filed May 25, 2021, which is a continuation of U.S. application Ser. No. 16/510,405, filed on Jul. 12, 2019, now U.S. Pat. No. 11,050,411, issued on Jun. 29, 2021, which is a continuation of U.S. application Ser. No. 14/625,795 filed Feb. 19, 2015, now U.S. Pat. No. 10,425,110, issued on Sep. 24, 2019, which claims priority from U.S. Provisional Application No. 61/941,708 filed on Feb. 19, 2014 and Korean Patent Application No. 10-2015-0024183 filed on Feb. 17, 2015, the disclosures of which are incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     1. Field 
     Apparatuses and methods consistent with exemplary embodiments relate to a transmitting apparatus and an interleaving method thereof, and more particularly, to a transmitting apparatus which processes and transmits data, and an interleaving method thereof. 
     2. Description of the Related Art 
     In the 21st century information-oriented society, broadcasting communication services are moving into the era of digitalization, multi-channel, wideband, and high quality. In particular, as high quality digital televisions, portable multimedia players and portable broadcasting equipment are increasingly used in recent years, there is an increasing demand for methods for supporting various receiving methods of digital broadcasting services. 
     In order to meet such demand, standard groups are establishing various standards and are providing a variety of services to satisfy users needs. Therefore, there is a need for a method for providing improved services to users with high decoding and receiving performance. 
     SUMMARY 
     Exemplary embodiments of the inventive concept may overcome the above disadvantages and other disadvantages not described above. However, it is understood that the exemplary embodiment are not required to overcome the disadvantages described above, and may not overcome any of the problems described above. 
     The exemplary embodiments provide a transmitting apparatus which can map a bit included in a predetermined bit group from among a plurality of bit groups of a low density parity check (LDPC) codeword onto a predetermined bit of a modulation symbol, and transmit the bit, and an interleaving method thereof. 
     According to an aspect of an exemplary embodiment, there is provided a transmitting apparatus including: an encoder configured to generate an LDPC codeword by LDPC encoding based on a parity check matrix; an interleaver configured to interleave the LDPC codeword; and a modulator configured to map the interleaved LDPC codeword onto a modulation symbol, wherein the modulator is further configured to map a bit included in a predetermined bit group from among a plurality of bit groups constituting the LDPC codeword onto a predetermined bit of the modulation symbol. 
     Each of the plurality of bit groups may be formed of M number of bits. M may be a common divisor of N ldpc  and K ldpc  and may be determined to satisfy Q ldpc =(N ldpc −K ldpc )/M. In this case, Q ldpc  may be a cyclic shift parameter value regarding columns in a column group of an information word submatrix of the parity check matrix, N ldpc  may be a length of the LDPC codeword, and K ldpc  may be a length of information word bits of the LDPC codeword. 
     The interleaver may include: a parity interleaver configured to interleave parity bits of the LDPC codeword; a group interleaver configured to divide the parity-interleaved LDPC codeword by the plurality of bit groups and rearrange an order of the plurality of bit groups in bit group wise; and a block interleaver configured to interleave the plurality of bit groups the order of which is rearranged. 
     The group interleaver may be configured to rearrange the order of the plurality of bit groups in bit group wise by using the following equation: 
         Y   j   =X   π(j) (0≤ j&lt;N   group ),
 
     where X j  is a j th  bit group before the plurality of bit groups are interleaved, Y j  is a j th  bit group after the plurality of bit groups are interleaved, N group  is a total number of the plurality of bit groups, and π(j) is a parameter indicating an interleaving order. 
     Here, π(j) may be determined based on at least one of a length of the LDPC codeword, a modulation method, and a code rate. 
     When the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 6/15, π(j) may be defined as in table 11. 
     When the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 10/15, π(j) may be defined as in table 14. 
     When the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 12/15, π(j) may be defined as in table 15. 
     When the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 6/15, π(j) may be defined as in table 17. 
     When the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 8/15, π(j) may be defined as in table 18. 
     When the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 12/15, π(j) may be defined as in table 21. 
     The block interleaver may be configured to interleave by writing the plurality of bit groups in each of a plurality of columns in bit group wise in a column direction, and reading each row of the plurality of columns in which the plurality of bit groups are written in bit group wise in a row direction. 
     The block interleaver may be configured to serially write, in the plurality of columns, at least some bit groups which are writable in the plurality of columns in bit group wise from among the plurality of bit groups, and then divide and write the other bit groups in an area which remains after the at least some bit groups are written in the plurality of columns in bit group wise. 
     According to an aspect of another exemplary embodiment, there is provided an interleaving method of a transmitting apparatus, including: generating an LDPC codeword by LDPC encoding based on a parity check matrix; interleaving the LDPC codeword; and mapping the interleaved LDPC codeword onto a modulation symbol, wherein the mapping comprises mapping a bit included in a predetermined bit group from among a plurality of bit groups constituting the LDPC codeword onto a predetermined bit of the modulation symbol. 
     Each of the plurality of bit groups may be formed of M number of bits, and M may be a common divisor of N ldpc  and K ldpc  and may be determined to satisfy Q ldpc =(N ldpc −K ldpc )/M. In this case, Q ldpc  may be a cyclic shift parameter value regarding columns in a column group of an information word submatrix of the parity check matrix, N ldpc  may be a length of the LDPC codeword, and K ldpc  may be a length of information word bits of the LDPC codeword. 
     The interleaving may include: interleaving parity bits of the LDPC codeword; dividing the parity-interleaved LDPC codeword by the plurality of bit groups and rearranging an order of the plurality of bit groups in bit group wise; and interleaving the plurality of bit groups the order of which is rearranged. 
     The rearranging in bit group wise may include rearranging the order of the plurality of bit groups in bit group wise by using the following equation: 
         Y   j   =X   π(j) (0≤1&lt; N   group ),
 
     where X j  is a j th  bit group before the plurality of bit groups are interleaved, Y j  is a j th  bit group after the plurality of bit groups are interleaved, N group  is a total number of the plurality of bit groups, and π(j) is a parameter indicating an interleaving order. 
     Here, π(j) may be determined based on at least one of a length of the LDPC codeword, a modulation method, and a code rate. 
     When the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 6/15, π(j) may be defined as in table 11. 
     When the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 10/15, π(j) may be defined as in table 14. 
     When the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 12/15, π(j) may be defined as in table 15. 
     When the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 6/15, π(j) may be defined as in table 17. 
     When the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 8/15, π(j) may be defined as in table 18. 
     When the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 12/15, π(j) may be defined as in table 21. 
     The interleaving the plurality of bit groups may include interleaving by writing the plurality of bit groups in each of a plurality of columns in bit group wise in a column direction, and reading each row of the plurality of columns in which the plurality of bit groups are written in bit group wise in a row direction. 
     The interleaving the plurality of bit groups may include serially writing, in the plurality of columns, at least some bit groups which are writable in the plurality of columns in bit group wise from among the plurality of bit groups, and then dividing and writing the other bit groups in an area which remains after the at least some bit groups are written in the plurality of columns in bit group wise. 
     According to various exemplary embodiments, improved decoding and receiving performance can be provided. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and/or other aspects will be more apparent by describing in detail exemplary embodiments, with reference to the accompanying drawings, in which: 
         FIG.  1    is a block diagram to illustrate a configuration of a transmitting apparatus, according to an exemplary embodiment; 
         FIGS.  2  to  4    are views to illustrate a configuration of a parity check matrix, according to exemplary embodiments; 
         FIG.  5    is a block diagram to illustrate a configuration of an interleaver, according to an exemplary embodiment; 
         FIGS.  6  to  8    are views to illustrate an interleaving method, according to exemplary embodiments; 
         FIGS.  9  to  14    are views to illustrate an interleaving method of a block interleaver, according to exemplary embodiments; 
         FIG.  15    is a view to illustrate an operation of a demultiplexer, according to an exemplary embodiment; 
         FIGS.  16  and  17    are views to illustrate a method for designing an interleaving pattern, according to exemplary embodiments; 
         FIG.  18    is a block diagram to illustrate a configuration of a receiving apparatus according to an exemplary embodiment; 
         FIG.  19    is a block diagram to illustrate a configuration of a deinterleaver, according to an exemplary embodiment; 
         FIG.  20    is a view to illustrate a deinterleaving method of a block deinterleaver, according to an exemplary embodiment; and 
         FIG.  21    is a flowchart to illustrate an interleaving method, according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION OF THE EXEMPLARY EMBODIMENTS 
     Hereinafter, various exemplary embodiments will be described in greater detail with reference to the accompanying drawings. 
     In the following description, same reference numerals are used for the same elements when they are depicted in different drawings. The matters defined in the description, such as detailed construction and elements, are provided to assist in a comprehensive understanding of the exemplary embodiments. Thus, it is apparent that the exemplary embodiments can be carried out without those specifically defined matters. Also, functions or elements known in the related art are not described in detail since they would obscure the exemplary embodiments with unnecessary detail. 
       FIG.  1    is a block diagram to illustrate a configuration of a transmitting apparatus according to an exemplary embodiment. Referring to  FIG.  1   , the transmitting apparatus  100  includes an encoder  110 , an interleaver  120 , and a modulator  130  (or a constellation mapper). 
     The encoder  110  generates a low density parity check (LDPC) codeword by performing LDPC encoding based on a parity check matrix. To achieve this, the encoder  110  may include an LDPC encoder (not shown) to perform the LDPC encoding. 
     Specifically, the encoder  110  LDPC-encodes information word(or information) bits to generate the LDPC codeword which is formed of information word bits and parity bits (that is, LDPC parity bits). Here, bits input to the encoder  110  may be used as the information word bits. Also, since an LDPC code is a systematic code, the information word bits may be included in the LDPC codeword as they are. 
     The LDPC codeword is formed of the information word bits and the parity bits. For example, the LDPC codeword is formed of N ldpc  number of bits, and includes K ldpc  number of information word bits and N parity =N ldpc −K ldpc  number of parity bits. 
     In this case, the encoder  110  may generate the LDPC codeword by performing the LDPC encoding based on the parity check matrix. That is, since the LDPC encoding is a process for generating an LDPC codeword to satisfy H·C T =0, the encoder  110  may use the parity check matrix when performing the LDPC encoding. Herein, H is a parity check matrix and C is an LDPC codeword. 
     For the LDPC encoding, the transmitting apparatus  100  may include a memory and may pre-store parity check matrices of various formats. 
     For example, the transmitting apparatus  100  may pre-store parity check matrices which are defined in Digital Video Broadcasting-Cable version 2 (DVB-C2), Digital Video Broadcasting-Satellite-Second Generation (DVB-S2), Digital Video Broadcasting-Second Generation Terrestrial (DVB-T2), etc., or may pre-store parity check matrices which are defined in the North America digital broadcasting standard system Advanced Television System Committee (ATSC) 3.0 standards, which are currently being established. However, this is merely an example and the transmitting apparatus  100  may pre-store parity check matrices of other formats in addition to these parity check matrices. 
     Hereinafter, a parity check matrix according to various exemplary embodiments will be explained in detail with reference to the drawings. In the parity check matrix, elements other than elements having 1 have 0. 
     For example, the parity check matrix according to an exemplary embodiment may have a configuration of  FIG.  2   . 
     Referring to  FIG.  2   , a parity check matrix  200  is formed of an information word submatrix(or an information submatrix)  210  corresponding to information word bits, and a parity submatrix  220  corresponding to parity bits. 
     The information word submatrix  210  includes K ldpc  number of columns and the parity submatrix  220  includes N parity =N ldpc −K ldpc  number of columns. The number of rows of the parity check matrix  200  is identical to the number of columns of the parity submatrix  220 , N parity =N ldpc −K ldpc . 
     In addition, in the parity check matrix  200 , N ldpc  is a length of an LDPC codeword, K ldpc  is a length of information word bits, and N parity =N ldpc −K ldpc  is a length of parity bits. The length of the LDPC codeword, the information word bits, and the parity bits mean the number of bits included in each of the LDPC codeword, the information word bits, and the parity bits. 
     Hereinafter, the configuration of the information word submatrix  210  and the parity submatrix  220  will be explained in detail. 
     The information word submatrix  210  includes K ldpc  number of columns (that is, 0 th  column to (K ldpc −1) th  column), and follows the following rules: 
     First, M number of columns from among K ldpc  number of columns of the information word submatrix  210  belong to the same group, and K ldpc  number of columns is divided into K ldpc /M number of column groups. In each column group, a column is cyclic-shifted from an immediately previous column by Q ldpc . That is, Q ldpc  may be a cyclic shift parameter value regarding columns in a column group of the information word submatrix  210  of the parity check matrix  200 . 
     Herein, M is an interval at which a pattern of a column group, which includes a plurality of columns, is repeated in the information word submatrix  210  (e.g., M=360), and Q ldpc  is a size by which one column is cyclic-shifted from an immediately previous column in a same column group in the information word submatrix  210 . Also, M is a common divisor of N ldpc  and K ldpc  and is determined to satisfy Q ldpc =(N ldpc −K ldpc )/M. Here, M and Q ldpc  are integers and K ldpc /M is also an integer. M and Q ldpc  may have various values according to a length of the LDPC codeword and a code rate (CR)(or, coding rate). 
     For example, when M=360 and the length of the LDPC codeword, N ldpc , is 64800, Q ldpc  may be defined as in table 1 presented below, and, when M=360 and the length N ldpc  of the LDPC codeword is 16200, Q ldpc  may be defined as in table 2 presented below. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Code Rate 
                 N ldpc   
                 M 
                 Q ldpc   
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 5/15 
                 64800 
                 360 
                 120 
               
               
                   
                 6/15 
                 64800 
                 360 
                 108 
               
               
                   
                 7/15 
                 64800 
                 360 
                 96 
               
               
                   
                 8/15 
                 64800 
                 360 
                 84 
               
               
                   
                 9/15 
                 64800 
                 360 
                 72 
               
               
                   
                 10/15  
                 64800 
                 360 
                 60 
               
               
                   
                 11/15  
                 64800 
                 360 
                 48 
               
               
                   
                 12/15  
                 64800 
                 360 
                 36 
               
               
                   
                 13/15  
                 64800 
                 360 
                 24 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 Code Rate 
                 N ldpc   
                 M 
                 Q ldpc   
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 5/15 
                 16200 
                 360 
                 30 
               
               
                   
                 6/15 
                 16200 
                 360 
                 27 
               
               
                   
                 7/15 
                 16200 
                 360 
                 24 
               
               
                   
                 8/15 
                 16200 
                 360 
                 21 
               
               
                   
                 9/15 
                 16200 
                 360 
                 18 
               
               
                   
                 10/15  
                 16200 
                 360 
                 15 
               
               
                   
                 11/15  
                 16200 
                 360 
                 12 
               
               
                   
                 12/15  
                 16200 
                 360 
                 9 
               
               
                   
                 13/15  
                 16200 
                 360 
                 6 
               
               
                   
                   
               
            
           
         
       
     
     Second, when the degree of the 0 th  column of the i th  column group (i=0, 1, . . . , K ldpc /M−1) is D i  (herein, the degree is the number of value 1 existing in each column and all columns belonging to the same column group have the same degree), and a position (or an index) of each row where 1 exists in the 0 th  column of the i th  column group is R i,0   (0) , R i,0   (1) , . . . , R i,0   (D     i     −1) , an index R i,j   (k)  of a row where k th  1 is located in the j th  column in the i th  column group is determined by following Equation 1: 
         R   i,j   (k)   =R   i,(j-1)   (k)   +Q   ldpc  mod( N   ldpc   −K   ldpc )  (1),
 
     where k=0, 1, 2, . . . D i −1; i=0, 1, . . . , K ldpc /M−1; and j=1, 2, . . . , M−1. 
     Equation 1 can be expressed as following Equation 2: 
         R   i,j   (k)   ={R   i,(j-1)   (k)   +Q   ldpc  mod( N   ldpc   −K   ldpc )  (2),
 
     where k=0, 1, 2, . . . D i −1; i=0, 1, . . . , K ldpc /M−1; and j=1, 2, . . . , M−1. Since j=1, 2, . . . , M−1, (j mod M) of Equation 2 may be regarded as j. 
     In the above equations, R i,j   (k)  is an index of a row where k th  1 is located in the j th  column in the i th  column group, N ldpc  is a length of an LDPC codeword, K ldpc  is a length of information word bits, D i  is a degree of columns belonging to the i th  column group, M is the number of columns belonging to a single column group, and Q ldpc  is a size by which each column in the column group is cyclic-shifted. 
     As a result, referring to these equations, when only R i,0   (k)  is known, the index R i,j   (k)  of the row where the k th  1 is located in the j th  column in the i th  column group can be known. Therefore, when the index value of the row where the k th  1 is located in the 0 th  column of each column group is stored, a position of column and row where 1 is located in the parity check matrix  200  having the configuration of  FIG.  2    (that is, in the information word submatrix  210  of the parity check matrix  200 ) can be known. 
     According to the above-described rules, all of the columns belonging to the i th  column group have the same degree D i . Accordingly, the LDPC codeword which stores information on the parity check matrix according to the above-described rules may be briefly expressed as follows. 
     For example, when N ldpc  is 30, K ldpc  is 15, and Q ldpc  is 3, position information of the row where 1 is located in the 0 th  column of the three column groups may be expressed by a sequence of Equations 3 and may be referred to as “weight-1 position sequence”. 
         R   1,0   (1) =1, R   1,0   (2) =2, R   1,0   (3) =8, R   1,0   (4) =10, 
         R   2,0   (1) =0, R   2,0   (2) =9, R   2,0   (3) =13, 
         R   3,0   (1) =0, R   3,0   (2) =14  (3),
 
     where R i,j   (k)  is an index of a row where k th  1 is located in the j th  column in the i th  column group. 
     The weight-1 position sequence like Equation 3 which expresses an index of a row where 1 is located in the 0 th  column of each column group may be briefly expressed as in Table 3 presented below: 
     
       
         
           
               
               
             
               
                   
                 TABLE 3 
               
               
                   
                   
               
             
            
               
                   
                 1 2 8 10 
               
               
                   
                 0 9 13 
               
               
                   
                 0 14 
               
               
                   
                   
               
            
           
         
       
     
     Table 3 shows positions of elements having value 1 in the parity check matrix, and the i th  weight-1 position sequence is expressed by indexes of rows where 1 is located in the 0 th  column belonging to the i th  column group. 
     The information word submatrix  210  of the parity check matrix according to an exemplary embodiment may be defined as in Tables 4 to 8 presented below, based on the above descriptions. 
     Specifically, Tables 4 to 8 show indexes of rows where 1 is located in the 0 th  column of the i th  column group of the information word submatrix  210 . That is, the information word submatrix  210  is formed of a plurality of column groups each including M number of columns, and positions of 1 in the 0 th  column of each of the plurality of column groups may be defined by Tables 4 to 8. 
     Herein, the indexes of the rows where 1 is located in the 0 th  column of the i th  column group mean “addresses of parity bit accumulators”. The “addresses of parity bit accumulators” have the same meaning as defined in the DVB-C2/S2/T2 standards or the ATSC 3.0 standards which are currently being established, and thus, a detailed explanation thereof is omitted. 
     For example, when the length N ldpc  of the LDPC codeword is 64800, the code rate is 6/15, and M is 360, the indexes of the rows where 1 is located in the 0 th  column of the i th  column group of the information word submatrix  210  are as shown in Table 4 presented below: 
     
       
         
           
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                 Index of row where 1 is located in the 0th column of the 
               
               
                 i 
                 ith column group 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 0 
                 1606 3402 4961 6751 7132 11516 12300 12482 12592 13342 13764 
               
               
                   
                 14123 21576 23946 24533 25376 25667 26836 31799 34173 
               
               
                   
                 35462 36153 36740 37085 37152 37468 37658 
               
               
                 1 
                 4621 5007 6910 8732 9757 11508 13099 15513 16335 18052 19512 
               
               
                   
                 21319 23663 25628 27208 31333 32219 33003 33239 33447 
               
               
                   
                 36200 36473 36938 37201 37283 37495 38642 
               
               
                 2 
                 16 1094 2020 3080 4194 5098 5631 6877 7889 8237 9804 10067 
               
               
                   
                 11017 11366 13136 13354 15379 18934 20199 24522 26172 
               
               
                   
                 28666 30386 32714 36390 37015 37162 
               
               
                 3 
                 700 897 1708 6017 6490 7372 7825 9546 10398 16605 18561 18745 
               
               
                   
                 21625 22137 23693 24340 24966 25015 26995 28586 28895 
               
               
                   
                 29687 33938 34520 34858 37056 38297 
               
               
                 4 
                 159 2010 2573 3617 4452 4958 5556 5832 6481 8227 9924 10836 
               
               
                   
                 14954 15594 16623 18065 19249 22394 22677 23408 23731 
               
               
                   
                 24076 24776 27007 28222 30343 38371 
               
               
                 5 
                 3118 3545 4768 4992 5227 6732 8170 9397 10522 11508 15536 
               
               
                   
                 20218 21921 28599 29445 29758 29968 31014 32027 33685 
               
               
                   
                 34378 35867 36323 36728 36870 38335 38623 
               
               
                 6 
                 1264 4254 6936 9165 9486 9950 10861 11653 13697 13961 15164 
               
               
                   
                 15665 18444 19470 20313 21189 24371 26431 26999 28086 
               
               
                   
                 28251 29261 31981 34015 35850 36129 37186 
               
               
                 7 
                 111 1307 1628 2041 2524 5358 7988 8191 10322 11905 12919 
               
               
                   
                 14127 15515 15711 17061 19024 21195 22902 23727 24401 
               
               
                   
                 24608 25111 25228 27338 35398 37794 38196 
               
               
                 8 
                 961 3035 7174 7948 13355 13607 14971 18189 18339 18665 18875 
               
               
                   
                 19142 20615 21136 21309 21758 23366 24745 25849 25982 
               
               
                   
                 27583 30006 31118 32106 36469 36583 37920 
               
               
                 9 
                 2990 3549 4273 4808 5707 6021 6509 7456 8240 10044 12262 
               
               
                   
                 12660 13085 14750 15680 16049 21587 23997 25803 28343 
               
               
                   
                 28693 34393 34860 35490 36021 37737 38296 
               
               
                 10 
                 955 4323 5145 6885 8123 9730 11840 12216 19194 20313 23056 
               
               
                   
                 24248 24830 25268 26617 26801 28557 29753 30745 31450 
               
               
                   
                 31973 32839 33025 33296 35710 37366 37509 
               
               
                 11 
                 264 605 4181 4483 5156 7238 8863 10939 11251 12964 16254 
               
               
                   
                 17511 20017 22395 22818 23261 23422 24064 26329 27723 
               
               
                   
                 28186 30434 31956 33971 34372 36764 38123 
               
               
                 12 
                 520 2562 2794 3528 3860 4402 5676 6963 8655 9018 9783 11933 
               
               
                   
                 16336 17193 17320 19035 20606 23579 23769 24123 24966 
               
               
                   
                 27866 32457 34011 34499 36620 37526 
               
               
                 13 
                 10106 10637 10906 34242 
               
               
                 14 
                 1856 15100 19378 21848 
               
               
                 15 
                 943 11191 27806 29411 
               
               
                 16 
                 4575 6359 13629 19383 
               
               
                 17 
                 4476 4953 18782 24313 
               
               
                 18 
                 5441 6381 21840 35943 
               
               
                 19 
                 9638 9763 12546 30120 
               
               
                 20 
                 9587 10626 11047 25700 
               
               
                 21 
                 4088 15298 28768 35047 
               
               
                 22 
                 2332 6363 8782 28863 
               
               
                 23 
                 4625 4933 28298 30289 
               
               
                 24 
                 3541 4918 18257 31746 
               
               
                 25 
                 1221 25233 26757 34892 
               
               
                 26 
                 8150 16677 27934 30021 
               
               
                 27 
                 8500 25016 33043 38070 
               
               
                 28 
                 7374 10207 16189 35811 
               
               
                 29 
                 611 18480 20064 38261 
               
               
                 30 
                 25416 27352 36089 38469 
               
               
                 31 
                 1667 17614 25839 32776 
               
               
                 32 
                 4118 12481 21912 37945 
               
               
                 33 
                 5573 13222 23619 31271 
               
               
                 34 
                 18271 26251 27182 30587 
               
               
                 35 
                 14690 26430 26799 34355 
               
               
                 36 
                 13688 16040 20716 34558 
               
               
                 37 
                 2740 14957 23436 32540 
               
               
                 38 
                 3491 14365 14681 36858 
               
               
                 39 
                 4796 6238 25203 27854 
               
               
                 40 
                 1731 12816 17344 26025 
               
               
                 41 
                 19182 21662 23742 27872 
               
               
                 42 
                 6502 13641 17509 34713 
               
               
                 43 
                 12246 12372 16746 27452 
               
               
                 44 
                 1589 21528 30621 34003 
               
               
                 45 
                 12328 20515 30651 31432 
               
               
                 46 
                 3415 22656 23427 36395 
               
               
                 47 
                 632 5209 25958 31085 
               
               
                 48 
                 619 3690 19648 37778 
               
               
                 49 
                 9528 13581 26965 36447 
               
               
                 50 
                 2147 26249 26968 28776 
               
               
                 51 
                 15698 18209 30683 
               
               
                 52 
                 1132 19888 34111 
               
               
                 53 
                 4608 25513 38874 
               
               
                 54 
                 475 1729 34100 
               
               
                 55 
                 7348 32277 38587 
               
               
                 56 
                 182 16473 33082 
               
               
                 57 
                 3865 9678 21265 
               
               
                 58 
                 4447 20151 27618 
               
               
                 59 
                 6335 14371 38711 
               
               
                 60 
                 704 9695 28858 
               
               
                 61 
                 4856 9757 30546 
               
               
                 62 
                 1993 19361 30732 
               
               
                 63 
                 756 28000 29138 
               
               
                 64 
                 3821 24076 31813 
               
               
                 65 
                 4611 12326 32291 
               
               
                 66 
                 7628 21515 34995 
               
               
                 67 
                 1246 13294 30068 
               
               
                 68 
                 6466 33233 35865 
               
               
                 69 
                 14484 23274 38150 
               
               
                 70 
                 21269 36411 37450 
               
               
                 71 
                 23129 26195 37653 
               
               
                   
               
            
           
         
       
     
     In another example, when the length N ldpc  of the LDPC codeword is 64800, the code rate is 8/15, and M is 360, the indexes of the rows where 1 is located in the 0 th  column of the i th  column group of the information word submatrix  210  are as shown in Table 5 presented below: 
     
       
         
           
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                   
                 Index of row where 1 is located in the 0th column of the 
               
               
                 i 
                 ith column group 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 0 
                 2768 3039 4059 5856 6245 7013 8157 9341 9802 10470 11521 
               
               
                   
                 12083 16610 18361 20321 24601 27420 28206 29788 
               
               
                 1 
                 2739 8244 8891 9157 12624 12973 15534 16622 16919 18402 
               
               
                   
                 18780 19854 20220 20543 22306 25540 27478 27678 28053 
               
               
                 2 
                 1727 2268 6246 7815 9010 9556 10134 10472 11389 14599 
               
               
                   
                 15719 16204 17342 17666 18850 22058 25579 25860 29207 
               
               
                 3 
                 28 1346 3721 5565 7019 9240 12355 13109 14800 16040 16839 
               
               
                   
                 17369 17631 19357 19473 19891 20381 23911 29683 
               
               
                 4 
                 869 2450 4386 5316 6160 7107 10362 11132 11271 13149 16397 
               
               
                   
                 16532 17113 19894 22043 22784 27383 28615 28804 
               
               
                 5 
                 508 4292 5831 8559 10044 10412 11283 14810 15888 17243 
               
               
                   
                 17538 19903 20528 22090 22652 27235 27384 28208 28485 
               
               
                 6 
                 389 2248 5840 6043 7000 9054 11075 11760 12217 12565 
               
               
                   
                 13587 15403 19422 19528 21493 25142 27777 28566 28702 
               
               
                 7 
                 1015 2002 5764 6777 9346 9629 11039 11153 12690 13068 
               
               
                   
                 13990 16841 17702 20021 24106 26300 29332 30081 30196 
               
               
                 8 
                 1480 3084 3467 4401 4798 5187 7851 11368 12323 14325 14546 
               
               
                   
                 16360 17158 18010 21333 25612 26556 26906 27005 
               
               
                 9 
                 6925 8876 12392 14529 15253 15437 19226 19950 20321 23021 
               
               
                   
                 23651 24393 24653 26668 27205 28269 28529 29041 29292 
               
               
                 10 
                 2547 3404 3538 4666 5126 5468 7695 8799 14732 15072 15881 
               
               
                   
                 17410 18971 19609 19717 22150 24941 27908 29018 
               
               
                 11 
                 888 1581 2311 5511 7218 9107 10454 12252 13662 15714 15894 
               
               
                   
                 17025 18671 24304 25316 25556 28489 28977 29212 
               
               
                 12 
                 1047 1494 1718 4645 5030 6811 7868 8146 10611 15767 17682 
               
               
                   
                 18391 22614 23021 23763 25478 26491 29088 29757 
               
               
                 13 
                 59 1781 1900 3814 4121 8044 8906 9175 11156 14841 15789 
               
               
                   
                 16033 16755 17292 18550 19310 22505 29567 29850 
               
               
                 14 
                 1952 3057 4399 9476 10171 10769 11335 11569 15002 19501 
               
               
                   
                 20621 22642 23452 24360 25109 25290 25828 28505 29122 
               
               
                 15 
                 2895 3070 3437 4764 4905 6670 9244 11845 13352 13573 13975 
               
               
                   
                 14600 15871 17996 19672 20079 20579 25327 27958 
               
               
                 16 
                 612 1528 2004 4244 4599 4926 5843 7684 10122 10443 12267 
               
               
                   
                 14368 18413 19058 22985 24257 26202 26596 27899 
               
               
                 17 
                 1361 2195 4146 6708 7158 7538 9138 9998 14862 15359 16076 
               
               
                   
                 18925 21401 21573 22503 24146 24247 27778 29312 
               
               
                 18 
                 5229 6235 7134 7655 9139 13527 15408 16058 16705 18320 
               
               
                   
                 19909 20901 22238 22437 23654 25131 27550 28247 29903 
               
               
                 19 
                 697 2035 4887 5275 6909 9166 11805 15338 16381 18403 20425 
               
               
                   
                 20688 21547 24590 25171 26726 28848 29224 29412 
               
               
                 20 
                 5379 17329 22659 23062 
               
               
                 21 
                 11814 14759 22329 22936 
               
               
                 22 
                 2423 2811 10296 12727 
               
               
                 23 
                 8460 15260 16769 17290 
               
               
                 24 
                 14191 14608 29536 30187 
               
               
                 25 
                 7103 10069 20111 22850 
               
               
                 26 
                 4285 15413 26448 29069 
               
               
                 27 
                 548 2137 9189 10928 
               
               
                 28 
                 4581 7077 23382 23949 
               
               
                 29 
                 3942 17248 19486 27922 
               
               
                 30 
                 8668 10230 16922 26678 
               
               
                 31 
                 6158 9980 13788 28198 
               
               
                 32 
                 12422 16076 24206 29887 
               
               
                 33 
                 8778 10649 18747 22111 
               
               
                 34 
                 21029 22677 27150 28980 
               
               
                 35 
                 7918 15423 27672 27803 
               
               
                 36 
                 5927 18086 23525 
               
               
                 37 
                 3397 15058 30224 
               
               
                 38 
                 24016 25880 26268 
               
               
                 39 
                 1096 4775 7912 
               
               
                 40 
                 3259 17301 20802 
               
               
                 41 
                 129 8396 15132 
               
               
                 42 
                 17825 28119 28676 
               
               
                 43 
                 2343 8382 28840 
               
               
                 44 
                 3907 18374 20939 
               
               
                 45 
                 1132 1290 8786 
               
               
                 46 
                 1481 4710 28846 
               
               
                 47 
                 2185 3705 26834 
               
               
                 48 
                 5496 15681 21854 
               
               
                 49 
                 12697 13407 22178 
               
               
                 50 
                 12788 21227 22894 
               
               
                 51 
                 629 2854 6232 
               
               
                 52 
                 2289 18227 27458 
               
               
                 53 
                 7593 21935 23001 
               
               
                 54 
                 3836 7081 12282 
               
               
                 55 
                 7925 18440 23135 
               
               
                 56 
                 497 6342 9717 
               
               
                 57 
                 11199 22046 30067 
               
               
                 58 
                 12572 28045 28990 
               
               
                 59 
                 1240 2023 10933 
               
               
                 60 
                 19566 20629 25186 
               
               
                 61 
                 6442 13303 28813 
               
               
                 62 
                 4765 10572 16180 
               
               
                 63 
                 552 19301 24286 
               
               
                 64 
                 6782 18480 21383 
               
               
                 65 
                 11267 12288 15758 
               
               
                 66 
                 771 5652 15531 
               
               
                 67 
                 16131 20047 25649 
               
               
                 68 
                 13227 23035 24450 
               
               
                 69 
                 4839 13467 27488 
               
               
                 70 
                 2852 4677 22993 
               
               
                 71 
                 2504 28116 29524 
               
               
                 72 
                 12518 17374 24267 
               
               
                 73 
                 1222 11859 27922 
               
               
                 74 
                 9660 17286 18261 
               
               
                 75 
                 232 11296 29978 
               
               
                 76 
                 9750 11165 16295 
               
               
                 77 
                 4894 9505 23622 
               
               
                 78 
                 10861 11980 14110 
               
               
                 79 
                 2128 15883 22836 
               
               
                 80 
                 6274 17243 21989 
               
               
                 81 
                 10866 13202 22517 
               
               
                 82 
                 11159 16111 21608 
               
               
                 83 
                 3719 18787 22100 
               
               
                 84 
                 1756 2020 23901 
               
               
                 85 
                 20913 29473 30103 
               
               
                 86 
                 2729 15091 26976 
               
               
                 87 
                 4410 8217 12963 
               
               
                 88 
                 5395 24564 28235 
               
               
                 89 
                 3859 17909 23051 
               
               
                 90 
                 5733 26005 29797 
               
               
                 91 
                 1935 3492 29773 
               
               
                 92 
                 11903 21380 29914 
               
               
                 93 
                 6091 10469 29997 
               
               
                 94 
                 2895 8930 15594 
               
               
                 95 
                 1827 10028 20070 
               
               
                   
               
            
           
         
       
     
     In another example, when the length N ldpc  of the LDPC codeword is 64800, the code rate is 10/15, and M is 360, the indexes of rows where 1 exists in the 0 th  column of the i th  column group of the information word submatrix  210  are defined as shown in Table 6 below. 
     
       
         
           
               
               
             
               
                 TABLE 6 
               
               
                   
               
               
                   
                 Index of row where 1 is located in the 0th column of the 
               
               
                 i 
                 ith column group 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 0 
                 979 1423 4166 4609 6341 8258 10334 10548 14098 14514 
               
               
                   
                 17051 17333 17653 17830 17990 
               
               
                 1 
                 2559 4025 6344 6510 9167 9728 11312 14856 17104 17721 18600 
               
               
                   
                 18791 19079 19697 19840 
               
               
                 2 
                 3243 6894 7950 10539 12042 13233 13938 14752 16449 16727 
               
               
                   
                 17025 18297 18796 19400 21577 
               
               
                 3 
                 3272 3574 6341 6722 9191 10807 10957 12531 14036 15580 
               
               
                   
                 16651 17007 17309 19415 19845 
               
               
                 4 
                 155 4598 10201 10975 11086 11296 12713 15364 15978 16395 
               
               
                   
                 17542 18164 18451 18612 20617 
               
               
                 5 
                 1128 1999 3926 4069 5558 6085 6337 8386 10693 12450 15438 
               
               
                   
                 16223 16370 17308 18634 
               
               
                 6 
                 2408 2929 3630 4357 5852 7329 8536 8695 10603 11003 14304 
               
               
                   
                 14937 15767 18402 21502 
               
               
                 7 
                 199 3066 6446 6849 8973 9536 10452 12857 13675 15913 16717 
               
               
                   
                 17654 19802 20115 21579 
               
               
                 8 
                 312 870 2095 2586 5517 6196 6757 7311 7368 13046 15384 
               
               
                   
                 18576 20349 21424 21587 
               
               
                 9 
                 985 1591 3248 3509 3706 3847 6174 6276 7864 9033 13618 
               
               
                   
                 15675 16446 18355 18843 
               
               
                 10 
                 975 3774 4083 5825 6166 7218 7633 9657 10103 13052 14240 
               
               
                   
                 17320 18126 19544 20208 
               
               
                 11 
                 1795 2005 2544 3418 6148 8051 9066 9725 10676 10752 11512 
               
               
                   
                 15171 17523 20481 21059 
               
               
                 12 
                 167 315 1824 2325 2640 2868 6070 6597 7016 8109 9815 
               
               
                   
                 11608 16142 17912 19625 
               
               
                 13 
                 1298 1896 3039 4303 4690 8787 12241 13600 14478 15492 
               
               
                   
                 16602 17115 17913 19466 20597 
               
               
                 14 
                 568 3695 6045 6624 8131 8404 8590 9059 9246 11570 14336 
               
               
                   
                 18657 18941 19218 21506 
               
               
                 15 
                 228 1889 1967 2299 3011 5074 7044 7596 7689 9534 10244 
               
               
                   
                 10697 11691 17902 21410 
               
               
                 16 
                 1330 1579 1739 2234 3701 3865 5713 6677 7263 11172 12143 
               
               
                   
                 12765 17121 20011 21436 
               
               
                 17 
                 303 1668 2501 4925 5778 5985 9635 10140 10820 11779 11849 
               
               
                   
                 12058 15650 20426 20527 
               
               
                 18 
                 698 2484 3071 3219 4054 4125 5663 5939 6928 7086 8054 
               
               
                   
                 12173 16280 17945 19302 
               
               
                 19 
                 232 1619 3040 4901 7438 8135 9117 9233 10131 13321 17347 
               
               
                   
                 17436 18193 18586 19929 
               
               
                 20 
                 12 3721 6254 6609 7880 8139 10437 12262 13928 14065 14149 
               
               
                   
                 15032 15694 16264 18883 
               
               
                 21 
                 482 915 1548 1637 6687 9338 10163 11768 11970 15524 15695 
               
               
                   
                 17386 18787 19210 19340 
               
               
                 22 
                 1291 2500 4109 4511 5099 5194 10014 13165 13256 13972 
               
               
                   
                 15409 16113 16214 18584 20998 
               
               
                 23 
                 1761 4778 7444 7740 8129 8341 8931 9136 9207 10003 10678 
               
               
                   
                 13959 17673 18194 20990 
               
               
                 24 
                 3060 3522 5361 5692 6833 8342 8792 11023 11211 11548 11914 
               
               
                   
                 13987 15442 15541 19707 
               
               
                 25 
                 1322 2348 2970 5632 6349 7577 8782 9113 9267 9376 12042 
               
               
                   
                 12943 16680 16970 21321 
               
               
                 26 
                 6785 11960 21455 
               
               
                 27 
                 1223 15672 19550 
               
               
                 28 
                 5976 11335 20385 
               
               
                 29 
                 2818 9387 15317 
               
               
                 30 
                 2763 3554 18102 
               
               
                 31 
                 5230 11489 18997 
               
               
                 32 
                 5809 15779 20674 
               
               
                 33 
                 2620 17838 18533 
               
               
                 34 
                 3025 9342 9931 
               
               
                 35 
                 3728 5337 12142 
               
               
                 36 
                 2520 6666 9164 
               
               
                 37 
                 12892 15307 20912 
               
               
                 38 
                 10736 12393 16539 
               
               
                 39 
                 1075 2407 12853 
               
               
                 40 
                 4921 5411 18206 
               
               
                 41 
                 5955 15647 16838 
               
               
                 42 
                 6384 10336 19266 
               
               
                 43 
                 429 10421 17266 
               
               
                 44 
                 4880 10431 12208 
               
               
                 45 
                 2910 11895 12442 
               
               
                 46 
                 7366 18362 18772 
               
               
                 47 
                 4341 7903 14994 
               
               
                 48 
                 4564 6714 7378 
               
               
                 49 
                 4639 8652 18871 
               
               
                 50 
                 15787 18048 20246 
               
               
                 51 
                 3241 11079 13640 
               
               
                 52 
                 1559 2936 15881 
               
               
                 53 
                 2737 6349 10881 
               
               
                 54 
                 10394 16107 17073 
               
               
                 55 
                 8207 9043 12874 
               
               
                 56 
                 7805 16058 17905 
               
               
                 57 
                 11189 15767 17764 
               
               
                 58 
                 5823 12923 14316 
               
               
                 59 
                 11080 20390 20924 
               
               
                 60 
                 568 8263 17411 
               
               
                 61 
                 1845 3557 6562 
               
               
                 62 
                 2890 10936 14756 
               
               
                 63 
                 9031 14220 21517 
               
               
                 64 
                 3529 12955 15902 
               
               
                 65 
                 413 6750 8735 
               
               
                 66 
                 6784 12092 16421 
               
               
                 67 
                 12019 13794 15308 
               
               
                 68 
                 12588 15378 17676 
               
               
                 69 
                 8067 14589 19304 
               
               
                 70 
                 1244 5877 6085 
               
               
                 71 
                 15897 19349 19993 
               
               
                 72 
                 1426 2394 12264 
               
               
                 73 
                 3456 8931 12075 
               
               
                 74 
                 13342 15273 20351 
               
               
                 75 
                 9138 13352 20798 
               
               
                 76 
                 7031 7626 14081 
               
               
                 77 
                 4280 4507 15617 
               
               
                 78 
                 4170 10569 14335 
               
               
                 79 
                 3839 7514 16578 
               
               
                 80 
                 4688 12815 18782 
               
               
                 81 
                 4861 7858 9435 
               
               
                 82 
                 605 5445 12912 
               
               
                 83 
                 2280 4734 7311 
               
               
                 84 
                 6668 8128 12638 
               
               
                 85 
                 3733 10621 19534 
               
               
                 86 
                 13933 18316 19341 
               
               
                 87 
                 1786 3037 21566 
               
               
                 88 
                 2202 13239 16432 
               
               
                 89 
                 4882 5808 9300 
               
               
                 90 
                 4580 8484 16754 
               
               
                 91 
                 14630 17502 18269 
               
               
                 92 
                 6889 11119 12447 
               
               
                 93 
                 8162 9078 16330 
               
               
                 94 
                 6538 17851 18100 
               
               
                 95 
                 17763 19793 20816 
               
               
                 96 
                 2183 11907 17567 
               
               
                 97 
                 6640 14428 15175 
               
               
                 98 
                 877 12035 14081 
               
               
                 99 
                 1336 6468 12328 
               
               
                 100 
                 5948 9146 12003 
               
               
                 101 
                 3782 5699 12445 
               
               
                 102 
                 1770 7946 8244 
               
               
                 103 
                 7384 12639 14989 
               
               
                 104 
                 1469 11586 20959 
               
               
                 105 
                 7943 10450 15907 
               
               
                 106 
                 5005 8153 10035 
               
               
                 107 
                 17750 18826 21513 
               
               
                 108 
                 4725 8041 10112 
               
               
                 109 
                 3837 16266 17376 
               
               
                 110 
                 11340 17361 17512 
               
               
                 111 
                 1269 4611 4774 
               
               
                 112 
                 2322 10813 16157 
               
               
                 113 
                 16752 16843 18959 
               
               
                 114 
                 70 4325 18753 
               
               
                 115 
                 3165 8153 15384 
               
               
                 116 
                 160 8045 16823 
               
               
                 117 
                 14112 16724 16792 
               
               
                 118 
                 4291 7667 18176 
               
               
                 119 
                 5943 19879 20721 
               
               
                   
               
            
           
         
       
     
     In another example, when the length N ldpc  of the LDPC codeword is 64800, the code rate is 10/15, and M is 360, the indexes of rows where 1 exists in the 0 th  column of the i th  column group of the information word submatrix  210  are defined as shown in Table 7 below. 
     
       
         
           
               
               
             
               
                 TABLE 7 
               
               
                   
               
               
                   
                 Index of row where 1 is located in the 0th column of the 
               
               
                 i 
                 ith column group 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 0 
                 316 1271 3692 9495 12147 12849 14928 16671 16938 
               
               
                   
                 17864 19108 20502 21097 21115 
               
               
                 1 
                 2341 2559 2643 2816 2865 5137 5331 7000 7523 8023 10439 
               
               
                   
                 10797 13208 15041 
               
               
                 2 
                 5556 6858 7677 10162 10207 11349 12321 12398 14787 
               
               
                   
                 15743 15859 15952 19313 20879 
               
               
                 3 
                 349 573 910 2702 3654 6214 9246 9353 10638 11772 14447 
               
               
                   
                 14953 16620 19888 
               
               
                 4 
                 204 1390 2887 3835 6230 6533 7443 7876 9299 10291 10896 
               
               
                   
                 13960 18287 20086 
               
               
                 5 
                 541 2429 2838 7144 8523 8637 10490 10585 11074 12074 
               
               
                   
                 15762 16812 17900 18548 
               
               
                 6 
                 733 1659 3838 5323 5805 7882 9429 10682 13697 16909 
               
               
                   
                 18846 19587 19592 20904 
               
               
                 7 
                 1134 2136 4631 4653 4718 5197 10410 11666 14996 15305 
               
               
                   
                 16048 17417 18960 20303 
               
               
                 8 
                 734 1001 1283 4959 10016 10176 10973 11578 12051 15550 
               
               
                   
                 15915 19022 19430 20121 
               
               
                 9 
                 745 4057 5855 9885 10594 10989 13156 13219 13351 13631 
               
               
                   
                 13685 14577 17713 20386 
               
               
                 10 
                 968 1446 2130 2502 3092 3787 5323 8104 8418 9998 11681 
               
               
                   
                 13972 17747 17929 
               
               
                 11 
                 3020 3857 5275 5786 6319 8608 11943 14062 17144 17752 
               
               
                   
                 18001 18453 19311 21414 
               
               
                 12 
                 709 747 1038 2181 5320 8292 10584 10859 13964 15009 
               
               
                   
                 15277 16953 20675 21509 
               
               
                 13 
                 1663 3247 5003 5760 7186 7360 10346 14211 14717 14792 
               
               
                   
                 15155 16128 17355 17970 
               
               
                 14 
                 516 578 1914 6147 9419 11148 11434 13289 13325 13332 
               
               
                   
                 19106 19257 20962 21556 
               
               
                 15 
                 5009 5632 6531 9430 9886 10621 11765 13969 16178 16413 
               
               
                   
                 18110 18249 20616 20759 
               
               
                 16 
                 457 2686 3318 4608 5620 5858 6480 7430 9602 12691 14664 
               
               
                   
                 18777 20152 20848 
               
               
                 17 
                 33 2877 5334 6851 7907 8654 10688 15401 16123 17942 
               
               
                   
                 17969 18747 18931 20224 
               
               
                 18 
                 87 897 7636 8663 11425 12288 12672 14199 16435 17615 
               
               
                   
                 17950 18953 19667 20281 
               
               
                 19 
                 1042 1832 2545 2719 2947 3672 3700 6249 6398 6833 11114 
               
               
                   
                 14283 17694 20477 
               
               
                 20 
                 326 488 2662 2880 3009 5357 6587 8882 11604 14374 18781 
               
               
                   
                 19051 19057 20508 
               
               
                 21 
                 854 1294 2436 2852 4903 6466 7761 9072 9564 10321 13638 
               
               
                   
                 15658 16946 19119 
               
               
                 22 
                 194 899 1711 2408 2786 5391 7108 8079 8716 11453 17303 
               
               
                   
                 19484 20989 21389 
               
               
                 23 
                 1631 3121 3994 5005 7810 8850 10315 10589 13407 17162 
               
               
                   
                 18624 18758 19311 20301 
               
               
                 24 
                 736 2424 4792 5600 6370 10061 16053 16775 18600 
               
               
                 25 
                 1254 8163 8876 9157 12141 14587 16545 17175 18191 
               
               
                 26 
                 388 6641 8974 10607 10716 14477 16825 17191 18400 
               
               
                 27 
                 5578 6082 6824 7360 7745 8655 11402 11665 12428 
               
               
                 28 
                 3603 8729 13463 14698 15210 19112 19550 20727 21052 
               
               
                 29 
                 48 1732 3805 5158 15442 16909 19854 21071 21579 
               
               
                 30 
                 11707 14014 21531 
               
               
                 31 
                 1542 4133 4925 
               
               
                 32 
                 10083 13505 21198 
               
               
                 33 
                 14300 15765 16752 
               
               
                 34 
                 778 1237 11215 
               
               
                 35 
                 1325 3199 14534 
               
               
                 36 
                 2007 14510 20599 
               
               
                 37 
                 1996 5881 16429 
               
               
                 38 
                 5111 15018 15980 
               
               
                 39 
                 4989 10681 12810 
               
               
                 40 
                 3763 10715 16515 
               
               
                 41 
                 2259 10080 15642 
               
               
                 42 
                 9032 11319 21305 
               
               
                 43 
                 3915 15213 20884 
               
               
                 44 
                 11150 15022 20201 
               
               
                 45 
                 1147 6749 19625 
               
               
                 46 
                 12139 12939 18870 
               
               
                 47 
                 3840 4634 10244 
               
               
                 48 
                 1018 10231 17720 
               
               
                 49 
                 2708 13056 13393 
               
               
                 50 
                 5781 11588 18888 
               
               
                 51 
                 1345 2036 5252 
               
               
                 52 
                 5908 8143 15141 
               
               
                 53 
                 1804 13693 18640 
               
               
                 54 
                 10433 13965 16950 
               
               
                 55 
                 9568 10122 15945 
               
               
                 56 
                 547 6722 14015 
               
               
                 57 
                 321 12844 14095 
               
               
                 58 
                 2632 10513 14936 
               
               
                 59 
                 6369 11995 20321 
               
               
                 60 
                 9920 19136 21529 
               
               
                 61 
                 1990 2726 10183 
               
               
                 62 
                 5763 12118 15467 
               
               
                 63 
                 503 10006 19564 
               
               
                 64 
                 9839 11942 19472 
               
               
                 65 
                 11205 13552 15389 
               
               
                 66 
                 8841 13797 19697 
               
               
                 67 
                 124 6053 18224 
               
               
                 68 
                 6477 14406 21146 
               
               
                 69 
                 1224 8027 16011 
               
               
                 70 
                 3046 4422 17717 
               
               
                 71 
                 739 12308 17760 
               
               
                 72 
                 4014 4130 7835 
               
               
                 73 
                 2266 5652 11981 
               
               
                 74 
                 2711 7970 18317 
               
               
                 75 
                 2196 15229 17217 
               
               
                 76 
                 8636 13302 16764 
               
               
                 77 
                 5612 15010 16657 
               
               
                 78 
                 615 1249 4639 
               
               
                 79 
                 3821 12073 18506 
               
               
                 80 
                 1066 16522 21536 
               
               
                 81 
                 11307 18363 19740 
               
               
                 82 
                 3240 8560 10391 
               
               
                 83 
                 3124 11424 20779 
               
               
                 84 
                 1604 8861 17394 
               
               
                 85 
                 2083 7400 8093 
               
               
                 86 
                 3218 7454 9155 
               
               
                 87 
                 9855 15998 20533 
               
               
                 88 
                 316 2850 20652 
               
               
                 89 
                 5583 9768 10333 
               
               
                 90 
                 7147 7713 18339 
               
               
                 91 
                 12607 17428 21418 
               
               
                 92 
                 14216 16954 18164 
               
               
                 93 
                 8477 15970 18488 
               
               
                 94 
                 1632 8032 9751 
               
               
                 95 
                 4573 9080 13507 
               
               
                 96 
                 11747 12441 13876 
               
               
                 97 
                 1183 15605 16675 
               
               
                 98 
                 4408 10264 17109 
               
               
                 99 
                 5495 7882 12150 
               
               
                 100 
                 1010 3763 5065 
               
               
                 101 
                 9828 18054 21599 
               
               
                 102 
                 6342 7353 15358 
               
               
                 103 
                 6362 9462 19999 
               
               
                 104 
                 7184 13693 17622 
               
               
                 105 
                 4343 4654 10995 
               
               
                 106 
                 7099 8466 18520 
               
               
                 107 
                 11505 14395 15138 
               
               
                 108 
                 6779 16691 18726 
               
               
                 109 
                 7146 12644 20196 
               
               
                 110 
                 5865 16728 19634 
               
               
                 111 
                 4657 8714 21246 
               
               
                 112 
                 4580 5279 18750 
               
               
                 113 
                 3767 6620 18905 
               
               
                 114 
                 9209 13093 17575 
               
               
                 115 
                 12486 15875 19791 
               
               
                 116 
                 8046 14636 17491 
               
               
                 117 
                 2120 4643 13206 
               
               
                 118 
                 6186 9675 12601 
               
               
                 119 
                 784 5770 21585 
               
               
                   
               
            
           
         
       
     
     In another example, when the length N ldpc  of the LDPC codeword is 64800, the code rate is 12/15, and M is 360, the indexes of rows where 1 exists in the 0 th  column of the i th  column group of the information word submatrix  210  are defined as shown in Table 8 below. 
     
       
         
           
               
               
             
               
                 TABLE 8 
               
               
                   
               
               
                   
                 Index of row where 1 is located in the 0th column of the 
               
               
                 i 
                 ith column group 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 0 
                 584 1472 1621 1867 3338 3568 3723 4185 5126 5889 
               
               
                   
                 7737 8632 8940 9725 
               
               
                 1 
                 221 445 590 3779 3835 6939 7743 8280 8448 8491 9367 
               
               
                   
                 10042 11242 12917 
               
               
                 2 
                 4662 4837 4900 5029 6449 6687 6751 8684 9936 11681 
               
               
                   
                 11811 11886 12089 12909 
               
               
                 3 
                 2418 3018 3647 4210 4473 7447 7502 9490 10067 11092 
               
               
                   
                 11139 11256 12201 12383 
               
               
                 4 
                 2591 2947 3349 3406 4417 4519 5176 6672 8498 8863 9201 
               
               
                   
                 11294 11376 12184 
               
               
                 5 
                 27 101 197 290 871 1727 3911 5411 6676 8701 9350 10310 
               
               
                   
                 10798 12439 
               
               
                 6 
                 1765 1897 2923 3584 3901 4048 6963 7054 7132 9165 
               
               
                   
                 10184 10824 11278 12669 
               
               
                 7 
                 2183 3740 4808 5217 5660 6375 6787 8219 8466 9037 
               
               
                   
                 10353 10583 11118 12762 
               
               
                 8 
                 73 1594 2146 2715 3501 3572 3639 3725 6959 7187 8406 
               
               
                   
                 10120 10507 10691 
               
               
                 9 
                 240 732 1215 2185 2788 2830 3499 3881 4197 4991 6425 
               
               
                   
                 7061 9756 10491 
               
               
                 10 
                 831 1568 1828 3424 4319 4516 4639 6018 9702 10203 
               
               
                   
                 10417 11240 11518 12458 
               
               
                 11 
                 2024 2970 3048 3638 3676 4152 5284 5779 5926 9426 9945 
               
               
                   
                 10873 11787 11837 
               
               
                 12 
                 1049 1218 1651 2328 3493 4363 5750 6483 7613 8782 9738 
               
               
                   
                 9803 11744 11937 
               
               
                 13 
                 1193 2060 2289 2964 3478 4592 4756 6709 7162 8231 
               
               
                   
                 8326 11140 11908 12243 
               
               
                 14 
                 978 2120 2439 3338 3850 4589 6567 8745 9656 9708 10161 
               
               
                   
                 10542 10711 12639 
               
               
                 15 
                 2403 2938 3117 3247 3711 5593 5844 5932 7801 10152 
               
               
                   
                 10226 11498 12162 12941 
               
               
                 16 
                 1781 2229 2276 2533 3582 3951 5279 5774 7930 9824 
               
               
                   
                 10920 11038 12340 12440 
               
               
                 17 
                 289 384 1980 2230 3464 3873 5958 8656 8942 9006 10175 
               
               
                   
                 11425 11745 12530 
               
               
                 18 
                 155 354 1090 1330 2002 2236 3559 3705 4922 5958 6576 
               
               
                   
                 8564 9972 12760 
               
               
                 19 
                 303 876 2059 2142 5244 5330 6644 7576 8614 9598 10410 
               
               
                   
                 10718 11033 12957 
               
               
                 20 
                 3449 3617 4408 4602 4727 6182 8835 8928 9372 9644 10237 
               
               
                   
                 10747 11655 12747 
               
               
                 21 
                 811 2565 2820 8677 8974 9632 11069 11548 11839 12107 
               
               
                   
                 12411 12695 12812 12890 
               
               
                 22 
                 972 4123 4943 6385 6449 7339 7477 8379 9177 9359 10074 
               
               
                   
                 11709 12552 12831 
               
               
                 23 
                 842 973 1541 2262 2905 5276 6758 7099 7894 8128 8325 
               
               
                   
                 8663 8875 10050 
               
               
                 24 
                 474 791 968 3902 4924 4965 5085 5908 6109 6329 7931 
               
               
                   
                 9038 9401 10568 
               
               
                 25 
                 1397 4461 4658 5911 6037 7127 7318 8678 8924 9000 9473 
               
               
                   
                 9602 10446 12692 
               
               
                 26 
                 1334 7571 12881 
               
               
                 27 
                 1393 1447 7972 
               
               
                 28 
                 633 1257 10597 
               
               
                 29 
                 4843 5102 11056 
               
               
                 30 
                 3294 8015 10513 
               
               
                 31 
                 1108 10374 10546 
               
               
                 32 
                 5353 7824 10111 
               
               
                 33 
                 3398 7674 8569 
               
               
                 34 
                 7719 9478 10503 
               
               
                 35 
                 2997 9418 9581 
               
               
                 36 
                 5777 6519 11229 
               
               
                 37 
                 1966 5214 9899 
               
               
                 38 
                 6 4088 5827 
               
               
                 39 
                 836 9248 9612 
               
               
                 40 
                 483 7229 7548 
               
               
                 41 
                 7865 8289 9804 
               
               
                 42 
                 2915 11098 11900 
               
               
                 43 
                 6180 7096 9481 
               
               
                 44 
                 1431 6786 8924 
               
               
                 45 
                 748 6757 8625 
               
               
                 46 
                 3312 4475 7204 
               
               
                 47 
                 1852 8958 11020 
               
               
                 48 
                 1915 2903 4006 
               
               
                 49 
                 6776 10886 12531 
               
               
                 50 
                 2594 9998 12742 
               
               
                 51 
                 159 2002 12079 
               
               
                 52 
                 853 3281 3762 
               
               
                 53 
                 5201 5798 6413 
               
               
                 54 
                 3882 6062 12047 
               
               
                 55 
                 4133 6775 9657 
               
               
                 56 
                 228 6874 11183 
               
               
                 57 
                 7433 10728 10864 
               
               
                 58 
                 7735 8073 12734 
               
               
                 59 
                 2844 4621 11779 
               
               
                 60 
                 3909 7103 12804 
               
               
                 61 
                 6002 9704 11060 
               
               
                 62 
                 5864 6856 7681 
               
               
                 63 
                 3652 5869 7605 
               
               
                 64 
                 2546 2657 4461 
               
               
                 65 
                 2423 4203 9111 
               
               
                 66 
                 244 1855 4691 
               
               
                 67 
                 1106 2178 6371 
               
               
                 68 
                 391 1617 10126 
               
               
                 69 
                 250 9259 10603 
               
               
                 70 
                 3435 4614 6924 
               
               
                 71 
                 1742 8045 9529 
               
               
                 72 
                 7667 8875 11451 
               
               
                 73 
                 4023 6108 6911 
               
               
                 74 
                 8621 10184 11650 
               
               
                 75 
                 6726 10861 12348 
               
               
                 76 
                 3228 6302 7388 
               
               
                 77 
                 1 1137 5358 
               
               
                 78 
                 381 2424 8537 
               
               
                 79 
                 3256 7508 10044 
               
               
                 80 
                 1980 2219 4569 
               
               
                 81 
                 2468 5699 10319 
               
               
                 82 
                 2803 3314 12808 
               
               
                 83 
                 8578 9642 11533 
               
               
                 84 
                 829 4585 7923 
               
               
                 85 
                 59 329 5575 
               
               
                 86 
                 1067 5709 6867 
               
               
                 87 
                 1175 4744 12219 
               
               
                 88 
                 109 2518 6756 
               
               
                 89 
                 2105 10626 11153 
               
               
                 90 
                 5192 10696 10749 
               
               
                 91 
                 6260 7641 8233 
               
               
                 92 
                 2998 3094 11214 
               
               
                 93 
                 3398 6466 11494 
               
               
                 94 
                 6574 10448 12160 
               
               
                 95 
                 2734 10755 12780 
               
               
                 96 
                 1028 7958 10825 
               
               
                 97 
                 8545 8602 10793 
               
               
                 98 
                 392 3398 11417 
               
               
                 99 
                 6639 9291 12571 
               
               
                 100 
                 1067 7919 8934 
               
               
                 101 
                 1064 2848 12753 
               
               
                 102 
                 6076 8656 12690 
               
               
                 103 
                 5504 6193 10171 
               
               
                 104 
                 1951 7156 7356 
               
               
                 105 
                 4389 4780 7889 
               
               
                 106 
                 526 4804 9141 
               
               
                 107 
                 1238 3648 10464 
               
               
                 108 
                 2587 5624 12557 
               
               
                 109 
                 5560 5903 11963 
               
               
                 110 
                 1134 2570 3297 
               
               
                 111 
                 10041 11583 12157 
               
               
                 112 
                 1263 9585 12912 
               
               
                 113 
                 3744 7898 10646 
               
               
                 114 
                 45 9074 10315 
               
               
                 115 
                 1051 6188 10038 
               
               
                 116 
                 2242 8394 12712 
               
               
                 117 
                 3598 9025 12651 
               
               
                 118 
                 2295 3540 5610 
               
               
                 119 
                 1914 4378 12423 
               
               
                 120 
                 1766 3635 12759 
               
               
                 121 
                 5177 9586 11143 
               
               
                 122 
                 943 3590 11649 
               
               
                 123 
                 4864 6905 10454 
               
               
                 124 
                 5852 6042 10421 
               
               
                 125 
                 6095 8285 12349 
               
               
                 126 
                 2070 7171 8563 
               
               
                 127 
                 718 12234 12716 
               
               
                 128 
                 512 10667 11353 
               
               
                 129 
                 3629 6485 7040 
               
               
                 130 
                 2880 8865 11466 
               
               
                 131 
                 4490 10220 11796 
               
               
                 132 
                 5440 8819 9103 
               
               
                 133 
                 5262 7543 12411 
               
               
                 134 
                 516 7779 10940 
               
               
                 135 
                 2515 5843 9202 
               
               
                 136 
                 4684 5994 10586 
               
               
                 137 
                 573 2270 3324 
               
               
                 138 
                 7870 8317 10322 
               
               
                 139 
                 6856 7638 12909 
               
               
                 140 
                 1583 7669 10781 
               
               
                 141 
                 8141 9085 12555 
               
               
                 142 
                 3903 5485 9992 
               
               
                 143 
                 4467 11998 12904 
               
               
                   
               
            
           
         
       
     
     In the above-described examples, the length of the LDPC codeword is 64800 and the code rate is 6/15, 8/15, 10/15, and 12/15. However, this is merely an example and the position of 1 in the information word submatrix  210  may be defined variously when the length of the LDPC codeword is 16200 or the code rate has different values. 
     According to an exemplary embodiment, even when the order of numbers in a sequence corresponding to the i th  column group of the parity check matrix  200  as shown in the above-described Tables 4 to 8 is changed, the changed parity check matrix is a parity check matrix used for the same code. Therefore, a case in which the order of numbers in the sequence corresponding to the i th  column group in Tables 4 to 8 is changed is covered by the inventive concept. 
     According to an exemplary embodiment, even when the arrangement order of sequences corresponding to each column group is changed in Tables 4 to 8, cycle characteristics on a graph of a code and algebraic characteristics such as degree distribution are not changed. Therefore, a case in which the arrangement order of the sequences shown in Tables 4 to 8 is changed is also covered by the inventive concept. 
     In addition, even when a multiple of Q ldpc  is equally added to all sequences corresponding to a certain column group in Tables 4 to 8, the cycle characteristics on the graph of the code or the algebraic characteristics such as degree distribution are not changed. Therefore, a result of equally adding a multiple of Q ldpc  to the sequences shown in Tables 4 to 8 is also covered by the inventive concept. However, it should be noted that, when the resulting value obtained by adding the multiple of Q ldpc  to a given sequence is greater than or equal to (N ldpc −K ldpc ), a value obtained by applying a modulo operation for (N ldpc −K ldpc ) to the resulting value should be applied instead. 
     Once positions of the rows where 1 exists in the 0 th  column of the i th  column group of the information word submatrix  210  are defined as shown in Tables 4 to 8, positions of rows where 1 exists in another column of each column group may be defined since the positions of the rows where 1 exists in the 0 th  column are cyclic-shifted by Q ldpc  in the next column. 
     For example, in the case of Table 4, in the 0 th  column of the 0 th  column group of the information word submatrix  210 , 1 exists in the 1606 th  row, 3402 nd  row, 4961 st  row, . . . . 
     In this case, since Q ldpc =(N ldpc −K ldpc )/M=(64800−25920)/360=108, the indexes of the rows where 1 is located in the 1st column of the 0 th  column group may be 1714(=1606+108), 3510(=3402+108), 5069(=4961+108), . . . , and the indexes of the rows where 1 is located in the 2 nd  column of the 0 th  column group may be 1822(=1714+108), 3618(=3510+108), 5177(=5069+108), . . . . 
     In the above-described method, the indexes of the rows where 1 is located in all rows of each column group may be defined. 
     The parity submatrix  220  of the parity check matrix  200  shown in  FIG.  2    may be defined as follows: 
     The parity submatrix  220  includes N ldpc −K ldpc  number of columns (that is, K ldpc   th  column to (N ldpc −1) th  column), and has a dual diagonal or staircase configuration. Accordingly, the degree of columns except the last column (that is, (N ldpc −1) th  column) from among the columns included in the parity submatrix  220  is 2, and the degree of the last column is 1. 
     As a result, the information word submatrix  210  of the parity check matrix  200  may be defined by Tables 4 to 8, and the parity submatrix  220  of the parity check matrix  200  may have a dual diagonal configuration. 
     When the columns and rows of the parity check matrix  200  shown in  FIG.  2    are permutated based on Equation 4 and Equation 5, the parity check matrix shown in  FIG.  2    may be changed to a parity check matrix  300  shown in  FIG.  3   . 
         Q   ldpc   ·i+j⇒M·j+i (0≤ i&lt;M, 0≤ j&lt;Q   ldpc )  (4)
 
         K   ldpc   +Q   ldpc   ·k+l⇒K   ldpc   +M·l+k (0≤ k&lt;M, 0≤ l&lt;Q   ldpc )  (5)
 
     The method for permutating based on Equation 4 and Equation 5 will be explained below. Since row permutation and column permutation apply the same principle, the row permutation will be explained by the way of an example. 
     In the case of the row permutation, regarding the X th  row, i and j satisfying X=Q ldpc ×i+j are calculated and the X th  row is permutated by assigning the calculated i and j to M×j+i. For example, regarding the 7 th  row, i and j satisfying 7=2×i+j are 3 and 1, respectively. Therefore, the 7 th  row is permutated to the 13 th  row (10×1+3=13). 
     When the row permutation and the column permutation are performed in the above-described method, the parity check matrix of  FIG.  2    may be converted into the parity check matrix of  FIG.  3   . 
     Referring to  FIG.  3   , the parity check matrix  300  is divided into a plurality of partial blocks, and a quasi-cyclic matrix of M×M corresponds to each partial block. 
     Accordingly, the parity check matrix  300  having the configuration of  FIG.  3    is formed of matrix units of M×M. That is, the submatrices of M×M are arranged in the plurality of partial blocks, constituting the parity check matrix  300 . 
     Since the parity check matrix  300  is formed of the quasi-cyclic matrices of M×M, M number of columns may be referred to as a column block and M number of rows may be referred to as a row block. Accordingly, the parity check matrix  300  having the configuration of  FIG.  3    is formed of N qc_column =N ldpc /M number of column blocks and N qc_row =N parity /M number of row blocks. 
     Hereinafter, the submatrix of M×M will be explained. 
     First, the (N qc_column −1) th  column block of the 0 th  row block has a form shown in Equation 6 presented below: 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     [ 
                     
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           … 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     As described above, A  330  is an M×M matrix, values of the 0 th  row and the (M−1) th  column are all “0”, and, regarding 0≤i≤(M−2), the (i+1) th  row of the i th  column is “1” and the other values are “0”. 
     Second, regarding 0≤i≤(N ldpc −K ldpc )/M−1 in the parity submatrix  320 , the row block of the (K ldpc /M+i) th  column block is configured by a unit matrix I M×M    340 . In addition, regarding 0≤i≤(N ldpc −K ldpc )/M−2, the (i+1) th  row block of the (K ldpc /M+i) th  column block is configured by a unit matrix I M×M    340 . 
     Third, a block  350  constituting the information word submatrix  310  may have a cyclic-shifted format of a cyclic matrix P, P α     ij   , or an added format of the cyclic-shifted matrix P α     ij    of the cyclic matrix P (or an overlapping format). 
     For example, a format in which the cyclic matrix P is cyclic-shifted to the right by 1 may be expressed by Equation 7 presented below: 
     
       
         
           
             
               
                 
                   P 
                   = 
                   
                     [ 
                     
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                             
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                             
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           1 
                         
                       
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                             
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The cyclic matrix P is a square matrix having an M×M size and is a matrix in which a weight of each of M number of rows is 1 and a weight of each of M number of columns is 1. When a ij  is 0, the cyclic matrix P, that is, P 0  indicates a unit matrix I M×M , and when a ij  is ∞, P ∞  is a zero matrix. 
     A submatrix existing where the i th  row block and the j th  column block intersect in the parity check matrix  300  of  FIG.  3    may be P α     ij   . Accordingly, i and j indicate the number of row blocks and the number of column blocks in the partial blocks corresponding to the information word. Accordingly, in the parity check matrix  300 , the total number of columns is N ldpc =M×N qc_column , and the total number of rows is N parity =M×N qc_row . That is, the parity check matrix  300  is formed of N qc_column  number of “column blocks” and N qc_row  number of “row blocks”. 
     Hereinafter, a method for performing LDPC encoding based on the parity check matrix  200  as shown in  FIG.  2    will be explained. An LDPC encoding process when the parity check matrix  200  is defined as shown in Table 4 by way of an example will be explained for the convenience of explanation. 
     First, when information word bits having a length of K ldpc  are [i 0 , i 1 , i 2 , . . . , i K     ldpc     −1 ], and parity bits having a length of N ldpc −K ldpc  are [p 0 , p 1 , p 2 , . . . p N     ldpc     −K     ldpc     −1 ], the LDPC encoding is performed by the following process. 
     Step 1) Parity bits are initialized as ‘0’. That is, p 0 =p 1 =p 2 = . . . =p N     ldpc     −K     ldpc     −1 =0. 
     Step 2) The 0 th  information word bit i 0  is accumulated in a parity bit having the address of the parity bit defined in the first row (that is, the row of i=0) of table 4 as the index of the parity bit. This may be expressed by Equation 8 presented below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           1606 
                         
                         = 
                         
                           
                             P 
                             1606 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           24533 
                         
                         = 
                         
                           
                             P 
                             24533 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           3402 
                         
                         = 
                         
                           
                             P 
                             3402 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           25376 
                         
                         = 
                         
                           
                             P 
                             25376 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           4961 
                         
                         = 
                         
                           
                             P 
                             4961 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           25667 
                         
                         = 
                         
                           
                             P 
                             25667 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           6751 
                         
                         = 
                         
                           
                             P 
                             6751 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           26836 
                         
                         = 
                         
                           
                             P 
                             26835 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           7132 
                         
                         = 
                         
                           
                             P 
                             7132 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           31799 
                         
                         = 
                         
                           
                             P 
                             31799 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           11516 
                         
                         = 
                         
                           
                             P 
                             11516 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           34173 
                         
                         = 
                         
                           
                             P 
                             34173 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           12300 
                         
                         = 
                         
                           
                             P 
                             12300 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           35462 
                         
                         = 
                         
                           
                             P 
                             35462 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           12482 
                         
                         = 
                         
                           
                             P 
                             12482 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           36153 
                         
                         = 
                         
                           
                             P 
                             36153 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           12592 
                         
                         = 
                         
                           
                             P 
                             12592 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           36740 
                         
                         = 
                         
                           
                             P 
                             36740 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           13342 
                         
                         = 
                         
                           
                             P 
                             13342 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37085 
                         
                         = 
                         
                           
                             P 
                             37085 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           13764 
                         
                         = 
                         
                           
                             P 
                             13764 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37152 
                         
                         = 
                         
                           
                             P 
                             37152 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           14123 
                         
                         = 
                         
                           
                             P 
                             14123 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37468 
                         
                         = 
                         
                           
                             P 
                             37468 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           21576 
                         
                         = 
                         
                           
                             P 
                             21576 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37658 
                         
                         = 
                         
                           
                             P 
                             37658 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           23946 
                         
                         = 
                         
                           
                             P 
                             23946 
                           
                           ⊕ 
                           
                             i 
                             0 
                           
                         
                       
                     
                     
                         
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Herein, i 0  is a 0 th  information word bit, p i  is an i th  parity bit, and ⊕ is a binary operation. According to the binary operation, 1⊕1 equals 0, 1⊕0 equals 1, 0⊕1 equals 1, 0⊕0 equals 0. 
     Step 3) The other 359 information word bits i m  (m=1, 2, . . . , 359) are accumulated in the parity bit. The other information word bits may belong to the same column group as that of i 0 . In this case, the address of the parity bit may be determined based on Equation 9 presented below: 
       ( x +( m  mod 360)× Q   ldpc )mod( N   ldpc   −K   ldpc )  (9)
 
     Herein, x is an address of a parity bit accumulator corresponding to the information word bit i 0 , and Q ldpc  is a size by which each column is cyclic-shifted in the information word submatrix, and may be 108 in the case of table 4. In addition, since m=1, 2, . . . , 359, (m mod 360) in Equation 9 may be regarded as m. 
     As a result, information word bits i m  (m=1, 2, . . . , 359) are accumulated in the parity bits having the address of the parity bit calculated based on Equation 9 as the index. For example, an operation as shown in Equation 10 presented below may be performed for the information word bit 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           1714 
                         
                         = 
                         
                           
                             P 
                             1714 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           24641 
                         
                         = 
                         
                           
                             P 
                             24641 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           3510 
                         
                         = 
                         
                           
                             P 
                             3510 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           25484 
                         
                         = 
                         
                           
                             P 
                             25484 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           5069 
                         
                         = 
                         
                           
                             P 
                             5069 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           25775 
                         
                         = 
                         
                           
                             P 
                             25775 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           6859 
                         
                         = 
                         
                           
                             P 
                             6859 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           26944 
                         
                         = 
                         
                           
                             P 
                             26944 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           7240 
                         
                         = 
                         
                           
                             P 
                             7240 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           31907 
                         
                         = 
                         
                           
                             P 
                             31907 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           11624 
                         
                         = 
                         
                           
                             P 
                             11624 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           34281 
                         
                         = 
                         
                           
                             P 
                             34281 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           12408 
                         
                         = 
                         
                           
                             P 
                             12408 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           35570 
                         
                         = 
                         
                           
                             P 
                             35570 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           12590 
                         
                         = 
                         
                           
                             P 
                             12590 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           36261 
                         
                         = 
                         
                           
                             P 
                             36261 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           12700 
                         
                         = 
                         
                           
                             P 
                             12700 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           36848 
                         
                         = 
                         
                           
                             P 
                             36848 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           13450 
                         
                         = 
                         
                           
                             P 
                             13450 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37193 
                         
                         = 
                         
                           
                             P 
                             37193 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           13872 
                         
                         = 
                         
                           
                             P 
                             13872 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37260 
                         
                         = 
                         
                           
                             P 
                             37260 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           14231 
                         
                         = 
                         
                           
                             P 
                             14231 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37576 
                         
                         = 
                         
                           
                             P 
                             37576 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           21684 
                         
                         = 
                         
                           
                             P 
                             21684 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           37766 
                         
                         = 
                         
                           
                             P 
                             37766 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           24054 
                         
                         = 
                         
                           
                             P 
                             24054 
                           
                           ⊕ 
                           
                             i 
                             1 
                           
                         
                       
                     
                     
                         
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Herein, i 1  is a 1 st  information word bit, p i  is an i th  parity bit, and ⊕ is a binary operation. According to the binary operation, 1δ1 equals 0, 1δ0 equals 1, 0δ1 equals 1, 0δ0 equals 0. 
     Step 4) The 360 th  information word bits i 360  is accumulated in a parity bit having the address of the parity bit defined in the 2 nd  row (that is, the row of i=1) of table 4 as the index of the parity bit. 
     Step 5) The other 359 information word bits belonging to the same group as that of the information word bit i 360  are accumulated in the parity bit. In this case, the address of the parity bit may be determined based on Equation 9. However, in this case, x is the address of the parity bit accumulator corresponding to the information word bit i 360 . 
     Step 6) Steps 4 and 5 described above are repeated for all of the column groups of table 4. 
     Step 7) As a result, a parity bit p i  is calculated based on Equation 11 presented below. In this case, i is initialized as 1. 
         p   i   =p   i   ⊕p   i-1   i= 1,2, . . . , N   ldpc   —K   ldpc −1  (11)
 
     In Equation 11, p i  is an i th  parity bit, N ldpc  is a length of an LDPC codeword, K ldpc  is a length of an information word of the LDPC codeword, and ⊕ is a binary operation. 
     As a result, the encoder  110  may calculate the parity bits according to the above-described method. 
     In another example, a parity check matrix according to an exemplary embodiment may have a configuration as shown in  FIG.  4   . 
     Referring to  FIG.  4   , the parity check matrix  400  may be formed of 5 matrices A, B, C, Z, and D. Hereinafter, the configuration of each matrix will be explained to explain the configuration of the parity check matrix  400 . 
     First, M 1 , M 2 , Q 1 , and Q 2 , which are parameter values related to the parity check matrix  400  as shown in  FIG.  4   , may be defined as shown in table 9 presented below according to the length and the code rate of the LDPC codeword. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 9 
               
             
            
               
                   
                   
               
               
                   
                 Sizes 
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 Rate 
                 Length 
                 M 1   
                 M 2   
                 Q 1   
                 Q 2   
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 1/15 
                 16200 
                 2520 
                 12600 
                 7 
                 35 
               
               
                   
                   
                 64800 
                 1080 
                 59400 
                 3 
                 165 
               
               
                   
                 2/15 
                 16200 
                 3240 
                 10800 
                 9 
                 30 
               
               
                   
                   
                 64800 
                 1800 
                 54360 
                 5 
                 151 
               
               
                   
                 3/15 
                 16200 
                 1080 
                 11880 
                 3 
                 33 
               
               
                   
                   
                 64800 
                 1800 
                 50040 
                 5 
                 139 
               
               
                   
                 4/15 
                 16200 
                 1080 
                 10800 
                 3 
                 30 
               
               
                   
                   
                 64800 
                 1800 
                 45720 
                 5 
                 127 
               
               
                   
                 5/15 
                 16200 
                 720 
                 10080 
                 2 
                 28 
               
               
                   
                   
                 64800 
                 1440 
                 41760 
                 4 
                 116 
               
               
                   
                 6/15 
                 16200 
                 1080 
                 8640 
                 3 
                 24 
               
               
                   
                   
                 64800 
                 1080 
                 37800 
                 3 
                 105 
               
               
                   
                   
               
            
           
         
       
     
     The matrix A is formed of K number of columns and g number of rows, and the matrix C is formed of K+g number of columns and N−K-g number of rows. Herein, K is a length of information word bits, and N is a length of the LDPC codeword. 
     Indexes of rows where 1 is located in the 0 th  column of the i th  column group in the matrix A and the matrix C may be defined based on table 10 according to the length and the code rate of the LDPC codeword. In this case, an interval at which a pattern of a column is repeated in each of the matrix A and the matrix C, that is, the number of columns belonging to the same group, may be 360. 
     For example, when the length N of the LDPC codeword is 64800 and the code rate is 6/15, the indexes of rows where 1 is located in the 0 th  column of the i th  column group in the matrix A and the matrix C are defined as shown in table 10 presented below: 
     
       
         
           
               
               
             
               
                 TABLE 10 
               
               
                   
               
               
                   
                 Index of row where 1 is located in the 0th column of the 
               
               
                 i 
                 ith column group 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 0 
                 71 276 856 6867 12964 17373 18159 26420 28460 28477 
               
               
                 1 
                 257 322 672 2533 5316 6578 9037 10231 13845 36497 
               
               
                 2 
                 233 765 904 1366 3875 13145 15409 18620 23910 30825 
               
               
                 3 
                 100 224 405 12776 13868 14787 16781 23886 29099 31419 
               
               
                 4 
                 23 496 891 2512 12589 14074 19392 20339 27658 28684 
               
               
                 5 
                 473 712 759 1283 4374 9898 12551 13814 24242 32728 
               
               
                 6 
                 511 567 815 11823 17106 17900 19338 22315 24396 26448 
               
               
                 7 
                 45 733 836 1923 3727 17468 25746 33806 35995 36657 
               
               
                 8 
                 17 487 675 2670 3922 5145 18009 23993 31073 36624 
               
               
                 9 
                 72 751 773 1937 17324 28512 30666 30934 31016 31849 
               
               
                 10 
                 257 343 594 14041 19141 24914 26864 28809 32055 34753 
               
               
                 11 
                 99 241 491 2650 9670 17433 17785 18988 22235 30742 
               
               
                 12 
                 198 299 655 6737 8304 10917 16092 19387 20755 37690 
               
               
                 13 
                 351 916 926 18151 21708 23216 30321 33578 34052 37949 
               
               
                 14 
                 54 332 373 2010 3332 5623 16301 34337 36451 37861 
               
               
                 15 
                 139 257 1068 11090 20289 29694 29732 32640 35133 36404 
               
               
                 16 
                 457 885 968 2115 4956 5422 5949 17570 26673 32387 
               
               
                 17 
                 137 570 619 5006 6099 7979 14429 16650 25443 32789 
               
               
                 18 
                 46 282 287 10258 18383 20258 27186 27494 28429 38266 
               
               
                 19 
                 445 486 1058 1868 9976 11294 20364 23695 30826 35330 
               
               
                 20 
                 134 900 931 12518 14544 17715 19623 21111 33868 34570 
               
               
                 21 
                 62 66 586 8020 20270 23831 31041 31965 32224 35189 
               
               
                 22 
                 174 290 784 6740 14673 17642 26286 27382 33447 34879 
               
               
                 23 
                 332 675 1033 1838 12004 15439 20765 31721 34225 38863 
               
               
                 24 
                 527 558 832 3867 6318 8317 10883 13466 18427 25377 
               
               
                 25 
                 431 780 1021 1112 2873 7675 13059 17793 20570 20771 
               
               
                 26 
                 339 536 1015 5725 6916 10846 14487 21156 28123 32614 
               
               
                 27 
                 456 830 1078 7511 11801 12362 12705 17401 28867 34032 
               
               
                 28 
                 222 538 989 5593 6022 8302 14008 23445 25127 29022 
               
               
                 29 
                 37 393 788 3025 7768 11367 22276 22761 28232 30394 
               
               
                 30 
                 234 257 1045 1307 2908 6337 26530 28142 34129 35997 
               
               
                 31 
                 35 46 978 9912 9978 12567 17843 24194 34887 35206 
               
               
                 32 
                 39 959 967 5027 10847 14657 18859 28075 28214 36325 
               
               
                 33 
                 275 477 823 11376 18073 28997 30521 31661 31941 32116 
               
               
                 34 
                 185 580 966 11733 12013 12760 13358 19372 32534 35504 
               
               
                 35 
                 760 891 1046 11150 20358 21638 29930 31014 33050 34840 
               
               
                 36 
                 360 389 1057 5316 5938 14186 16404 32445 34021 35722 
               
               
                 37 
                 306 344 679 5224 6674 10305 18753 25583 30585 36943 
               
               
                 38 
                 103 171 1016 8780 11741 12144 19470 20955 22495 27377 
               
               
                 39 
                 818 832 894 3883 14279 14497 22505 28129 28719 31246 
               
               
                 40 
                 215 411 760 5886 25612 28556 32213 32704 35901 36130 
               
               
                 41 
                 229 489 1067 2385 8587 20565 23431 28102 30147 32859 
               
               
                 42 
                 288 664 980 8138 8531 21676 23787 26708 28798 34490 
               
               
                 43 
                 89 552 847 6656 9889 23949 26226 27080 31236 35823 
               
               
                 44 
                 66 142 443 3339 3813 7977 14944 15464 19186 25983 
               
               
                 45 
                 605 876 931 16682 17669 25800 28220 33432 35738 37382 
               
               
                 46 
                 346 423 806 5669 7668 8789 9928 19724 24039 27893 
               
               
                 47 
                 48 460 1055 3512 7389 7549 20216 22180 28221 35437 
               
               
                 48 
                 187 636 824 1678 4508 13588 19683 21750 30311 33480 
               
               
                 49 
                 25 768 935 2856 8187 9052 21850 29941 33217 34293 
               
               
                 50 
                 349 624 716 2698 6395 6435 8974 10649 15932 17378 
               
               
                 51 
                 336 410 871 3582 9830 10885 13892 18027 19203 36659 
               
               
                 52 
                 176 849 1078 17302 19379 27964 28164 28720 32557 35495 
               
               
                 53 
                 234 890 1075 9431 9605 9700 10113 11332 12679 24268 
               
               
                 54 
                 516 638 733 8851 19871 22740 25791 30152 32659 35568 
               
               
                 55 
                 253 830 879 2086 16885 22952 23765 25389 34656 37293 
               
               
                 56 
                 94 954 998 2003 3369 6870 7321 29856 31373 34888 
               
               
                 57 
                 79 350 933 4853 6252 11932 12058 21631 24552 24876 
               
               
                 58 
                 246 647 778 4036 10391 10656 13194 32335 32360 34179 
               
               
                 59 
                 149 339 436 6971 8356 8715 11577 22376 28684 31249 
               
               
                 60 
                 36 149 220 6936 18408 19192 19288 23063 28411 35312 
               
               
                 61 
                 273 683 1042 6327 10011 18041 21704 29097 30791 31425 
               
               
                 62 
                 46 138 722 2701 10984 13002 19930 26625 28458 28965 
               
               
                 63 
                 12 1009 1040 1990 2930 5302 21215 22625 23011 29288 
               
               
                 64 
                 125 241 819 2245 3199 8415 21133 26786 27226 38838 
               
               
                 65 
                 45 476 1075 7393 15141 20414 31244 33336 35004 38391 
               
               
                 66 
                 432 578 667 1343 10466 11314 11507 23314 27720 34465 
               
               
                 67 
                 248 291 556 1971 3989 8992 18000 19998 23932 34652 
               
               
                 68 
                 68 694 837 2246 7472 7873 11078 12868 20937 35591 
               
               
                 69 
                 272 924 949 2030 4360 6203 9737 19705 19902 38039 
               
               
                 70 
                 21 314 979 2311 2632 4109 19527 21920 31413 34277 
               
               
                 71 
                 197 253 804 1249 4315 10021 14358 20559 27099 30525 
               
               
                 72 
                 9802 16164 17499 22378 22403 22704 26742 29908 
               
               
                 73 
                 9064 10904 12305 14057 16156 26000 32613 34536 
               
               
                 74 
                 5178 6319 10239 19343 25628 30577 31110 32291 
               
               
                   
               
            
           
         
       
     
     In the above-described example, the length of the LDPC codeword is 64800 and the code rate 6/15. However, this is merely an example and the indexes of rows where 1 is located in the 0 th  column of the i th  column group in the matrix A and the matrix C may be defined variously when the length of the LDPC codeword is 16200 or the code rate has different values. 
     Hereinafter, positions of rows where 1 exists in the matrix A and the matrix C will be explained with reference to table 10 by way of an example. 
     Since the length N of the LDPC codeword is 64800 and the code rate is 6/15 in table 10, M 1 =1080, M 2 =37800, Q 1 =3, and Q 2 =105 in the parity check matrix  400  defined by table 10 with reference to table 9. 
     Herein, Q 1  is a size by which columns of the same column group are cyclic-shifted in the matrix A, and Q 2  is a size by which columns of the same column group are cyclic-shifted in the matrix C. 
     In addition, Q 1 =M 1 /L, Q 2 =M 2 /L, M 1 =g, and M 2 =N−K−g, and L is an interval at which a pattern of a column is repeated in the matrix A and the matrix C, and for example, may be 360. 
     The index of the row where 1 is located in the matrix A and the matrix C may be determined based on the M 1  value. 
     For example, since M 1 =1080 in the case of table 10, the positions of the rows where 1 exists in the 0 th  column of the i th  column group in the matrix A may be determined based on values smaller than 1080 from among the index values of table 10, and the positions of the rows where 1 exists in the 0 th  column of the i th  column group in the matrix C may be determined based on values greater than or equal to 1080 from among the index values of table 10. 
     Specifically, in table 10, the sequence corresponding to the 0 th  column group is “71, 276, 856, 6867, 12964, 17373, 18159, 26420, 28460, 28477”. Accordingly, in the case of the 0 th  column of the 0 th  column group of the matrix A, 1 may be located in the 71 st  row, 276th row, and 856 th  row, and, in the case of the 0 th  column of the 0 th  column group of the matrix C, 1 may be located in the 6867 th  row, 12964 th  row, 17373 rd  row, 18159 th  row, 26420 th  row, 28460 th  row, and 28477 th  row. 
     Once positions of 1 in the 0 th  column of each column group of the matrix A are defined, positions of rows where 1 exists in another column of each column group may be defined by cyclic-shifting from the previous column by Q 1 . Once positions of 1 in the 0 th  column of each column group of the matrix C are defined, position of rows where 1 exists in another column of each column group may be defined by cyclic-shifting from the previous column by Q 2 . 
     In the above-described example, in the case of the 0 th  column of the 0 th  column group of the matrix A, 1 exists in the 71 st  row, 276 th  row, and 856 th  row. In this case, since Q 1 =3, the indexes of rows where 1 exists in the 1st column of the 0 th  column group are 74(=71+3), 279(=276+3), and 859(=856+3), and the index of rows where 1 exists in the 2 nd  column of the 0 th  column group are 77(=74+3), 282 (=279+3), and 862(=859+3). 
     In the case of the 0 th  column of the 0 th  column group of the matrix C, 1 exists in the 6867 th  row, 12964 th  row, 17373 rd  row, 18159 th  row, 26420 th  row, 28460 th  row, and 28477th row. In this case, since Q 2 =105, the index of rows where 1 exists in the 1st column of the 0 th  column group are 6972(=6867+105), 13069(=12964+105), 17478(=17373+105), 18264(=18159+105), 26525(=26420+105), 28565(=28460+105), 28582(=28477+105), and the indexes of rows where 1 exists in the 2 nd  column of the 0 th  column group are 7077(=6972+105), 13174(=13069+105), 17583(=17478+105), 18369(=18264+105), 26630(=26525+105), 28670(=28565+105), 28687(=28582+105). 
     In this method, the positions of rows where 1 exists in all column groups of the matrix A and the matrix C are defined. 
     The matrix B may have a dual diagonal configuration, the matrix D may have a diagonal configuration (that is, the matrix D is an identity matrix), and the matrix Z may be a zero matrix. 
     As a result, the parity check matrix  400  shown in  FIG.  4    may be defined by the matrices A, B, C, D, and Z having the above-described configurations. 
     Hereinafter, a method for performing LDPC encoding based on the parity check matrix  400  shown in  FIG.  4    will be explained. An LDPC encoding process when the parity check matrix  400  is defined as shown in Table 10 by way of an example will be explained for the convenience of explanation. 
     For example, when an information word block S=(s 0 , s 1 , . . . , s K−1 ) is LDPC-encoded, an LDPC codeword Λ=(λ 0 , λ 1 , . . . , λ N-1 )=(s 0 , s 1 , . . . , S K−1 , p 0 , p 1 , . . . , P M     1     +M     2     −1 ) including a parity bit P=(p 0 , p 1 , . . . , P M     1     +M     2     −1 ) may be generated. 
     M 1  and M 2  indicate the size of the matrix B having the dual diagonal configuration and the size of the matrix C having the diagonal configuration, respectively, and M 1 =g, M 2 =N−K−g. 
     A process of calculating a parity bit is as follows. In the following explanation, the parity check matrix  400  is defined as shown in table 10 by way of an example, for the convenience of explanation. 
     Step 1) λ and p are initialized as λ i =s i  (i=0, 1, . . . , K−1), p j =0 (j=0, 1, . . . , M 1 +M 2 −1). 
     Step 2) The 0 th  information word bit λ 0  is accumulated in the address of the parity bit defined in the first row (that is, the row of i=0) of table 10. This may be expressed by Equation 12 presented below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           71 
                         
                         = 
                         
                           
                             P 
                             71 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           17373 
                         
                         = 
                         
                           
                             P 
                             17373 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           276 
                         
                         = 
                         
                           
                             P 
                             276 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           18159 
                         
                         = 
                         
                           
                             P 
                             18159 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           856 
                         
                         = 
                         
                           
                             P 
                             856 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           26420 
                         
                         = 
                         
                           
                             P 
                             26420 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           6867 
                         
                         = 
                         
                           
                             P 
                             6867 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           28460 
                         
                         = 
                         
                           
                             P 
                             28460 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           12964 
                         
                         = 
                         
                           
                             P 
                             12964 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           28477 
                         
                         = 
                         
                           
                             P 
                             28477 
                           
                           ⊕ 
                           
                             λ 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Step 3) Regarding the next L−1 number of information word bits λ m  (m=1, 2, . . . , L−1), λ m  is accumulated in the parity bit address calculated based on Equation 13 presented below: 
       (χ+ m×Q   1 )mod  M   1 (if χ&lt; M   1 )
 
         M   1 +{(χ× M   1   +m×Q   2 )mod  M   2 }(if χ≤ M   1 )  (13)
 
     Herein, x is an address of a parity bit accumulator corresponding to the 0 th  information word bit λ 0 . 
     In addition, Q 1 =M 1 /L and Q 2 =M 2 /L. In addition, since the length N of the LDPC codeword is 64800 and the code rate is 6/15 in table 10, M 1 =1080, M 2 =37800, Q 1 =3, Q 2 =105, and L=360 with reference to table 9. 
     Accordingly, an operation as shown in Equation 14 presented below may be performed for the 1st information word bit λ 1 : 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           74 
                         
                         = 
                         
                           
                             P 
                             74 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           17478 
                         
                         = 
                         
                           
                             P 
                             17478 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           279 
                         
                         = 
                         
                           
                             P 
                             279 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           18264 
                         
                         = 
                         
                           
                             P 
                             18264 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           859 
                         
                         = 
                         
                           
                             P 
                             859 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           26525 
                         
                         = 
                         
                           
                             P 
                             26525 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           6972 
                         
                         = 
                         
                           
                             P 
                             6972 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           28565 
                         
                         = 
                         
                           
                             P 
                             28565 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           13069 
                         
                         = 
                         
                           
                             P 
                             13069 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           P 
                           28582 
                         
                         = 
                         
                           
                             P 
                             28582 
                           
                           ⊕ 
                           
                             λ 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Step 4) Since the same address of the parity bit as in the second row (that is the row of i=1) of table 10 is given to the Lth information word bit λ L , in a similar method to the above-described method, the address of the parity bit regarding the next L−1 number of information word bits λ m  (m=L+1, L+2, 2L−1) is calculated based on Equation 13. In this case, x is the address of the parity bit accumulator corresponding to the information word bit λ L , and may be obtained based on the second row of table 10. 
     Step 5) The above-described processes are repeated for L number of new information word bits of each group by considering new rows of table 10 as the address of the parity bit accumulator. 
     Step 6) After the above-described processes are repeated for the codeword bits λ 0  to λ K−1 , values regarding Equation 15 presented below are calculated in sequence from i=1: 
         P   i   =P   i   ⊕P   i-1 ( i= 1,2, . . . , M   1 −1)  (15)
 
     Step 7) Parity bits λ K  to λ K+M     1     −1  corresponding to the matrix B having the dual diagonal configuration are calculated based on Equation 16 presented below: 
       λ K+L×t+s   =P   Q     1     ×S+t (0≤ s&lt;L, 0≤ t&lt;Q   1 )  (16)
 
     Step 8) The address of the parity bit accumulator regarding L number of new codeword bits λ K  to λ K+M     1     −1  of each group is calculated based on table 10 and Equation 13. 
     Step 9) After the codeword bits λ K  to λ K+M     1     −1  are calculated, parity bits λ K+M     1    to λ K+M     1     +M     2     −1  corresponding to the matrix C having the diagonal configuration are calculated based on Equation 17 presented below: 
       λ K+M     1     +L×t+s   =p   M     1     +Q     2     ×S+t (0≤ s&lt;L, 0 t≤Q   2 )  (17)
 
     As a result, the parity bits may be calculated in the above-described method. 
     Referring back to  FIG.  1   , the encoder  110  may perform the LDPC encoding by using various code rates such as 3/15, 4/15, 5/15, 6/15, 7/15, 8/15, 9/15, 10/15, 11/15, 12/15, 13/15, etc. In addition, the encoder  110  may generate an LDPC codeword having various lengths such as 16200, 64800, etc., based on the length of the information word bits and the code rate. 
     In this case, the encoder  110  may perform the LDPC encoding by using the parity check matrix, and the parity check matrix is configured as shown in  FIGS.  2  to  4   . 
     In addition, the encoder  110  may perform Bose, Chaudhuri, Hocquenghem (BCH) encoding as well as LDPC encoding. To achieve this, the encoder  110  may further include a BCH encoder (not shown) to perform BCH encoding. 
     In this case, the encoder  110  may perform encoding in an order of BCH encoding and LDPC encoding. Specifically, the encoder  110  may add BCH parity bits to input bits by performing BCH encoding and LDPC-encodes the information word bits including the input bits and the BCH parity bits, thereby generating the LDPC codeword. 
     The interleaver  120  interleaves the LDPC codeword. That is, the interleaver  120  receives the LDPC codeword from the encoder  110 , and interleaves the LDPC codeword based on various interleaving rules. 
     In particular, the interleaver  120  may interleave the LDPC codeword such that a bit included in a predetermined bit group from among a plurality of bit groups constituting the LDPC codeword (that is, a plurality of groups or a plurality of blocks) is mapped onto a predetermined bit of a modulation symbol. Accordingly, the modulator  130  may map a bit included in a predetermined group from among the plurality of groups of the LDPC codeword onto a predetermined bit of the modulation symbol. 
     To achieve this, as shown in  FIG.  5   , the interleaver  120  may include a parity interleaver  121 , a group interleaver (or a group-wise interleaver  122 ), a group twist interleaver  123  and a block interleaver  124 . 
     The parity interleaver  121  interleaves the parity bits constituting the LDPC codeword. 
     Specifically, when the LDPC codeword is generated based on the parity check matrix  200  having the configuration of  FIG.  2   , the parity interleaver  121  may interleave only the parity bits of the LDPC codeword by using Equations 18 presented below: 
         u   i   =c   i  for 0≤ i≤K   ldpc , and
 
         u   K     ldpc     +M·t+s   =c   K     ldpc     +Q     ldpc     ·s+t  for 0≤ s≤M, 0≤ t≤Q   ldpc   (18),
 
     where M is an interval at which a pattern of a column group is repeated in the information word submatrix  210 , that is, the number of columns included in a column group (for example, M=360), and Q ldpc  is a size by which each column is cyclic-shifted in the information word submatrix  210 . That is, the parity interleaver  121  performs parity interleaving with respect to the LDPC codeword c=(c 0 , c 1 , . . . c N     ldpc     −1 ), and outputs U=(u 0 , u 1 , . . . , u N     ldpc     −1 ). 
     The LDPC codeword parity-interleaved in the above-described method may be configured such that a predetermined number of continuous bits of the LDPC codeword have similar decoding characteristics (cycle distribution, a degree of a column, etc.). 
     For example, the LDPC codeword may have the same characteristics on the basis of M number of continuous bits. Herein, M is an interval at which a pattern of a column group is repeated in the information word submatrix  210  and, for example, may be 360. 
     Specifically, a product of the LDPC codeword bits and the parity check matrix should be “0”. This means that a sum of products of the i th  LDPC codeword bit, c i (i=0, 1, . . . , N ldpc −1) and the column of the parity check matrix should be a “0” vector. Accordingly, the LDPC codeword bit may be regarded as corresponding to the i th  column of the parity check matrix. 
     In the case of the parity check matrix  200  of  FIG.  2   , M number of columns in the information word submatrix  210  belong to the same group and the information word submatrix  210  has the same characteristics on the basis of a column group (for example, the columns belonging to the same column group have the same degree distribution and the same cycle characteristic). 
     In this case, since M number of continuous bits in the information word bits correspond to the same column group of the information word submatrix  210 , the information word bits may be formed of M number of continuous bits having the same codeword characteristics. When the parity bits of the LDPC codeword are interleaved by the parity interleaver  121 , the parity bits of the LDPC codeword may be formed of M number of continuous bits having the same codeword characteristics. 
     However, regarding the LDPC codeword encoded based on the parity check matrix  300  of  FIG.  3    and the parity check matrix  400  of  FIG.  4   , parity interleaving may not be performed. In this case, the parity interleaver  121  may be omitted. 
     The group interleaver  122  may divide the parity-interleaved LDPC codeword into a plurality of bit groups and rearrange the order of the plurality of bit groups in bit group wise (or bit group unit). That is, the group interleaver  122  may interleave the plurality of bit groups in bit group wise. 
     To achieve this, the group interleaver  122  divides the parity-interleaved LDPC codeword into a plurality of bit groups by using Equation 19 or Equation 20 presented below. 
     
       
         
           
             
               
                 
                   
                     X 
                     j 
                   
                   = 
                   
                     
                       
                         { 
                         
                           
                             
                               
                                 u 
                                 k 
                               
                               | 
                               j 
                             
                             = 
                             
                               ⌊ 
                               
                                 k 
                                 360 
                               
                               ⌋ 
                             
                           
                           , 
                           
                             0 
                             ≤ 
                             k 
                             ≤ 
                             
                               N 
                               ldpc 
                             
                           
                         
                         } 
                       
                       ⁢ 
                           
                       for 
                       ⁢ 
                           
                       0 
                     
                     ≤ 
                     j 
                     &lt; 
                     
                       N 
                       group 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     X 
                     j 
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             u 
                             k 
                           
                           | 
                           
                             
                               360 
                               × 
                               j 
                             
                             ≤ 
                             k 
                             &lt; 
                             
                               360 
                               × 
                               
                                 ( 
                                 
                                   j 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                         , 
                         
                           0 
                           ≤ 
                           k 
                           &lt; 
                           
                             N 
                             ldpc 
                           
                         
                       
                       } 
                     
                     ⁢ 
                         
                     for 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
       
         
           
             0 
             ≤ 
             j 
             &lt; 
             
               N 
               group 
             
           
         
       
     
     where N group  is the total number of bit groups, N is the j th  bit group, and u k  is the k th  LDPC codeword bit input to the group interleaver  122 . In addition, 
     
       
         
           
             ⌊ 
             
               k 
               360 
             
             ⌋ 
           
         
       
     
     is the largest integer below k/360. 
     Since 360 in these equations indicates an example of the interval M at which the pattern of a column group is repeated in the information word submatrix, 360 in these equations can be changed to M. 
     The LDPC codeword which is divided into the plurality of bit groups may be as shown in  FIG.  6   . 
     Referring to  FIG.  6   , the LDPC codeword is divided into the plurality of bit groups and each bit group is formed of M number of continuous bits. When M is 360, each of the plurality of bit groups may be formed of 360 bits. Accordingly, the bit groups may be formed of bits corresponding to the column groups of the parity check matrix. 
     Specifically, since the LDPC codeword is divided by M number of continuous bits, K ldpc  number of information word bits are divided into (K ldpc /M) number of bit groups and N ldpc −K ldpc  number of parity bits are divided into (N ldpc −K ldpc )/M number of bit groups. Accordingly, the LDPC codeword may be divided into (N ldpc /M) number of bit groups in total. 
     For example, when M=360 and the length N ldpc  of the LDPC codeword is 16200, the number of groups N groups  constituting the LDPC codeword is 45(=16200/360), and, when M=360 and the length N ldpc  of the LDPC codeword is 64800, the number of bit groups N group  constituting the LDPC codeword is 180(=64800/360). 
     As described above, the group interleaver  122  divides the LDPC codeword such that M number of continuous bits are included in a same group since the LDPC codeword has the same codeword characteristics on the basis of M number of continuous bits. Accordingly, when the LDPC codeword is grouped by M number of continuous bits, the bits having the same codeword characteristics belong to the same group. 
     In the above-described example, the number of bits constituting each bit group is M. However, this is merely an example and the number of bits constituting each bit group is variable. 
     For example, the number of bits constituting each bit group may be an aliquot part of M. That is, the number of bits constituting each bit group may be an aliquot part of the number of columns constituting a column group of the information word submatrix of the parity check matrix. In this case, each bit group may be formed of aliquot part of M number of bits. For example, when the number of columns constituting a column group of the information word submatrix is 360, that is, M=360, the group interleaver  122  may divide the LDPC codeword into a plurality of bit groups such that the number of bits constituting each bit group is one of the aliquot parts of 360. 
     In the following explanation, the number of bits constituting a bit group is M by way of an example, for the convenience of explanation. 
     Thereafter, the group interleaver  122  interleaves the LDPC codeword in bit group wise. Specifically, the group interleaver  122  may group the LDPC codeword into the plurality of bit groups and rearrange the plurality of bit groups in bit group wise. That is, the group interleaver  122  changes positions of the plurality of bit groups constituting the LDPC codeword and rearranges the order of the plurality of bit groups constituting the LDPC codeword in bit group wise. 
     Herein, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise such that bit groups including bits mapped onto the same modulation symbol from among the plurality of bit groups are spaced apart from one another at predetermined intervals. 
     In this case, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by considering at least one of the number of rows and columns of the block interleaver  124 , the number of bit groups of the LDPC codeword, and the number of bits included in each bit group, such that bit groups including bits mapped onto the same modulation symbol are spaced apart from one another at predetermined intervals. 
     To achieve this, the group interleaver  122  may rearrange the order of the plurality of groups in bit group wise by using Equation 21 presented below: 
         Y   j   =X   π(j) (0≤ j&lt;N   group )  (21),
 
     where X j  is the j th  bit group before group interleaving, and Y j  is the j th  bit group after group interleaving. In addition, π(j) is a parameter indicating an interleaving order and is determined by at least one of a length of an LDPC codeword, a modulation method, and a code rate. That is, π(j) denotes a permutation order for group wise interleaving. 
     Accordingly, X π(j)  is a π(j) th  bit group before group interleaving, and Equation 21 means that the pre-interleaving π(j) th  bit group is interleaved into the j th  bit group. 
     According to an exemplary embodiment, an example of π(j) may be defined as in Tables 11 to 22 presented below. 
     In this case, π(j) is defined according to a length of an LPDC codeword and a code rate, and a parity check matrix is also defined according to a length of an LDPC codeword and a code rate. Accordingly, when LDPC encoding is performed based on a specific parity check matrix according to a length of an LDPC codeword and a code rate, the LDPC codeword may be interleaved in bit group wise based on π(j) satisfying the corresponding length of the LDPC codeword and code rate. 
     For example, when the encoder  110  performs LDPC encoding at a code rate of 6/15 to generate an LDPC codeword of a length of 64800, the group interleaver  122  may perform interleaving by using π(j) which is defined according to the length of the LDPC codeword of 16200 and the code rate of 6/15 in tables 11 to 22 presented below. 
     For example, when the length of the LDPC codeword is 64800, the code rate is 6/15, and the modulation method(or modulation format) is 16-Quadrature Amplitude Modulation (QAM), π(j) may be defined as in table 11 presented below. In particular, table 11 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 4. 
     
       
         
           
               
               
             
               
                   
                 TABLE 11 
               
               
                   
                   
               
             
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
               
               
                 block of 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
               
               
                 group- 
                 46 
                 47 
                 48 
                 49 
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
               
               
                 wise 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
               
               
                 interleaver 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
                 102 
                 103 
               
               
                 output 
                 115 
                 116 
                 117 
                 118 
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
               
               
                   
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
               
               
                   
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
                 170 
                 171 
                 172 
               
               
                 π(j)-th 
                 55 
                 146 
                 83 
                 52 
                 62 
                 176 
                 160 
                 68 
                 53 
                 56 
                 81 
                 97 
               
               
                 block of 
                 144 
                 57 
                 67 
                 116 
                 59 
                 70 
                 156 
                 172 
                 65 
                 149 
                 155 
                 82 
               
               
                 group- 
                 14 
                 37 
                 54 
                 44 
                 63 
                 43 
                 18 
                 47 
                 7 
                 25 
                 34 
                 29 
               
               
                 wise 
                 27 
                 38 
                 48 
                 33 
                 22 
                 49 
                 51 
                 60 
                 46 
                 21 
                 4 
                 3 
               
               
                 interleaver 
                 127 
                 101 
                 94 
                 115 
                 105 
                 31 
                 19 
                 177 
                 74 
                 10 
                 145 
                 162 
               
               
                 input 
                 78 
                 171 
                 8 
                 142 
                 178 
                 154 
                 85 
                 107 
                 75 
                 12 
                 9 
                 151 
               
               
                   
                 150 
                 110 
                 175 
                 166 
                 131 
                 119 
                 103 
                 139 
                 148 
                 157 
                 114 
                 147 
               
               
                   
                 11 
                 92 
                 165 
                 84 
                 168 
                 124 
                 169 
                 2 
                 130 
                 167 
                 153 
                 137 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 j-th 
                 12 
                 13 
                 14 
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
               
               
                   
                 block of 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
               
               
                   
                 group- 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
                 68 
               
               
                   
                 wise 
                 81 
                 82 
                 83 
                 84 
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
               
               
                   
                 interleaver 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
               
               
                   
                 output 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
                 136 
                 137 
               
               
                   
                   
                 150 
                 151 
                 152 
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
               
               
                   
                   
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                   
                 π(j)-th 
                 79 
                 113 
                 163 
                 61 
                 58 
                 69 
                 133 
                 108 
                 66 
                 71 
                 86 
               
               
                   
                 block of 
                 138 
                 136 
                 141 
                 111 
                 96 
                 170 
                 90 
                 140 
                 64 
                 159 
                 15 
               
               
                   
                 group- 
                 30 
                 26 
                 39 
                 16 
                 41 
                 45 
                 36 
                 0 
                 23 
                 32 
                 28 
               
               
                   
                 wise 
                 20 
                 13 
                 50 
                 35 
                 24 
                 40 
                 17 
                 42 
                 6 
                 112 
                 93 
               
               
                   
                 interleaver 
                 102 
                 120 
                 126 
                 95 
                 73 
                 152 
                 129 
                 174 
                 125 
                 72 
                 128 
               
               
                   
                 input 
                 77 
                 117 
                 109 
                 80 
                 106 
                 134 
                 98 
                 1 
                 122 
                 173 
                 161 
               
               
                   
                   
                 87 
                 158 
                 121 
                 164 
                 104 
                 89 
                 179 
                 123 
                 118 
                 99 
                 88 
               
               
                   
                   
                 143 
                 91 
                 100 
                 5 
                 76 
                 132 
                 135 
               
               
                   
                   
               
            
           
         
       
     
     In the case of Table 11, Equation 21 may be expressed as Y 0 =X π(0) =X 55 , Y 1 =X π(1) =X 146 , Y 2 =X π(2) =X 83 , . . . , Y 178 =X π(178) =X 132 , and Y 179 =X π(179) =X 135 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 55 th  bit group to the 0 th  bit group, the 146 th  bit group to the 1 st  bit group, the 83 rd  bit group to the 2 nd  bit group, . . . , the 132 nd  bit group to the 178 th  bit group, and the 135 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 8/15, and the modulation method is 16-QAM, π(j) may be defined as in table 12 presented below. In particular, table 12 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 5. 
     
       
         
           
               
               
             
               
                   
                 TABLE 12 
               
               
                   
                   
               
             
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
               
               
                 block of 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
               
               
                 group- 
                 46 
                 47 
                 48 
                 49 
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
               
               
                 wise 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
               
               
                 interleaver 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
                 102 
                 103 
               
               
                 output 
                 115 
                 116 
                 117 
                 118 
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
               
               
                   
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
               
               
                   
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
                 170 
                 171 
                 172 
               
               
                 π(j)-th 
                 58 
                 55 
                 111 
                 73 
                 150 
                 87 
                 110 
                 71 
                 172 
                 45 
                 41 
                 113 
               
               
                 block of 
                 118 
                 38 
                 89 
                 99 
                 85 
                 79 
                 75 
                 83 
                 68 
                 63 
                 40 
                 77 
               
               
                 group- 
                 92 
                 17 
                 65 
                 60 
                 25 
                 29 
                 23 
                 28 
                 61 
                 59 
                 74 
                 57 
               
               
                 wise 
                 33 
                 24 
                 91 
                 47 
                 10 
                 52 
                 50 
                 20 
                 31 
                 48 
                 0 
                 39 
               
               
                 interleaver 
                 7 
                 5 
                 11 
                 136 
                 35 
                 165 
                 8 
                 3 
                 106 
                 159 
                 138 
                 19 
               
               
                 input 
                 100 
                 142 
                 96 
                 34 
                 161 
                 170 
                 134 
                 156 
                 12 
                 154 
                 174 
                 2 
               
               
                   
                 116 
                 130 
                 177 
                 160 
                 129 
                 108 
                 125 
                 147 
                 97 
                 148 
                 162 
                 173 
               
               
                   
                 105 
                 126 
                 109 
                 119 
                 101 
                 121 
                 107 
                 131 
                 152 
                 164 
                 175 
                 151 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 j-th 
                 12 
                 13 
                 14 
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
               
               
                   
                 block of 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
               
               
                   
                 group- 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
                 68 
               
               
                   
                 wise 
                 81 
                 82 
                 83 
                 84 
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
               
               
                   
                 interleaver 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
               
               
                   
                 output 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
                 136 
                 137 
               
               
                   
                   
                 150 
                 151 
                 152 
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
               
               
                   
                   
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                   
                 π(j)-th 
                 115 
                 69 
                 120 
                 95 
                 88 
                 178 
                 123 
                 80 
                 66 
                 53 
                 82 
               
               
                   
                 block of 
                 117 
                 70 
                 81 
                 112 
                 43 
                 94 
                 37 
                 72 
                 46 
                 67 
                 51 
               
               
                   
                 group- 
                 49 
                 62 
                 78 
                 86 
                 30 
                 93 
                 42 
                 44 
                 90 
                 22 
                 26 
               
               
                   
                 wise 
                 27 
                 54 
                 15 
                 32 
                 76 
                 21 
                 36 
                 56 
                 84 
                 18 
                 169 
               
               
                   
                 interleaver 
                 4 
                 128 
                 168 
                 166 
                 144 
                 149 
                 1 
                 179 
                 141 
                 6 
                 13 
               
               
                   
                 input 
                 9 
                 145 
                 146 
                 14 
                 124 
                 16 
                 102 
                 133 
                 176 
                 132 
                 135 
               
               
                   
                   
                 163 
                 122 
                 104 
                 64 
                 143 
                 167 
                 103 
                 140 
                 158 
                 139 
                 98 
               
               
                   
                   
                 127 
                 114 
                 137 
                 157 
                 153 
                 171 
                 155 
               
               
                   
                   
               
            
           
         
       
     
     In the case of Table 12, Equation 21 may be expressed as Y 0 =X π(0) =X 58 , Y 1 =X π(1) =X 55 , Y 2 =X π(2) =X 111 , . . . , Y 178 =X π(178) =X 171 , and Y 179 =X π(179) =X 155 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 58 th  bit group to the 0 th  bit group, the 55 th  bit group to the 1 st  bit group, the 111 th  bit group to the 2 nd  bit group, . . . , the 171 st  bit group to the 178 th  bit group, and the 155 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 10/15, and the modulation method is 16-QAM, π(j) may be defined as in table 13 presented below. In particular, table 13 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 6. 
     
       
         
           
               
               
             
               
                   
                 TABLE 13 
               
               
                   
                   
               
             
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
               
               
                 block of 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
               
               
                 group- 
                 46 
                 47 
                 48 
                 49 
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
               
               
                 wise 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
               
               
                 interleaver 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
                 102 
                 103 
               
               
                 output 
                 115 
                 116 
                 117 
                 118 
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
               
               
                   
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
               
               
                   
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
                 170 
                 171 
                 172 
               
               
                 π(j)-th 
                 74 
                 53 
                 84 
                 109 
                 28 
                 103 
                 99 
                 1 
                 65 
                 41 
                 50 
                 12 
               
               
                 block of 
                 34 
                 77 
                 81 
                 58 
                 113 
                 44 
                 49 
                 45 
                 33 
                 40 
                 91 
                 17 
               
               
                 group- 
                 107 
                 86 
                 38 
                 72 
                 31 
                 83 
                 76 
                 61 
                 54 
                 73 
                 102 
                 42 
               
               
                 wise 
                 56 
                 105 
                 4 
                 119 
                 39 
                 32 
                 70 
                 7 
                 101 
                 114 
                 52 
                 47 
               
               
                 interleaver 
                 160 
                 78 
                 18 
                 59 
                 23 
                 64 
                 19 
                 79 
                 134 
                 63 
                 24 
                 20 
               
               
                 input 
                 147 
                 11 
                 161 
                 162 
                 123 
                 138 
                 173 
                 177 
                 100 
                 22 
                 87 
                 137 
               
               
                   
                 146 
                 144 
                 139 
                 176 
                 164 
                 155 
                 175 
                 170 
                 125 
                 171 
                 152 
                 154 
               
               
                   
                 165 
                 149 
                 136 
                 145 
                 130 
                 120 
                 150 
                 167 
                 126 
                 178 
                 140 
                 133 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 j-th 
                 12 
                 13 
                 14 
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
               
               
                   
                 block of 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
               
               
                   
                 group- 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
                 68 
               
               
                   
                 wise 
                 81 
                 82 
                 83 
                 84 
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
               
               
                   
                 interleaver 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
               
               
                   
                 output 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
                 136 
                 137 
               
               
                   
                   
                 150 
                 151 
                 152 
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
               
               
                   
                   
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                   
                 π(j)-th 
                 95 
                 115 
                 29 
                 48 
                 25 
                 35 
                 89 
                 62 
                 80 
                 71 
                 8 
               
               
                   
                 block of 
                 94 
                 82 
                 16 
                 46 
                 93 
                 104 
                 36 
                 92 
                 111 
                 57 
                 116 
               
               
                   
                 group- 
                 108 
                 85 
                 110 
                 97 
                 14 
                 30 
                 60 
                 27 
                 66 
                 118 
                 69 
               
               
                   
                 wise 
                 15 
                 117 
                 13 
                 55 
                 37 
                 96 
                 88 
                 112 
                 68 
                 106 
                 5 
               
               
                   
                 interleaver 
                 156 
                 3 
                 90 
                 2 
                 10 
                 75 
                 21 
                 98 
                 26 
                 9 
                 128 
               
               
                   
                 input 
                 132 
                 6 
                 169 
                 158 
                 0 
                 43 
                 51 
                 67 
                 168 
                 143 
                 131 
               
               
                   
                   
                 157 
                 127 
                 124 
                 129 
                 142 
                 135 
                 172 
                 151 
                 153 
                 122 
                 166 
               
               
                   
                   
                 121 
                 174 
                 141 
                 148 
                 179 
                 159 
                 163 
               
               
                   
                   
               
            
           
         
       
     
     In the case of Table 13, Equation 21 may be expressed as Y 0 =X π(0) =X 74 , Y 1 =X π(1) =X 53 , Y 2 =X π(2) =X 84 , Y 178 =X π(178) =X 159 , and Y 179 = π(179) =X 163 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 74 th  bit group to the 0 th  bit group, the 53 rd  bit group to the 1 st  bit group, the 84 th  bit group to the 2 nd  bit group, . . . , the 159 th  bit group to the 178 th  bit group, and the 163 rd  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 10/15, and the modulation method is 16-QAM, π(j) may be defined as in table 14 presented below. In particular, table 14 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 7. 
     
       
         
           
               
               
             
               
                   
                 TABLE 14 
               
               
                   
                   
               
             
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
               
               
                 block of 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
               
               
                 group- 
                 46 
                 47 
                 48 
                 49 
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
               
               
                 wise 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
               
               
                 interleaver 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
                 102 
                 103 
               
               
                 output 
                 115 
                 116 
                 117 
                 118 
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
               
               
                   
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
               
               
                   
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
                 170 
                 171 
                 172 
               
               
                 π(j)-th 
                 68 
                 71 
                 54 
                 19 
                 25 
                 21 
                 102 
                 32 
                 105 
                 29 
                 16 
                 79 
               
               
                 block of 
                 57 
                 28 
                 76 
                 31 
                 26 
                 96 
                 65 
                 119 
                 114 
                 109 
                 9 
                 125 
               
               
                 group- 
                 38 
                 77 
                 115 
                 56 
                 87 
                 113 
                 100 
                 75 
                 72 
                 60 
                 47 
                 92 
               
               
                 wise 
                 104 
                 8 
                 34 
                 0 
                 84 
                 111 
                 35 
                 30 
                 64 
                 55 
                 80 
                 40 
               
               
                 interleaver 
                 50 
                 33 
                 7 
                 175 
                 51 
                 131 
                 106 
                 134 
                 88 
                 140 
                 117 
                 132 
               
               
                 input 
                 41 
                 158 
                 5 
                 120 
                 12 
                 52 
                 99 
                 146 
                 144 
                 78 
                 155 
                 128 
               
               
                   
                 11 
                 27 
                 160 
                 178 
                 133 
                 142 
                 121 
                 168 
                 173 
                 123 
                 13 
                 15 
               
               
                   
                 152 
                 177 
                 137 
                 149 
                 167 
                 1 
                 14 
                 169 
                 124 
                 148 
                 164 
                 130 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 j-th 
                 12 
                 13 
                 14 
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
               
               
                   
                 block of 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
               
               
                   
                 group- 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
                 68 
               
               
                   
                 wise 
                 81 
                 82 
                 83 
                 84 
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
               
               
                   
                 interleaver 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
               
               
                   
                 output 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
                 136 
                 137 
               
               
                   
                   
                 150 
                 151 
                 152 
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
               
               
                   
                   
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                   
                 π(j)-th 
                 53 
                 82 
                 107 
                 91 
                 67 
                 94 
                 85 
                 48 
                 83 
                 58 
                 42 
               
               
                   
                 block of 
                 81 
                 43 
                 103 
                 93 
                 70 
                 46 
                 89 
                 112 
                 61 
                 45 
                 66 
               
               
                   
                 group- 
                 36 
                 98 
                 4 
                 59 
                 6 
                 44 
                 20 
                 86 
                 3 
                 73 
                 95 
               
               
                   
                 wise 
                 97 
                 101 
                 2 
                 69 
                 63 
                 74 
                 62 
                 118 
                 110 
                 159 
                 18 
               
               
                   
                 interleaver 
                 147 
                 153 
                 116 
                 161 
                 10 
                 39 
                 126 
                 136 
                 90 
                 37 
                 174 
               
               
                   
                 input 
                 165 
                 141 
                 179 
                 150 
                 157 
                 171 
                 143 
                 108 
                 170 
                 22 
                 49 
               
               
                   
                   
                 154 
                 127 
                 139 
                 151 
                 163 
                 172 
                 138 
                 176 
                 145 
                 129 
                 162 
               
               
                   
                   
                 17 
                 156 
                 122 
                 23 
                 166 
                 135 
                 24 
               
               
                   
                   
               
            
           
         
       
     
     In the case of Table 14, Equation 21 may be expressed as Y 0 =X π(0) =X 68 , Y 1 =X π(1) =X 71 , Y 2 =X π(2) =X 54 , Y 178 =X π(178) =X 135 , and Y 179 =X π(179) =X 24 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 68 th  bit group to the 0 th  bit group, the 71 st  bit group to the 1 st  bit group, the 54 th  bit group to the 2 nd  bit group, . . . , the 135 th  bit group to the 178 th  bit group, and the 24 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 12/15, and the modulation method is 16-QAM, π(j) may be defined as in table 15 presented below. In particular, table 15 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 8. 
     
       
         
           
               
               
             
               
                   
                 TABLE 15 
               
               
                   
                   
               
             
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
               
               
                 block of 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
                 34 
               
               
                 group- 
                 46 
                 47 
                 48 
                 49 
                 50 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
               
               
                 wise 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
               
               
                 interleaver 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
                 102 
                 103 
               
               
                 output 
                 115 
                 116 
                 117 
                 118 
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
               
               
                   
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
               
               
                   
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
                 170 
                 171 
                 172 
               
               
                 π(j)-th 
                 120 
                 32 
                 38 
                 113 
                 71 
                 31 
                 65 
                 109 
                 36 
                 106 
                 134 
                 66 
               
               
                 block of 
                 125 
                 107 
                 44 
                 99 
                 75 
                 64 
                 78 
                 51 
                 95 
                 88 
                 49 
                 60 
               
               
                 group- 
                 67 
                 92 
                 98 
                 42 
                 77 
                 28 
                 121 
                 87 
                 18 
                 21 
                 93 
                 72 
               
               
                 wise 
                 85 
                 104 
                 124 
                 52 
                 20 
                 118 
                 34 
                 5 
                 94 
                 41 
                 68 
                 80 
               
               
                 interleaver 
                 173 
                 165 
                 175 
                 166 
                 169 
                 174 
                 159 
                 148 
                 158 
                 155 
                 145 
                 178 
               
               
                 input 
                 163 
                 152 
                 146 
                 177 
                 103 
                 160 
                 147 
                 76 
                 172 
                 144 
                 150 
                 132 
               
               
                   
                 119 
                 114 
                 117 
                 115 
                 84 
                 57 
                 62 
                 13 
                 47 
                 24 
                 0 
                 7 
               
               
                   
                 102 
                 15 
                 55 
                 23 
                 25 
                 11 
                 56 
                 45 
                 58 
                 128 
                 43 
                 135 
               
               
                   
               
            
           
           
               
               
            
               
                   
                 Order of bit groups to be block interleaved 
               
               
                   
                 π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 j-th 
                 12 
                 13 
                 14 
                 15 
                 16 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
               
               
                   
                 block of 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
               
               
                   
                 group- 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
                 68 
               
               
                   
                 wise 
                 81 
                 82 
                 83 
                 84 
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
               
               
                   
                 interleaver 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
               
               
                   
                 output 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
                 136 
                 137 
               
               
                   
                   
                 150 
                 151 
                 152 
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
               
               
                   
                   
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                   
                 π(j)-th 
                 29 
                 86 
                 136 
                 108 
                 83 
                 70 
                 79 
                 81 
                 105 
                 48 
                 30 
               
               
                   
                 block of 
                 54 
                 122 
                 140 
                 137 
                 89 
                 74 
                 129 
                 82 
                 164 
                 59 
                 3 
               
               
                   
                 group- 
                 2 
                 142 
                 112 
                 9 
                 50 
                 8 
                 90 
                 139 
                 14 
                 97 
                 63 
               
               
                   
                 wise 
                 110 
                 12 
                 133 
                 131 
                 53 
                 116 
                 123 
                 96 
                 61 
                 111 
                 33 
               
               
                   
                 interleaver 
                 126 
                 100 
                 154 
                 156 
                 179 
                 157 
                 46 
                 149 
                 171 
                 37 
                 153 
               
               
                   
                 input 
                 176 
                 168 
                 167 
                 162 
                 170 
                 138 
                 151 
                 161 
                 40 
                 26 
                 130 
               
               
                   
                   
                 10 
                 69 
                 19 
                 127 
                 17 
                 16 
                 27 
                 91 
                 4 
                 73 
                 35 
               
               
                   
                   
                 1 
                 143 
                 141 
                 6 
                 22 
                 101 
                 39 
               
               
                   
                   
               
            
           
         
       
     
     In the case of Table 15, Equation 21 may be expressed as Y 0 =X π(0) =X 120 , Y 1 =X π(1) =X 32 , Y 2 =X π(2) =X 38 , Y 178 =X π(178) =X 101 , and Y 179 =X π(l79) =X 39 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 120 th  bit group to the 0 th  bit group, the 32 nd  bit group to the 1 st  bit group, the 38 th  bit group to the 2 nd  bit group, . . . , the 101 st  bit group to the 178 th  bit group, and the 39 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 6/15, and the modulation method is 16-QAM, π(j) may be defined as in table 16 presented below. In particular, table 16 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 10. 
     
       
         
           
               
               
             
               
                   
                 TABLE 16 
               
               
                   
                   
               
               
                   
                 Order of bit groups to be block interleaved π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
               
               
                 block of 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
               
               
                 group-wise 
                 34 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
               
               
                 interleaver 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
               
               
                 output 
                 68 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
               
               
                   
                 102 
                 103 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
               
               
                   
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
               
               
                   
                 136 
                 137 
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
                 150 
                 151 
                 152 
               
               
                   
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
               
               
                   
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                 π(j)-th 
                 163 
                 160 
                 138 
                 143 
                 85 
                 108 
                 128 
                 121 
                 91 
                 147 
                 140 
                 142 
                 131 
                 79 
                 109 
                 126 
                 111 
               
               
                 block of 
                 162 
                 144 
                 75 
                 110 
                 118 
                 97 
                 81 
                 168 
                 157 
                 167 
                 90 
                 103 
                 80 
                 150 
                 125 
                 105 
                 129 
               
               
                 group-wise 
                 146 
                 141 
                 152 
                 164 
                 130 
                 114 
                 123 
                 134 
                 107 
                 96 
                 173 
                 20 
                 44 
                 64 
                 19 
                 6 
                 8 
               
               
                 interleaver 
                 113 
                 45 
                 116 
                 11 
                 5 
                 63 
                 66 
                 84 
                 39 
                 10 
                 69 
                 88 
                 135 
                 25 
                 55 
                 54 
                 58 
               
               
                 input 
                 61 
                 33 
                 57 
                 59 
                 3 
                 16 
                 18 
                 155 
                 21 
                 56 
                 36 
                 29 
                 48 
                 62 
                 154 
                 43 
                 51 
               
               
                   
                 34 
                 0 
                 27 
                 12 
                 24 
                 17 
                 42 
                 145 
                 1 
                 38 
                 2 
                 28 
                 112 
                 31 
                 60 
                 179 
                 13 
               
               
                   
                 30 
                 50 
                 95 
                 14 
                 15 
                 9 
                 26 
                 71 
                 132 
                 40 
                 104 
                 89 
                 106 
                 46 
                 4 
                 166 
                 47 
               
               
                   
                 161 
                 174 
                 49 
                 23 
                 41 
                 139 
                 68 
                 52 
                 99 
                 149 
                 115 
                 101 
                 127 
                 22 
                 158 
                 7 
                 169 
               
               
                   
                 153 
                 122 
                 117 
                 159 
                 93 
                 100 
                 82 
                 151 
                 171 
                 67 
                 94 
                 136 
                 72 
                 73 
                 74 
                 70 
                 86 
               
               
                   
                 76 
                 137 
                 35 
                 37 
                 32 
                 177 
                 87 
                 170 
                 178 
                 77 
                 175 
                 120 
                 165 
                 53 
                 172 
                 133 
                 176 
               
               
                   
                 65 
                 83 
                 124 
                 92 
                 78 
                 119 
                 102 
                 156 
                 148 
                 98 
               
               
                   
               
            
           
         
       
     
     In the case of Table 16, Equation 21 may be expressed as Y 0 =X π(0) =X 163 , Y 1 =X π(1) =X 160 , Y 2 =X π(2) =X 138 , . . . , Y 178 =X π(178) =X 148 , and Y 179 =X π(l79) =X 98 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 163 rd  bit group to the 0 th  bit group, the 160 th  bit group to the 1 st  bit group, the 138th bit group to the 2 nd  bit group, . . . , the 148 th  bit group to the 178 th  bit group, and the 98 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 6/15, and the modulation method is 64-QAM, π(j) may be defined as in table 17 presented below. In particular, table 17 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 4. 
     
       
         
           
               
               
             
               
                   
                 TABLE 17 
               
               
                   
                   
               
               
                   
                 Order of bit groups to be block interleaved π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
               
               
                 block of 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
               
               
                 group-wise 
                 34 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
               
               
                 interleaver 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
               
               
                 output 
                 68 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
               
               
                   
                 102 
                 103 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
               
               
                   
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
               
               
                   
                 136 
                 137 
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
                 150 
                 151 
                 152 
               
               
                   
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
               
               
                   
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                 π(j)-th 
                 29 
                 17 
                 38 
                 37 
                 27 
                 43 
                 31 
                 35 
                 16 
                 46 
                 44 
                 9 
                 23 
                 1 
                 34 
                 45 
                 14 
               
               
                 block of 
                 18 
                 156 
                 19 
                 22 
                 40 
                 50 
                 24 
                 56 
                 49 
                 26 
                 42 
                 69 
                 47 
                 59 
                 61 
                 66 
                 52 
               
               
                 group-wise 
                 64 
                 65 
                 67 
                 54 
                 170 
                 68 
                 132 
                 51 
                 70 
                 41 
                 21 
                 5 
                 160 
                 7 
                 13 
                 55 
                 62 
               
               
                 interleaver 
                 53 
                 63 
                 58 
                 3 
                 167 
                 71 
                 57 
                 151 
                 60 
                 36 
                 25 
                 74 
                 39 
                 32 
                 72 
                 85 
                 86 
               
               
                 input 
                 107 
                 113 
                 48 
                 88 
                 2 
                 129 
                 137 
                 20 
                 73 
                 166 
                 75 
                 77 
                 142 
                 174 
                 15 
                 149 
                 28 
               
               
                   
                 145 
                 92 
                 169 
                 30 
                 133 
                 163 
                 119 
                 82 
                 176 
                 152 
                 134 
                 139 
                 148 
                 164 
                 99 
                 173 
                 104 
               
               
                   
                 83 
                 106 
                 112 
                 135 
                 153 
                 0 
                 128 
                 144 
                 98 
                 171 
                 94 
                 97 
                 143 
                 110 
                 118 
                 127 
                 84 
               
               
                   
                 79 
                 108 
                 126 
                 131 
                 93 
                 111 
                 91 
                 4 
                 125 
                 162 
                 157 
                 158 
                 109 
                 140 
                 123 
                 154 
                 150 
               
               
                   
                 80 
                 11 
                 12 
                 146 
                 96 
                 81 
                 165 
                 8 
                 89 
                 138 
                 105 
                 141 
                 103 
                 6 
                 100 
                 161 
                 172 
               
               
                   
                 78 
                 101 
                 115 
                 179 
                 147 
                 116 
                 136 
                 122 
                 87 
                 33 
                 130 
                 124 
                 175 
                 120 
                 90 
                 102 
                 10 
               
               
                   
                 114 
                 159 
                 76 
                 177 
                 178 
                 121 
                 168 
                 95 
                 117 
                 155 
               
               
                   
               
            
           
         
       
     
     In the case of Table 17, Equation 21 may be expressed as Y 0 =X π(0) =X 29 , Y 1 =X π(1) =X 17 , Y 2 =X π(2) =X 38 , . . . , Y 178 =X π(178) =X 117 , and Y 179 =X π(179) =X 155 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 29 th  bit group to the 0 th  bit group, the 17 th  bit group to the 1 st  bit group, the 38 th  bit group to the 2 nd  bit group, . . . , the 117 th  bit group to the 178 th  bit group, and the 155 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 8/15, and the modulation method is 64-QAM, π(j) may be defined as in table 18 presented below. In particular, table 18 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 5. 
     
       
         
           
               
               
             
               
                   
                 TABLE 18 
               
               
                   
                   
               
               
                   
                 Order of bit groups to be block interleaved π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
               
               
                 block of 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
               
               
                 group-wise 
                 34 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
               
               
                 interleaver 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
               
               
                 output 
                 68 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
               
               
                   
                 102 
                 103 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
               
               
                   
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
               
               
                   
                 136 
                 137 
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
                 150 
                 151 
                 152 
               
               
                   
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
               
               
                   
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                 π(j)-th 
                 86 
                 71 
                 51 
                 48 
                 89 
                 94 
                 46 
                 81 
                 67 
                 49 
                 80 
                 37 
                 55 
                 61 
                 36 
                 57 
                 52 
               
               
                 block of 
                 92 
                 60 
                 82 
                 76 
                 72 
                 44 
                 42 
                 91 
                 62 
                 50 
                 90 
                 40 
                 78 
                 53 
                 58 
                 47 
                 85 
               
               
                 group-wise 
                 70 
                 4 
                 69 
                 43 
                 54 
                 84 
                 93 
                 38 
                 8 
                 64 
                 6 
                 18 
                 77 
                 95 
                 66 
                 59 
                 83 
               
               
                 interleaver 
                 73 
                 17 
                 87 
                 3 
                 75 
                 65 
                 88 
                 79 
                 14 
                 151 
                 117 
                 32 
                 22 
                 123 
                 30 
                 33 
                 162 
               
               
                 input 
                 144 
                 9 
                 121 
                 108 
                 139 
                 142 
                 24 
                 34 
                 20 
                 157 
                 159 
                 138 
                 143 
                 29 
                 140 
                 163 
                 150 
               
               
                   
                 175 
                 114 
                 31 
                 12 
                 35 
                 145 
                 28 
                 27 
                 26 
                 16 
                 98 
                 102 
                 103 
                 133 
                 161 
                 21 
                 25 
               
               
                   
                 107 
                 153 
                 45 
                 156 
                 23 
                 125 
                 141 
                 56 
                 166 
                 5 
                 1 
                 170 
                 119 
                 68 
                 134 
                 41 
                 74 
               
               
                   
                 179 
                 2 
                 129 
                 169 
                 101 
                 99 
                 109 
                 127 
                 168 
                 176 
                 11 
                 0 
                 122 
                 110 
                 113 
                 146 
                 132 
               
               
                   
                 165 
                 19 
                 13 
                 39 
                 7 
                 164 
                 106 
                 172 
                 154 
                 149 
                 10 
                 173 
                 131 
                 167 
                 63 
                 147 
                 155 
               
               
                   
                 100 
                 171 
                 158 
                 160 
                 15 
                 178 
                 148 
                 152 
                 104 
                 124 
                 177 
                 97 
                 130 
                 118 
                 137 
                 111 
                 126 
               
               
                   
                 120 
                 105 
                 115 
                 136 
                 112 
                 96 
                 135 
                 116 
                 174 
                 128 
               
               
                   
               
            
           
         
       
     
     In the case of Table 18, Equation 21 may be expressed as Y 0 =X π(0) =X 86 , Y 1 =X π(1) =X 71 , Y 2 =X π(2) =X 51 , . . . , Y 178 =X π(178) =X 174 , and Y 179 =X π(l79) =X 128 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 86 th  bit group to the 0 th  bit group, the 71 st  bit group to the 1 st  bit group, the 51 st  bit group to the 2 nd  bit group, . . . , the 174 th  bit group to the 178 th  bit group, and the 128 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 10/15, and the modulation method is 64-QAM, π(j) may be defined as in table 19 presented below. In particular, table 19 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 6. 
     
       
         
           
               
               
             
               
                   
                 TABLE 19 
               
               
                   
                   
               
               
                   
                 Order of bit groups to be block interleaved π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
               
               
                 block of 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
               
               
                 group-wise 
                 34 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
               
               
                 interleaver 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
               
               
                 output 
                 68 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
               
               
                   
                 102 
                 103 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
               
               
                   
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
               
               
                   
                 136 
                 137 
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
                 150 
                 151 
                 152 
               
               
                   
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
               
               
                   
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                 π(j)-th 
                 73 
                 36 
                 21 
                 53 
                 37 
                 78 
                 102 
                 119 
                 82 
                 75 
                 40 
                 77 
                 104 
                 59 
                 58 
                 41 
                 18 
               
               
                 block of 
                 46 
                 45 
                 93 
                 30 
                 49 
                 114 
                 79 
                 1 
                 97 
                 66 
                 33 
                 115 
                 112 
                 99 
                 107 
                 26 
                 39 
               
               
                 group-wise 
                 23 
                 70 
                 89 
                 116 
                 62 
                 55 
                 50 
                 96 
                 108 
                 57 
                 51 
                 86 
                 28 
                 88 
                 52 
                 69 
                 74 
               
               
                 interleaver 
                 113 
                 84 
                 109 
                 65 
                 101 
                 8 
                 111 
                 61 
                 44 
                 105 
                 83 
                 35 
                 130 
                 27 
                 106 
                 90 
                 92 
               
               
                 input 
                 6 
                 54 
                 7 
                 87 
                 38 
                 31 
                 85 
                 63 
                 117 
                 67 
                 110 
                 72 
                 94 
                 32 
                 118 
                 47 
                 48 
               
               
                   
                 68 
                 76 
                 60 
                 91 
                 64 
                 17 
                 142 
                 156 
                 24 
                 12 
                 42 
                 56 
                 4 
                 170 
                 172 
                 5 
                 16 
               
               
                   
                 34 
                 152 
                 29 
                 11 
                 20 
                 136 
                 158 
                 134 
                 43 
                 98 
                 141 
                 160 
                 95 
                 0 
                 154 
                 81 
                 169 
               
               
                   
                 71 
                 171 
                 162 
                 139 
                 175 
                 129 
                 25 
                 167 
                 9 
                 131 
                 123 
                 165 
                 140 
                 124 
                 178 
                 10 
                 14 
               
               
                   
                 145 
                 164 
                 3 
                 15 
                 143 
                 173 
                 176 
                 22 
                 161 
                 153 
                 2 
                 19 
                 151 
                 150 
                 174 
                 144 
                 157 
               
               
                   
                 163 
                 122 
                 147 
                 132 
                 137 
                 128 
                 159 
                 155 
                 127 
                 138 
                 103 
                 100 
                 120 
                 146 
                 168 
                 166 
                 148 
               
               
                   
                 13 
                 125 
                 177 
                 133 
                 126 
                 121 
                 179 
                 80 
                 149 
                 135 
               
               
                   
               
            
           
         
       
     
     In the case of Table 19, Equation 21 may be expressed as Y 0 =X π(0) =X 73 , Y 1 =X π(1) =X 36 , Y 2 =X π(2) =X 21 , . . . , Y 178 =X π(178) =X 149 , and Y 179 =X π(179) =X 135 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 73 rd  bit group to the 0 th  bit group, the 36 th  bit group to the 1 st  bit group, the 21 st  bit group to the 2 nd  bit group, . . . , the 149 th  bit group to the 178 th  bit group, and the 135 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 10/15, and the modulation method is 64-QAM, π(j) may be defined as in table 20 presented below. In particular, table 20 may be applied when LDPC encoding is performed ‘based on the parity check matrix defined by table 7. 
     
       
         
           
               
               
             
               
                   
                 TABLE 20 
               
               
                   
                   
               
               
                   
                 Order of bit groups to be block interleaved π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
               
               
                 block of 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
               
               
                 group-wise 
                 34 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
               
               
                 interleaver 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
               
               
                 output 
                 68 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
               
               
                   
                 102 
                 103 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
               
               
                   
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
               
               
                   
                 136 
                 137 
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
                 150 
                 151 
                 152 
               
               
                   
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
               
               
                   
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                 π(j)-th 
                 113 
                 115 
                 47 
                 111 
                 35 
                 84 
                 34 
                 83 
                 31 
                 88 
                 109 
                 86 
                 46 
                 177 
                 103 
                 57 
                 77 
               
               
                 block of 
                 73 
                 95 
                 150 
                 52 
                 107 
                 98 
                 43 
                 66 
                 55 
                 56 
                 49 
                 72 
                 118 
                 78 
                 27 
                 39 
                 97 
               
               
                 group-wise 
                 40 
                 6 
                 75 
                 79 
                 68 
                 93 
                 59 
                 119 
                 20 
                 10 
                 51 
                 108 
                 65 
                 114 
                 69 
                 1 
                 116 
               
               
                 interleaver 
                 7 
                 30 
                 101 
                 50 
                 14 
                 38 
                 99 
                 71 
                 96 
                 128 
                 82 
                 92 
                 166 
                 60 
                 178 
                 117 
                 45 
               
               
                 input 
                 157 
                 48 
                 129 
                 67 
                 37 
                 94 
                 89 
                 53 
                 100 
                 54 
                 91 
                 173 
                 169 
                 63 
                 149 
                 104 
                 70 
               
               
                   
                 61 
                 102 
                 110 
                 124 
                 80 
                 29 
                 18 
                 19 
                 24 
                 0 
                 36 
                 22 
                 58 
                 62 
                 33 
                 64 
                 42 
               
               
                   
                 28 
                 8 
                 26 
                 112 
                 85 
                 74 
                 13 
                 21 
                 105 
                 44 
                 5 
                 87 
                 76 
                 106 
                 25 
                 81 
                 90 
               
               
                   
                 11 
                 3 
                 168 
                 121 
                 153 
                 140 
                 152 
                 135 
                 174 
                 23 
                 139 
                 12 
                 16 
                 9 
                 146 
                 164 
                 142 
               
               
                   
                 15 
                 147 
                 2 
                 161 
                 120 
                 133 
                 155 
                 123 
                 158 
                 167 
                 154 
                 148 
                 137 
                 160 
                 145 
                 159 
                 4 
               
               
                   
                 17 
                 126 
                 143 
                 151 
                 162 
                 156 
                 172 
                 171 
                 131 
                 41 
                 179 
                 132 
                 136 
                 32 
                 175 
                 163 
                 165 
               
               
                   
                 141 
                 138 
                 122 
                 127 
                 125 
                 144 
                 170 
                 134 
                 130 
                 176 
               
               
                   
               
            
           
         
       
     
     In the case of Table 20, Equation 21 may be expressed as Y 0 =X π(0) =X 113 , Y 1 =X π(1) =X 115 , Y 2 =X π(2) =X 47 , . . . , Y 178 =X π(178) =X 130 , and Y 179 =X π(l79) =X 176 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 113 th  bit group to the 0 th  bit group, the 115 th  bit group to the 1 st  bit group, the 47th bit group to the 2 nd  bit group, . . . , the 130 th  bit group to the 178 th  bit group, and the 176 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 12/15, and the modulation method is 64-QAM, π(j) may be defined as in table 21 presented below. In particular, table 21 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 8. 
     
       
         
           
               
               
             
               
                   
                 TABLE 21 
               
               
                   
                   
               
               
                   
                 Order of bit groups to be block interleaved π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
               
               
                 block of 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
               
               
                 group-wise 
                 34 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
               
               
                 interleaver 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
               
               
                 output 
                 68 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
               
               
                   
                 102 
                 103 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
               
               
                   
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
               
               
                   
                 136 
                 137 
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
                 150 
                 151 
                 152 
               
               
                   
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
               
               
                   
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                 π(j)-th 
                 83 
                 93 
                 94 
                 47 
                 55 
                 40 
                 38 
                 77 
                 110 
                 124 
                 87 
                 61 
                 102 
                 76 
                 33 
                 35 
                 92 
               
               
                 block of 
                 59 
                 74 
                 11 
                 138 
                 72 
                 67 
                 37 
                 10 
                 95 
                 139 
                 131 
                 44 
                 57 
                 97 
                 53 
                 142 
                 0 
               
               
                 group-wise 
                 136 
                 9 
                 143 
                 86 
                 100 
                 21 
                 15 
                 75 
                 62 
                 19 
                 65 
                 129 
                 101 
                 79 
                 22 
                 68 
                 73 
               
               
                 interleaver 
                 23 
                 18 
                 81 
                 98 
                 112 
                 8 
                 128 
                 103 
                 25 
                 43 
                 126 
                 54 
                 90 
                 28 
                 109 
                 46 
                 91 
               
               
                 input 
                 41 
                 82 
                 113 
                 134 
                 52 
                 105 
                 78 
                 27 
                 135 
                 96 
                 56 
                 140 
                 64 
                 66 
                 89 
                 34 
                 120 
               
               
                   
                 108 
                 63 
                 45 
                 69 
                 121 
                 88 
                 39 
                 29 
                 133 
                 106 
                 117 
                 127 
                 32 
                 42 
                 58 
                 71 
                 118 
               
               
                   
                 51 
                 84 
                 85 
                 80 
                 104 
                 132 
                 111 
                 30 
                 26 
                 48 
                 50 
                 31 
                 141 
                 116 
                 123 
                 114 
                 70 
               
               
                   
                 107 
                 178 
                 145 
                 173 
                 36 
                 144 
                 130 
                 176 
                 171 
                 175 
                 125 
                 99 
                 162 
                 159 
                 20 
                 164 
                 115 
               
               
                   
                 169 
                 172 
                 165 
                 161 
                 151 
                 119 
                 122 
                 152 
                 157 
                 4 
                 137 
                 148 
                 153 
                 170 
                 154 
                 166 
                 13 
               
               
                   
                 150 
                 16 
                 167 
                 174 
                 163 
                 49 
                 6 
                 168 
                 147 
                 146 
                 1 
                 149 
                 158 
                 179 
                 12 
                 5 
                 160 
               
               
                   
                 177 
                 60 
                 24 
                 156 
                 7 
                 155 
                 17 
                 3 
                 2 
                 14 
               
               
                   
               
            
           
         
       
     
     In the case of Table 21, Equation 21 may be expressed as Y 0 =X π(0) =X 83 , Y 1 =X π(1) =X 93 , Y 2 =X π(2) =X 94 , . . . , Y 178 =X π(178) =X 2 , and Y 179 =X π(179) =X 14 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 83 rd  bit group to the 0 th  bit group, the 93 rd  bit group to the 1 st  bit group, the 94 th  bit group to the 2 nd  bit group, . . . , the 2 nd  bit group to the 178 th  bit group, and the 14 th  bit group to the 179 th  bit group. 
     In another example, when the length of the LDPC codeword is 64800, the code rate is 6/15, and the modulation method is 64-QAM, π(j) may be defined as in table 22 presented below. In particular, table 22 may be applied when LDPC encoding is performed based on the parity check matrix defined by table 10. 
     
       
         
           
               
               
             
               
                   
                 TABLE 22 
               
               
                   
                   
               
               
                   
                 Order of bit groups to be block interleaved π(j) (0 ≤ j &lt; 180) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 j-th 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
                 16 
               
               
                 block of 
                 17 
                 18 
                 19 
                 20 
                 21 
                 22 
                 23 
                 24 
                 25 
                 26 
                 27 
                 28 
                 29 
                 30 
                 31 
                 32 
                 33 
               
               
                 group-wise 
                 34 
                 35 
                 36 
                 37 
                 38 
                 39 
                 40 
                 41 
                 42 
                 43 
                 44 
                 45 
                 46 
                 47 
                 48 
                 49 
                 50 
               
               
                 interleaver 
                 51 
                 52 
                 53 
                 54 
                 55 
                 56 
                 57 
                 58 
                 59 
                 60 
                 61 
                 62 
                 63 
                 64 
                 65 
                 66 
                 67 
               
               
                 output 
                 68 
                 69 
                 70 
                 71 
                 72 
                 73 
                 74 
                 75 
                 76 
                 77 
                 78 
                 79 
                 80 
                 81 
                 82 
                 83 
                 84 
               
               
                   
                 85 
                 86 
                 87 
                 88 
                 89 
                 90 
                 91 
                 92 
                 93 
                 94 
                 95 
                 96 
                 97 
                 98 
                 99 
                 100 
                 101 
               
               
                   
                 102 
                 103 
                 104 
                 105 
                 106 
                 107 
                 108 
                 109 
                 110 
                 111 
                 112 
                 113 
                 114 
                 115 
                 116 
                 117 
                 118 
               
               
                   
                 119 
                 120 
                 121 
                 122 
                 123 
                 124 
                 125 
                 126 
                 127 
                 128 
                 129 
                 130 
                 131 
                 132 
                 133 
                 134 
                 135 
               
               
                   
                 136 
                 137 
                 138 
                 139 
                 140 
                 141 
                 142 
                 143 
                 144 
                 145 
                 146 
                 147 
                 148 
                 149 
                 150 
                 151 
                 152 
               
               
                   
                 153 
                 154 
                 155 
                 156 
                 157 
                 158 
                 159 
                 160 
                 161 
                 162 
                 163 
                 164 
                 165 
                 166 
                 167 
                 168 
                 169 
               
               
                   
                 170 
                 171 
                 172 
                 173 
                 174 
                 175 
                 176 
                 177 
                 178 
                 179 
               
               
                 π(j)-th 
                 175 
                 177 
                 173 
                 125 
                 89 
                 37 
                 165 
                 85 
                 82 
                 34 
                 17 
                 162 
                 92 
                 161 
                 88 
                 137 
                 149 
               
               
                 block of 
                 115 
                 113 
                 172 
                 123 
                 43 
                 4 
                 152 
                 76 
                 143 
                 98 
                 139 
                 20 
                 150 
                 13 
                 52 
                 50 
                 25 
               
               
                 group-wise 
                 24 
                 153 
                 133 
                 122 
                 55 
                 10 
                 83 
                 18 
                 27 
                 51 
                 15 
                 8 
                 46 
                 164 
                 155 
                 81 
                 102 
               
               
                 interleaver 
                 53 
                 11 
                 21 
                 47 
                 61 
                 7 
                 126 
                 1 
                 57 
                 26 
                 148 
                 56 
                 171 
                 22 
                 166 
                 101 
                 67 
               
               
                 input 
                 119 
                 178 
                 121 
                 118 
                 96 
                 80 
                 99 
                 68 
                 167 
                 90 
                 62 
                 147 
                 36 
                 140 
                 103 
                 5 
                 87 
               
               
                   
                 157 
                 176 
                 59 
                 66 
                 3 
                 64 
                 91 
                 29 
                 71 
                 107 
                 77 
                 111 
                 42 
                 35 
                 38 
                 23 
                 100 
               
               
                   
                 45 
                 69 
                 40 
                 129 
                 33 
                 163 
                 49 
                 112 
                 145 
                 54 
                 105 
                 117 
                 0 
                 108 
                 94 
                 79 
                 12 
               
               
                   
                 19 
                 106 
                 60 
                 39 
                 104 
                 28 
                 2 
                 73 
                 97 
                 75 
                 154 
                 84 
                 58 
                 144 
                 95 
                 136 
                 16 
               
               
                   
                 170 
                 44 
                 151 
                 70 
                 63 
                 48 
                 128 
                 114 
                 41 
                 174 
                 116 
                 9 
                 86 
                 6 
                 141 
                 109 
                 78 
               
               
                   
                 127 
                 142 
                 159 
                 65 
                 130 
                 124 
                 93 
                 30 
                 110 
                 32 
                 160 
                 135 
                 132 
                 134 
                 14 
                 146 
                 74 
               
               
                   
                 120 
                 158 
                 138 
                 179 
                 169 
                 156 
                 131 
                 168 
                 31 
                 72 
               
               
                   
               
            
           
         
       
     
     In the case of Table 22, Equation 21 may be expressed as Y 0 =X π(0) =X 175 , Y 1 =X π(1) =X 177 , Y 2 =X π(2) =X 173 , . . . , Y 178 =X π(178) =X 31 , and Y 179 =X π(l79) =X 72 . Accordingly, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by changing the 175 th  bit group to the 0 th  bit group, the 177 th  bit group to the 1 st  bit group, the 173 rd  bit group to the 2 nd  bit group, . . . , the 31 st  bit group to the 178 th  bit group, and the 72 nd  bit group to the 179 th  bit group. 
     In the above-described examples, the length of the LDPC codeword is 64800 and the code rate is 6/15, 8/15, 10/15, and 12/15. However, this is merely an example and the interleaving pattern may be defined variously when the length of the LDPC codeword is 16200 or the code rate has different values. 
     As described above, the group interleaver  122  may rearrange the order of the plurality of bit groups in bit group wise by using Equation 21 and Tables 11 to 22. 
     “j-th block of Group-wise Interleaver output” in tables 11 to 22 indicates the j-th bit group output from the group interleaver  122  after interleaving, and “π(j)-th block of Group-wise Interleaver input” indicates the π(j)-th bit group input to the group interleaver  122 . 
     In addition, since the order of the bit groups constituting the LDPC codeword is rearranged by the group interleaver  122  in bit group wise, and then the bit groups are block-interleaved by the block interleaver  124 , which will be described below, “Order of bit groups to be block interleaved” is set forth in Tables 11 to 22 in relation to π(j). 
     The LDPC codeword which is group-interleaved in the above-described method is illustrated in  FIG.  7   . Comparing the LDPC codeword of  FIG.  7    and the LDPC codeword of  FIG.  6    before group interleaving, it can be seen that the order of the plurality of bit groups constituting the LDPC codeword is rearranged. 
     That is, as shown in  FIGS.  6  and  7   , the groups of the LDPC codeword are arranged in order of bit group X 0 , bit group X 1 , . . . , bit group X Ngroup-1  before being group-interleaved, and are arranged in an order of bit group Y 0 , bit group Y 1 , . . . , bit group Y Ngroup-1  after being group-interleaved. In this case, the order of arranging the bit groups by the group interleaving may be determined based on Tables 11 to 22. 
     The group twist interleaver  123  interleaves bits in a same group. That is, the group twist interleaver  123  may rearrange the order of the bits in the same bit group by changing the order of the bits in the same bit group. 
     In this case, the group twist interleaver  123  may rearrange the order of the bits in the same bit group by cyclic-shifting a predetermined number of bits from among the bits in the same bit group. 
     For example, as shown in  FIG.  8   , the group twist interleaver  123  may cyclic-shift bits included in the bit group Y 1  to the right by 1 bit. In this case, the bits located in the 0 th  position, the 1 st  position, the 2 nd  position, . . . , the 358 th  position, and the 359 th  position in the bit group Y 1  as shown in  FIG.  8    are cyclic-shifted to the right by 1 bit. As a result, the bit located in the 359 th  position before being cyclic-shifted is located in the front of the bit group Y 1  and the bits located in the 0 th  position, the 1 st  position, the 2 nd  position, . . . , the 358 th  position before being cyclic-shifted are shifted to the right serially by 1 bit and located. 
     In addition, the group twist interleaver  123  may rearrange the order of bits in each bit group by cyclic-shifting a different number of bits in each bit group. 
     For example, the group twist interleaver  123  may cyclic-shift the bits included in the bit group Y 1  to the right by 1 bit, and may cyclic-shift the bits included in the bit group Y 2  to the right by 3 bits. 
     However, the above-described group twist interleaver  123  may be omitted according to circumstances. 
     In addition, the group twist interleaver  123  is placed after the group interleaver  122  in the above-described example. However, this is merely an example. That is, the group twist interleaver  123  changes only the order of bits in a certain bit group and does not change the order of the bit groups. Therefore, the group twist interleaver  123  may be placed before the group interleaver  122 . 
     The block interleaver  124  interleaves the plurality of bit groups the order of which has been rearranged. Specifically, the block interleaver  124  may interleave the plurality of bit groups the order of which has been rearranged by the group interleaver  122  in bit group wise (or bits group unit). The block interleaver  124  is formed of a plurality of columns each including a plurality of rows and may interleave by dividing the plurality of rearranged bit groups based on a modulation order determined according to a modulation method. 
     In this case, the block interleaver  124  may interleave the plurality of bit groups the order of which has been rearranged by the group interleaver  122  in bit group wise. Specifically, the block interleaver  124  may interleave by dividing the plurality of rearranged bit groups according to a modulation order by using a first part and a second part. 
     Specifically, the block interleaver  124  interleaves by dividing each of the plurality of columns into a first part and a second part, writing the plurality of bit groups in the plurality of columns of the first part serially in bit group wise, dividing the bits of the other bit groups into groups (or sub bit groups) each including a predetermined number of bits based on the number of columns, and writing the sub bit groups in the plurality of columns of the second part serially. 
     Herein, the number of bit groups which are interleaved in bit group wise may be determined by at least one of the number of rows and columns constituting the block interleaver  124 , the number of bit groups and the number of bits included in each bit group. In other words, the block interleaver  124  may determine the bit groups which are to be interleaved in bit group wise considering at least one of the number of rows and columns constituting the block interleaver  124 , the number of bit groups and the number of bits included in each bit group, interleave the corresponding bit groups in bit group wise, and divide bits of the other bit groups into sub bit groups and interleave the sub bit groups. For example, the block interleaver  124  may interleave at least part of the plurality of bit groups in bit group wise using the first part, and divide bits of the other bit groups into sub bit groups and interleave the sub bit groups using the second part. 
     Meanwhile, interleaving bit groups in bit group wise means that the bits included in the same bit group are written in the same column. In other words, the block interleaver  124 , in case of bit groups which are interleaved in bit group wise, may not divide the bits included in the same bit groups and write the bits in the same column, and in case of bit groups which are not interleaved in bit group wise, may divide the bits in the bit groups and write the bits in different columns. 
     Accordingly, the number of rows constituting the first part is a multiple of the number of bits included in one bit group (for example, 360), and the number of rows constituting the second part may be less than the number of bits included in one bit group. 
     In addition, in all bit groups interleaved by the first part, the bits included in the same bit group are written and interleaved in the same column of the first part, and in at least one group interleaved by the second part, the bits are divided and written in at least two columns of the second part. 
     The specific interleaving method will be described later. 
     Meanwhile, the group twist interleaver  123  changes only the order of bits in the bit group and does not change the order of bit groups by interleaving. Accordingly, the order of the bit groups to be block-interleaved by the block interleaver  124 , that is, the order of the bit groups to be input to the block interleaver  124 , may be determined by the group interleaver  122 . Specifically, the order of the bit groups to be block-interleaved by the block interleaver  124  may be determined by π(j) defined in Tables 11 to 22. 
     As described above, the block interleaver  124  may interleave the plurality of bit groups the order of which has been rearranged in bit group wise by using the plurality of columns each including the plurality of rows. 
     In this case, the block interleaver  124  may interleave the LDPC codeword by dividing the plurality of columns into at least two parts. For example, the block interleaver  124  may divide each of the plurality of columns into the first part and the second part and interleave the plurality of bit groups constituting the LDPC codeword. 
     In this case, the block interleaver  124  may divide each of the plurality of columns into N number of parts (N is an integer greater than or equal to 2) according to whether the number of bit groups constituting the LDPC codeword is an integer multiple of the number of columns constituting the block interleaver  124 , and may perform interleaving. 
     When the number of bit groups constituting the LDPC codeword is an integer multiple of the number of columns constituting the block interleaver  124 , the block interleaver  124  may interleave the plurality of bit groups constituting the LDPC codeword in bit group wise without dividing each of the plurality of columns into parts. 
     Specifically, the block interleaver  124  may interleave by writing the plurality of bit groups of the LDPC codeword on each of the columns in bit group wise in a column direction, and reading each row of the plurality of columns in which the plurality of bit groups are written in bit group wise in a row direction. 
     In this case, the block interleaver  124  may interleave by writing bits included in a predetermined number of bit groups, which corresponds to a quotient obtained by dividing the number of bit groups of the LDPC codeword by the number of columns of the block interleaver  124 , on each of the plurality of columns serially in a column direction, and reading each row of the plurality of columns in which the bits are written in a row direction. 
     Hereinafter, the group located in the j th  position after being interleaved by the group interleaver  122  will be referred to as group Y j . 
     For example, it is assumed that the block interleaver  124  is formed of C number of columns each including R 1  number of rows. In addition, it is assumed that the LDPC codeword is formed of N group  number of bit groups and the number of bit groups N group  is a multiple of C. 
     In this case, when the quotient obtained by dividing N group  number of bit groups constituting the LDPC codeword by C number of columns constituting the block interleaver  124  is A (=N group /C) (A is an integer greater than 0), the block interleaver  124  may interleave by writing A (=N group /C) number of bit groups on each column serially in a column direction and reading bits written on each column in a row direction. 
     For example, as shown in  FIG.  9   , the block interleaver  124  writes bits included in bit group Y 0 , bit group Y 1 , . . . , bit group Y A−1 , . . . , in the 1 st  column from the 1 st  row to the R 1   th  row, writes bits included in bit group Y A , bit group Y A+1 , . . . , bit group Y 2A-1  in the 2nd column from the 1 st  row to the R 1   th  row, . . . , and writes bits included in bit group Y CA-A , bit group Y CA-A+1 , . . . , bit group Y CA-1  in the column C from the 1 st  row to the R 1   th  row. The block interleaver  124  may read the bits written in each row of the plurality of columns in a row direction. 
     Accordingly, the block interleaver  124  interleaves all bit groups constituting the LDPC codeword in bit group wise. 
     However, when the number of bit groups of the LDPC codeword is not an integer multiple of the number of columns of the block interleaver  124 , the block interleaver  124  may divide each column into 2 parts and interleave a part of the plurality of bit groups of the LDPC codeword in bit group wise, and divide bits of the other bit groups into sub bit groups and interleave the sub bit groups. In this case, the bits included in the other bit groups, that is, the bits included in the number of groups which correspond to the remainder when the number of bit groups constituting the LDPC codeword is divided by the number of columns are not interleaved in bit group wise, but interleaved by being divided according to the number of columns. 
     Specifically, the block interleaver  124  may interleave the LDPC codeword by dividing each of the plurality of columns into two parts. 
     In this case, the block interleaver  124  may divide the plurality of columns into the first part and the second part based on at least one of the number of columns of the block interleaver  124 , the number of bit groups of the LDPC codeword, and the number of bits of bit groups. 
     Here, each of the plurality of bit groups may be formed of 360 bits. In addition, the number of bit groups of the LDPC codeword is determined based on the length of the LDPC codeword and the number of bits included in the bit group. For example, when an LDPC codeword in the length of 16200 is divided such that each bit group has 360 bits, the LDPC codeword is divided into 45 bit groups. Alternatively, when an LDPC codeword in the length of 64800 is divided such that each bit group has 360 bits, the LDPC codeword may be divided into 180 bit groups. Further, the number of columns constituting the block interleaver  124  may be determined according to a modulation method. This will be explained in detail below. 
     Accordingly, the number of rows constituting each of the first part and the second part may be determined based on the number of columns constituting the block interleaver  124 , the number of bit groups constituting the LDPC codeword, and the number of bits constituting each of the plurality of bit groups. 
     Specifically, in each of the plurality of columns, the first part may be formed of as many rows as the number of bits included in at least one bit group which can be written in each column in bit group wise from among the plurality of bit groups of the LDPC codeword, according to the number of columns constituting the block interleaver  124 , the number of bit groups constituting the LDPC codeword, and the number of bits constituting each bit group. 
     In each of the plurality of columns, the second part may be formed of rows excluding as many rows as the number of bits included in at least some bit groups which can be written in each of the plurality of columns in bit group wise. Specifically, the number rows of the second part may be the same value as a quotient when the number of bits included in all bit groups excluding bit groups corresponding to the first part is divided by the number of columns constituting the block interleaver  124 . In other words, the number of rows of the second part may be the same value as a quotient when the number of bits included in the remaining bit groups which are not written in the first part from among bit groups constituting the LDPC codeword is divided by the number of columns. 
     That is, the block interleaver  124  may divide each of the plurality of columns into the first part including as many rows as the number of bits included in bit groups which can be written in each column in bit group wise, and the second part including the other rows. 
     Accordingly, the first part may be formed of as many rows as the number of bits included in bit groups, that is, as many rows as an integer multiple of M. However, since the number of codeword bits constituting each bit group may be an aliquot part of M as described above, the first part may be formed of as many rows as an integer multiple of the number of bits constituting each bit group. 
     In this case, the block interleaver  124  may interleave by writing and reading the LDPC codeword in the first part and the second part in the same method. 
     Specifically, the block interleaver  124  may interleave by writing the LDPC codeword in the plurality of columns constituting each of the first part and the second part in a column direction, and reading the plurality of columns constituting the first part and the second part in which the LDPC codeword is written in a row direction. 
     That is, the block interleaver  124  may interleave by writing the bits included in at least some bit groups which can be written in each of the plurality of columns in bit group wise in each of the plurality of columns of the first part serially, dividing the bits included in the other bit groups except the at least some bit groups and writing in each of the plurality of columns of the second part in a column direction, and reading the bits written in each of the plurality of columns constituting each of the first part and the second part in a row direction. 
     In this case, the block interleaver  124  may interleave by dividing the other bit groups except the at least some bit groups from among the plurality of bit groups based on the number of columns constituting the block interleaver  124 . 
     Specifically, the block interleaver  124  may interleave by dividing the bits included in the other bit groups by the number of a plurality of columns, writing each of the divided bits in each of a plurality of columns constituting the second part in a column direction, and reading the plurality of columns constituting the second part, where the divided bits are written, in a row direction. 
     That is, the block interleaver  124  may divide the bits included in the other bit groups except the bit groups written in the first part from among the plurality of bit groups of the LDPC codeword, that is, the bits in the number of bit groups which correspond to the remainder when the number of bit groups constituting the LDPC codeword is divided by the number of columns, by the number of columns, and may write the divided bits in each column of the second part serially in a column direction. 
     For example, it is assumed that the block interleaver  124  is formed of C number of columns each including R 1  number of rows. In addition, it is assumed that the LDPC codeword is formed of N group  number of bit groups, the number of bit groups N group  is not a multiple of C, and A×C+1=N group  (A is an integer greater than 0). In other words, it is assumed that when the number of bit groups constituting the LDPC codeword is divided by the number of columns, the quotient is A and the remainder is 1. 
     In this case, as shown in  FIGS.  10  and  11   , the block interleaver  124  may divide each column into a first part including R 1  number of rows and a second part including R 2  number of rows. In this case, R 1  may correspond to the number of bits included in bit groups which can be written in each column in bit group wise, and R 2  may be R 1  subtracted from the number of rows of each column. 
     That is, in the above-described example, the number of bit groups which can be written in each column in bit group wise is A, and the first part of each column may be formed of as many rows as the number of bits included in A number of bit groups, that is, may be formed of as many rows as A×M number. 
     In this case, the block interleaver  124  writes the bits included in the bit groups which can be written in each column in bit group wise, that is, A number of bit groups, in the first part of each column in the column direction. 
     That is, as shown in  FIGS.  10  and  11   , the block interleaver  124  writes the bits included in each of bit group Y 0 , bit group Y 1 , . . . , group Y A−1  in the 1 st  to R 1   th  rows of the first part of the 1 st  column, writes bits included in each of bit group Y A , bit group Y A+1 , . . . , bit group Y 2A4-1  in the 1 st  to R 1   th  rows of the first part of the 2 nd  column, . . . , writes bits included in each of bit group Y CA-A , bit group Y CA-A+1 , . . . , bit group Y CA-1  in the 1 st  to Rid rows of the first part of the column C. 
     As described above, the block interleaver  124  writes the bits included in the bit groups which can be written in each column in bit group wise in the first part of each column. 
     In other words, in the above exemplary embodiment, the bits included in each of bit group (Y 0 ), bit group (Y 1 ), . . . , bit group (Y A−1 ) may not be divided and all of the bits may be written in the first column, the bits included in each of bit group (Y A ), bit group (Y A+1 ), . . . , bit group (Y 2A-1 ) may not be divided and all of the bits may be written in the second column, . . . , and the bits included in each of bit group (Y CA-A ), bit group (Y CA-A+1 ), . . . , group (Y CA-1 ) may not be divided and all of the bits may be written in the C column. As such, all bit groups interleaved by the first part are written in the same column of the first part. 
     Thereafter, the block interleaver  124  divides bits included in the other bit groups except the bit groups written in the first part of each column from among the plurality of bit groups, and writes the bits in the second part of each column in the column direction. In this case, the block interleaver  124  divides the bits included in the other bit groups except the bit groups written in the first part of each column by the number of columns, so that the same number of bits are written in the second part of each column, and writes the divided bits in the second part of each column in the column direction. 
     In the above-described example, since A×C+1=N group when the bit groups constituting the LDPC codeword are written in the first part serially, the last bit group Y Ngroup-1  of the LDPC codeword is not written in the first part and remains. Accordingly, the block interleaver  124  divides the bits included in the bit group Y Ngroup-1  into C number of sub bit groups as shown in  FIG.  10   , and writes the divided bits (that is, the bits corresponding to the quotient when the bits included in the last group (Y Ngroup-1 ) are divided by C) in the second part of each column serially. 
     The bits divided based on the number of columns may be referred to as sub bit groups. In this case, each of the sub bit groups may be written in each column of the second part. That is, the bits included in the bit groups may be divided and may form the sub bit groups. 
     That is, the block interleaver  124  writes the bits in the 1 st  to R 2   th  rows of the second part of the 1 st  column, writes the bits in the 1 st  to R 2   th  rows of the second part of the 2 nd  column, . . . , and writes the bits in the 1 st  to R 2   th  rows of the second part of the column C. In this case, the block interleaver  124  may write the bits in the second part of each column in the column direction as shown in  FIG.  10   . 
     That is, in the second part, the bits constituting the bit group may not be written in the same column and may be written in the plurality of columns. In other words, in the above example, the last bit group (Y Ngroup-1 ) is formed of M number of bits and thus, the bits included in the last bit group (Y Ngroup-1 ) may be divided by M/C and written in each column. That is, the bits included in the last bit group (Y Ngroup-1 ) are divided by M/C, forming M/C number of sub bit groups, and each of the sub bit groups may be written in each column of the second part. 
     Accordingly, in at least one bit group which is interleaved by the second part, the bits included in the at least one bit group are divided and written in at least two columns constituting the second part. 
     In the above-described example, the block interleaver  124  writes the bits in the second part in the column direction. However, this is merely an example. That is, the block interleaver  124  may write the bits in the plurality of columns of the second part in the row direction. In this case, the block interleaver  124  may write the bits in the first part in the same method as described above. 
     Specifically, referring to  FIG.  11   , the block interleaver  124  writes the bits from the 1 st  row of the second part in the 1 st  column to the 1 st  row of the second part in the column C, writes the bits from the 2 nd  row of the second part in the 1st column to the 2 nd  row of the second part in the column C, . . . , etc., and writes the bits from the R 2   th  row of the second part in the 1 st  column to the R 2   th  row of the second part in the column C. 
     On the other hand, the block interleaver  124  reads the bits written in each row of each part serially in the row direction. That is, as shown in  FIGS.  10  and  11   , the block interleaver  124  reads the bits written in each row of the first part of the plurality of columns serially in the row direction, and reads the bits written in each row of the second part of the plurality of columns serially in the row direction. 
     Accordingly, the block interleaver  124  may interleave a part of the plurality of bit groups constituting the LDPC codeword in bit group wise, and divide and interleave some of the remaining bit groups. That is, the block interleaver  124  may interleave by writing the LDPC codeword constituting a predetermined number of bit groups from among the plurality of bit groups in the plurality of columns of the first part in bit group wise, dividing the bits of the other bit groups and writing the bits in each of the columns of the second part, and reading the plurality of columns of the first and second parts in the row direction. 
     As described above, the block interleaver  124  may interleave the plurality of bit groups in the methods described above with reference to  FIGS.  9  to  11   . 
     In particular, in the case of  FIG.  10   , the bits included in the bit group which does not belong to the first part are written in the second part in the column direction and read in the row direction. In view of this, the order of the bits included in the bit group which does not belong to the first part is rearranged. Since the bits included in the bit group which does not belong to the first part are interleaved as described above, bit error rate (BER)/frame error rate (FER) performance can be improved in comparison with a case in which such bits are not interleaved. 
     However, the bit group which does not belong to the first part may not be interleaved as shown in  FIG.  11   . That is, since the block interleaver  124  writes and reads the bits included in the group which does not belong to the first part in and from the second part in the row direction, the order of the bits included in the group which does not belong to the first part is not changed and the bits are output to the modulator  130  serially. In this case, the bits included in the group which does not belong to the first part may be output serially and mapped onto a modulation symbol. 
     In  FIGS.  10  and  11   , the last single bit group of the plurality of bit groups is written in the second part. However, this is merely an example. The number of bit groups written in the second part may vary according to the total number of bit groups of the LDPC codeword, the number of columns and rows, the number of transmission antennas, etc. 
     The block interleaver  124  may have a configuration as shown in tables 23 and 24 presented below: 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 23 
               
             
            
               
                   
                   
               
               
                   
                 N ldpc  = 64800 
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 16 
                 64 
                 256 
                 1024 
                 4096 
               
               
                   
                 QPSK 
                 QAM 
                 QAM 
                 QAM 
                 QAM 
                 QAM 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 C 
                 2 
                 4 
                 6 
                 8 
                 10 
                 12 
               
               
                 R 1   
                 32400 
                 16200 
                 10800 
                 7920 
                 6480 
                 5400 
               
               
                 R 2   
                 0 
                 0 
                 0 
                 180 
                 0 
                 0 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                   
                 TABLE 24 
               
             
            
               
                   
                   
               
               
                   
                 N ldpc  = 16200 
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 16 
                 64 
                 256 
                 1024 
                 4096 
               
               
                   
                 QPSK 
                 QAM 
                 QAM 
                 QAM 
                 QAM 
                 QAM 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                   
                 C 
                 2 
                 4 
                 6 
                 8 
                 10 
                 12 
               
               
                   
                 R 1   
                 7920 
                 3960 
                 2520 
                 1800 
                 1440 
                 1080 
               
               
                   
                 R 2   
                 180 
                 90 
                 180 
                 225 
                 180 
                 270 
               
               
                   
                   
               
            
           
         
       
     
     Herein, C (or N C ) is the number of columns of the block interleaver  124 , R 1  is the number of rows constituting the first part in each column, and R 2  is the number of rows constituting the second part in each column. 
     Referring to Tables 23 and 24, the number of columns has the same value as a modulation order according to a modulation method, and each of a plurality of columns is formed of rows corresponding to the number of bits constituting the LDPC codeword divided by the number of a plurality of columns. 
     For example, when the length N ldpc  of the LDPC codeword is 64800 and the modulation method is 16-QAM, the block interleaver  124  is formed of 4 columns as the modulation order is 4 in the case of 16-QAM, and each column is formed of rows as many as R 1 +R 2 =16200(=64800/4). In another example, when the length N ldpc  of the LDPC codeword is 64800 and the modulation method is 64-QAM, the block interleaver  124  is formed of 6 columns as the modulation order is 6 in the case of 64-QAM, and each column is formed of rows as many as R 1 +R 2 =10800(=64800/6). 
     Meanwhile, referring to Tables 23 and 24, when the number of bit groups constituting an LDPC codeword is an integer multiple of the number of columns, the block interleaver  124  interleaves without dividing each column. Therefore, R 1  corresponds to the number of rows constituting each column, and R 2  is 0. In addition, when the number of bit groups constituting an LDPC codeword is not an integer multiple of the number of columns, the block interleaver  124  interleaves the groups by dividing each column into the first part formed of R 1  number of rows, and the second part formed of R 2  number of rows. 
     When the number of columns of the block interleaver  124  is equal to the number of bits constituting a modulation symbol, bits included in a same bit group are mapped onto a single bit of each modulation symbol as shown in Tables 23 and 24. 
     For example, when N ldpc =64800 and the modulation method is 16-QAM, the block interleaver  124  may be formed of four (4) columns each including 16200 rows. In this case, the bits included in each of the plurality of bit groups are written in the four (4) columns and the bits written in the same row in each column are output serially. In this case, since four (4) bits constitute a single modulation symbol in the modulation method of 16-QAM, bits included in the same bit group, that is, bits output from a single column, may be mapped onto a single bit of each modulation symbol. For example, bits included in a bit group written in the 1 st  column may be mapped onto the first bit of each modulation symbol. 
     In another example, when N ldpc =64800 and the modulation method is 64-QAM, the block interleaver  124  may be formed of six (6) columns each including 10800 rows. In this case, the bits included in each of the plurality of bit groups are written in the six (6) columns and the bits written in the same row in each column are output serially. In this case, since six (6) bits constitute a single modulation symbol in the modulation method of 64-QAM, bits included in the same bit group, that is, bits output from a single column, may be mapped onto a single bit of each modulation symbol. For example, bits included in a bit group written in the 1 st  column may be mapped onto the first bit of each modulation symbol. 
     Referring to Tables 23 and 24, the total number of rows of the block interleaver  124 , that is, R 1 +R 2 , is N ldpc /C. 
     In addition, the number of rows of the first part, R 1 , is an integer multiple of the number of bits included in each group, M (e.g., M=360), and maybe expressed as └N group /C┘×M, and the number of rows of the second part, R 2 , may be N ldpc /C−R 1 . Herein, └N group /C┘ is the largest integer below N group /C. Since R 1  is an integer multiple of the number of bits included in each group, M, bits may be written in R 1  in bit groups wise. 
     In addition, when the number of bit groups of the LDPC codeword is not a multiple of the number of columns, it can be seen from Tables 23 and 24 that the block interleaver  124  interleaves by dividing each column into two parts. 
     Specifically, the length of the LDPC codeword divided by the number of columns is the total number of rows included in the each column. In this case, when the number of bit groups of the LDPC codeword is a multiple of the number of columns, each column is not divided into two parts. However, when the number of bit groups of the LDPC codeword is not a multiple of the number of columns, each column is divided into two parts. 
     For example, it is assumed that the number of columns of the block interleaver  124  is identical to the number of bits constituting a modulation symbol, and an LDPC codeword is formed of 64800 bits as shown in Table 28. In this case, each bit group of the LDPC codeword is formed of 360 bits, and the LDPC codeword is formed of 64800/360(=180) bit groups. 
     When the modulation method is 16-QAM, the block interleaver  124  may be formed of four (4) columns and each column may have 64800/4(=16200) rows. 
     In this case, since the number of bit groups of the LDPC codeword divided by the number of columns is 180/4(=45), bits can be written in each column in bit group wise without dividing each column into two parts. That is, bits included in 45 bit groups which is the quotient when the number of bit groups constituting the LDPC codeword is divided by the number of columns, that is, 45×360(=16200) bits can be written in each column. 
     However, when the modulation method is 256-QAM, the block interleaver  124  may be formed of eight (8) columns and each column may have 64800/8(=8100) rows. 
     In this case, since the number of bit groups of the LDPC codeword divided by the number of columns is 180/8=22.5, the number of bit groups constituting the LDPC codeword is not an integer multiple of the number of columns. Accordingly, the block interleaver  124  divides each of the eight (8) columns into two parts to perform interleaving in bit group wise. 
     In this case, since the bits should be written in the first part of each column in bit group wise, the number of bit groups which can be written in the first part of each column in bit group wise is 22, which is the quotient when the number of bit groups constituting the LDPC codeword is divided by the number of columns, and accordingly, the first part of each column has 22×360(=7920) rows. Accordingly, 7920 bits included in 22 bit groups may be written in the first part of each column. 
     The second part of each column has rows which are the rows of the first part subtracted from the total rows of each column. Accordingly, the second part of each column includes 8100−7920(=180) rows. 
     In this case, the bits included in the other bit groups which have not been written in the first part are divided and written in the second part of each column. 
     Specifically, since 22×8(=176) bit groups are written in the first part, the number of bit groups to be written in the second part is 180−176 (=4) (for example, bit group Y 176 , bit group Y 177 , bit group Y 178 , and bit group Y 179  from among bit group Y 0 , bit group Y 1 , bit group Y 2 , . . . , bit group Y 178 , and bit group Y 179  constituting the LDPC codeword). 
     Accordingly, the block interleaver  124  may write the four (4) bit groups which have not been written in the first part and remains from among the groups constituting the LDPC codeword in the second part of each column serially. 
     That is, the block interleaver  124  may write 180 bits of the 360 bits included in the bit group Y 176  in the 1st row to the 180 th  row of the second part of the 1st column in the column direction, and may write the other 180 bits in the Pt row to the 180 th  row of the second part of the 2 nd  column in the column direction. In addition, the block interleaver  124  may write 180 bits of the 360 bits included in the bit group Y 177  in the 1st row to the 180 th  row of the second part of the 3 rd  column in the column direction, and may write the other 180 bits in the Pt row to the 180 th  row of the second part of the 4 th  column in the column direction. In addition, the block interleaver  124  may write 180 bits of the 360 bits included in the bit group Y 178  in the Pt row to the 180 th  row of the second part of the 5 th  column in the column direction, and may write the other 180 bits in the Pt row to the 180 th  row of the second part of the 6 th  column in the column direction. In addition, the block interleaver  124  may write 180 bits of the 360 bits included in the bit group Y 179  in the 1st row to the 180 th  row of the second part of the 7 th  column in the column direction, and may write the other 180 bits in the Pt row to the 180 th  row of the second part of the 8 th  column in the column direction. 
     Accordingly, the bits included in the bit group which has not been written in the first part and remains are not written in the same column in the second part and may be divided and written in the plurality of columns. 
     Hereinafter, the block interleaver  124  of  FIG.  5    according to an exemplary embodiment will be explained in detail with reference to  FIG.  12   . 
     In a group-interleaved LDPC codeword (v 0 , v 1 , . . . , v N     ldpc     −1 ), Y 1  is continuously arranged like V={Y 0 , Y 1 , . . . Y N     group     −1 }. 
     The LDPC codeword after group interleaving may be interleaved by the block interleaver  124  as shown in  FIG.  12   . In this case, the block interleaver  124  divide a plurality of columns into the first part(Part 1) and the second part(Part 2) based on the number of columns of the block interleaver  124  and the number of bits of bit groups. In this case, in the first part, the bits constituting the bit groups may be written in the same column, and in the second part, the bits constituting the bit groups may be written in a plurality of columns(i.e. the bits constituting the bit groups may be written in at least two columns). 
     Specifically, input bits vi are written serially from the first part to the second part column wise, and then read out serially from the first part to the second part row wise. That is, the data bits V, are written serially into the block interleaver column-wise starting in the first aprt and continuing column-wise finishing in the second part, and then read out serially row-wise from the first part and then row-wise from the second part. Accordingly, the bit included in the same bit group in the first part may be mapped onto a single bit of each modulation symbol. 
     In this case, the number of columns and the number of rows of the first part and the second part of the block interleaver  124  vary according to a modulation format and a length of the LDPC codeword as in Table 25 presented below. That is, the first part and the second part block interleaving configurations for each modulation format and code length are specified in Table 25 presented below. Herein, the number of columns of the block interleaver  124  may be equal to the number of bits constituting a modulation symbol. In addition, a sum of the number of rows of the first part, N r1  and the number of rows of the second part, N r2 , is equal to N ldpc /N C  (herein, N C  is the number of columns). In addition, since N r1 (=└N group /N c ┘×360) is a multiple of 360, a multiple of bit groups may be written in the first part. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 25 
               
             
            
               
                   
                   
               
               
                   
                 Rows in Part 1 N r1   
                 Rows in Part 2 N r2   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 N ldpc  = 
                 N ldpc  = 
                 N ldpc  = 
                 N ldpc  = 
                 Columns 
               
               
                 Modulation 
                 64800 
                 16200 
                 64800 
                 16200 
                 N c   
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 QPSK 
                 32400 
                 7920 
                 0 
                 180 
                 2 
               
               
                 16-QAM 
                 16200 
                 3960 
                 0 
                 90 
                 4 
               
               
                 64-QAM 
                 10800 
                 2520 
                 0 
                 180 
                 6 
               
               
                 256-QAM 
                 7920 
                 1800 
                 180 
                 225 
                 8 
               
               
                 1024-QAM 
                 6480 
                 1440 
                 0 
                 180 
                 10 
               
               
                 4096-QAM 
                 5400 
                 1080 
                 0 
                 270 
                 12 
               
               
                   
               
            
           
         
       
     
     Hereinafter, an operation of the block interleaver  124  will be explained in detail. 
     Specifically, as shown in  FIG.  12   , the input bit v i  (0≤i&lt;N C ×N r1 ) is written in r i  row of c i  column of the first part of the block interleaver  124 . Herein, c i  and r i  are 
     
       
         
           
             
               c 
               i 
             
             = 
             
               ⌊ 
               
                 i 
                 
                   N 
                   
                     r 
                     ⁢ 
                     1 
                   
                 
               
               ⌋ 
             
           
         
       
     
     and r i =(i mod N r1 ), respectively. 
     In addition, the input bit v i  (N C ×N r1 ≤i&lt;N ldpc ) is written in r row of c i  column of the second part of the block interleaver  124 . Herein, c i  and r i  satisfy 
     
       
         
           
             
               c 
               i 
             
             = 
             
               ⌊ 
               
                 
                   ( 
                   
                     i 
                     - 
                     
                       
                         N 
                         C 
                       
                       × 
                       
                         N 
                         
                           r 
                           ⁢ 
                           1 
                         
                       
                     
                   
                   ) 
                 
                 
                   N 
                   
                     r 
                     ⁢ 
                     2 
                   
                 
               
               ⌋ 
             
           
         
       
     
     and r i =N r1 +{(i−N C ×N r1 )mod N r2 }, respectively. 
     An output bit q j (0≤j&lt;N ldpc ) is read from c j  column of r j  row. Herein, r j  and c j  satisfy 
     
       
         
           
             
               r 
               j 
             
             = 
             
               ⌊ 
               
                 j 
                 
                   N 
                   c 
                 
               
               ⌋ 
             
           
         
       
     
     and c j =(j mod N C ), respectively. 
     For example, when the length N ldpc  of an LDPC codeword is 64800 and the modulation method is 256-QAM, the order of bits output from the block interleaver  124  may be (q 0 ,q 1 ,q 2 , . . . , q 63357 ,q 63358 ,q 63359 ,q 63360 ,q 63361 , . . . ,q 64799 )=(v 0 ,v 7920 ,v 15840 , . . . ,v 47519 ,v 55439 ,v 63359 ,v 63360 ,v 63540 , . . . ,v 64799 ). Herein, the indexes of the right side of the foregoing equation may be specifically expressed for the eight (8) columns as 0, 7920, 15840, 23760, 31680, 39600, 47520, 55440, 1, 7921, 15841, 23761, 31681, 39601, 47521, 55441, . . . , 7919, 15839, 23759, 31679, 39599, 47519, 55439, 63359, 63360, 63540, 63720, 63900, 64080, 64260, 64440, 64620, . . . , 63539, 63719, 63899, 64079, 64259, 64439, 64619, 64799. 
     Hereinafter, the interleaving operation of the block interleaver  124  will be explained in detail. 
     The block interleaver  124  may interleave by writing a plurality of bit groups in each column in bit group wise in a column direction, and reading each row of the plurality of columns in which the plurality of bit groups are written in bit group wise in a row direction. 
     In this case, the number of columns constituting the block interleaver  124  may vary according to a modulation method, and the number of rows may be the length of the LDPC codeword/the number of columns. 
     For example, when the modulation method is 16-QAM, the block interleaver  124  may be formed of 4 columns. In this case, when the length N ldpc  of the LDPC codeword is 16200, the number of rows is 16200 (=64800/4). In another example, when the modulation method is 64-QAM, the block interleaver  124  may be formed of 6 columns. In this case, when the length N ldpc  of the LDPC codeword is 64800, the number of rows is 10800 (=64800/6). 
     Hereinafter, the method for interleaving the plurality of bit groups in bit group wise by the block interleaver  124  will be explained in detail. 
     When the number of bit groups constituting the LDPC codeword is an integer multiple of the number of columns, the block interleaver  124  may interleave by writing the bit groups as many as the number of bit groups divided by the number of columns in each column serially in bit group wise. 
     For example, when the modulation method is 16-QAM and the length N ldpc  of the LDPC codeword is 64800, the block interleaver  124  may be formed of four (4) columns each including 16200 rows. In this case, since the LDPC codeword is divided into (64800/360=180) number of bit groups when the length N ldpc  of the LDPC codeword is 64800, the number of bit groups (=180) of the LDPC codeword may be an integer multiple of the number of columns (=4) when the modulation method is 16-QAM. That is, no remainder is generated when the number of bit groups of the LDPC codeword is divided by the number of columns. 
     In this case, as shown in  FIG.  13   , the block interleaver  124  writes the bits included in each of the bit group Y 0 , bit group Y 1 , . . . , bit group Y 44  in the 1 st  row to 16200 th  row of the first column, writes the bits included in each of the bit group Y 45 , the bit group Y 46 , . . . , the bit group Y 89  in the 1st row to 16200 th  row of the second column, writes the bits included in each of the bit group Y 90 , the bit group Y 91 , . . . , the bit group Y 134  in the 1st row to 16200 th  row of the third column, and writes the bits included in each of the bit group Y 135 , the bit group Y 136 , . . . , the bit group Y 179  in the 1st row to 16200 th  row of the fourth column. In addition, the block interleaver  124  may read the bits written in each row of the two columns serially in the row direction. 
     In another, when the modulation method is 64-QAM and the length N ldpc  of the LDPC codeword is 64800, the block interleaver  124  may be formed of six (6) columns each including 10800 rows. In this case, since the LDPC codeword is divided into (64800/360=180) number of bit groups when the length N ldpc  of the LDPC codeword is 64800, the number of bit groups (=180) of the LDPC codeword may be an integer multiple of the number of columns (=4) when the modulation method is 64-QAM. That is, no remainder is generated when the number of bit groups of the LDPC codeword is divided by the number of columns. 
     In this case, as shown in  FIG.  14   , the block interleaver  124  writes the bits included in each of the bit group Y 0 , bit group Y 1 , . . . , bit group Y 29  in the 1st row to 10800 th  row of the first column, writes the bits included in each of the bit group Y 30 , the bit group Y 31 , . . . , the bit group Y 59  in the 1st row to 10800 th  row of the second column, writes the bits included in each of the bit group Y 60 , the bit group Y 61 , . . . , the bit group Y 89  in the 1st row to 10800 th  row of the third column, writes the bits included in each of the bit group Y 90 , the bit group Y 91 , . . . , the bit group Y 119  in the 1st row to 10800 th  row of the fourth column, writes the bits included in each of the bit group Y 120 , the bit group Y 121 , . . . , the bit group Y 149  in the 1st row to 10800 th  row of the fifth column, and writes the bits included in each of the bit group Y 150 , the bit group Y 151 , . . . , the bit group Y 179  in the 1st row to 10800 th  row of the sixth column. In addition, the block interleaver  124  may read the bits written in each row of the two columns serially in the row direction. 
     As described above, when the number of bit groups constituting the LDPC codeword is an integer multiple of the number of columns of the block interleaver  124 , the block interleaver  124  may interleave the plurality of bit groups in bit group wise, and accordingly, the bits belonging to the same bit group may be written in the same column. 
     As described above, the block interleaver  124  may interleave the plurality of bit groups of the LDPC codeword in the method described above with reference to  FIGS.  13  and  14   . 
     The modulator  130  maps the interleaved LDPC codeword onto a modulation symbol. Specifically, the modulator  130  may demultiplex the interleaved LDPC codeword, modulate the demultiplexed LDPC codeword, and map the LDPC codeword onto a constellation. 
     In this case, the modulator  130  may generate a modulation symbol using the bits included in each of a plurality of bit groups. 
     In other words, as described above, the bits included in different bit groups are written in each column of the block interleaver  124 , and the block interleaver  124  reads the bits written in each column in the row direction. In this case, the modulator  130  generates a modulation symbol by mapping the bits read in each column onto each bit of the modulation symbol. Accordingly, each bit of the modulation symbol belongs to a different bit group. 
     For example, it is assumed that the modulation symbol consists of C number of bits. In this case, the bits which are read from each row of C number of columns of the block interleaver  124  may be mapped onto each bit of the modulation symbol and thus, each bit of the modulation symbol consisting of C number of bits belong to C number of different bit groups. 
     Hereinbelow, the above feature will be described in greater detail. 
     First, the modulator  130  demultiplexes the interleaved LDPC codeword. To achieve this, the modulator  130  may include a demultiplexer (not shown) to demultiplex the interleaved LDPC codeword. 
     The demultiplexer (not shown) demultiplexes the interleaved LDPC codeword. Specifically, the demultiplexer (not shown) performs serial-to-parallel conversion with respect to the interleaved LDPC codeword, and demultiplexes the interleaved LDPC codeword into a cell having a predetermined number of bits (or a data cell). 
     For example, as shown in  FIG.  15   , the demultiplexer (not shown) receives the LDPC codeword Q=(q 0 , q 1 , q 2 , . . . ) output from the interleaver  120 , outputs the received LDPC codeword bits to a plurality of substreams serially, converts the input LDPC codeword bits into cells, and outputs the cells. 
     In this case, the bits having the same index in each of the plurality of substreams may constitute the same cell. Accordingly, the cells may be configured like (y 0,0 , y 1,0 , . . . , y ηMOD−1,0 )=(q 0 , q 1 , q ηMOD−1 ), (y 0,1 , y 1,1 , . . . , y ηMOD−1,1 )=(q ηMOD , q ηMOD+1 , . . . , q 2xηMOD−1 ), . . . . 
     Herein, the number of substreams, N substreams , may be equal to the number of bits constituting a modulation symbol, η MOD . Accordingly, the number of bits constituting each cell may be equal to the number of bits constituting a modulation symbol (that is, a modulationorder). 
     For example, when the modulation method is 16-QAM, the number of bits constituting the modulation symbol, η MOD , is 4 and thus the number of substreams, N substreams , is 4, and the cells may be configured like (y 0,0 , y 1,0 , y 2,0 , y 3,0 )=(q 0 , q 1 , q 2 , q 3 ), (y 0,1 , y 1,1 , y 2,1 , y 3,1 )=(q 4 ,q 5 , q 6 ,q 7 ), (Y 0,2 , Y 1,2 , Y 2,2 , Y 3,2 )=(q 8 , q 9 , q 10 , q 11 ), . . . . 
     In another example, when the modulation method is 64-QAM, the number of bits constituting the modulation symbol, η MOD , is 6 and thus the number of substreams, N substreams , is 6, and the cells may be configured like (y 0,0 , y 1,0 , y 2,0 , y 3,0 , y 4,0 , y 5,0 )=(q 0 , q 1 , q 2 , q 3 , q 4 , q 5 ), (y 0,1 , y 1,1 , Y 2,1 , Y 3,1 , y 4,1 , y 5,1 )=(q 6 ,q 7 , q 8 ,q 9 , q 10 ,q 11 ), (Y 0,2 , Y 1,2 , y 2,2 , y 3,2 , y 4,2 , y 5,2 )=(q 12 , q 13 , q 14 , q 15 , q 16 , q 17 ), . . . . 
     The modulator  130  may map the demultiplexed LDPC codeword onto modulation symbols. 
     Specifically, the modulator  130  may modulate bits (that is, cells) output from the demultiplexer (not shown) in various modulation methods such as Quadrature Phase Shift Keying (QPSK), 16-QAM, 64-QAM, 256-QAM, 1024-QAM, 4096-QAM, etc. For example, when the modulation method is QPSK, 16-QAM, 64-QAM, 256-QAM, 1024-QAM, and 4096-QAM, the number of bits constituting the modulation symbol, η MOD  (that is, the modulation order), may be 2, 4, 6, 8, 10 and 12, respectively. 
     In this case, since each cell output from the demultiplexer (not shown) is formed of as many bits as the number of bits constituting the modulation symbol, the modulator  130  may generate the modulation symbol by mapping each cell output from the demultiplexer (not shown) onto a constellation point serially. Herein, the modulation symbol corresponds to a constellation point on the constellation. 
     However, the above-described demultiplexer (not shown) may be omitted according to circumstances. In this case, the modulator  130  may generate modulation symbols by grouping a predetermined number of bits from interleaved bits serially and mapping the predetermined number of bits onto constellation points. In this case, the modulator  130  may generate the modulation symbols by mapping η MOD  number of bits onto the constellation points serially according to a modulation method. 
     The modulator  130  may modulate by mapping cells output from the demultiplexer (not shown) onto constellation points in a non-uniform constellation (NUC) method. 
     In the non-uniform constellation method, once a constellation point of the first quadrant is defined, constellation points in the other three quadrants may be determined as follows. For example, when a set of constellation points defined for the first quadrant is X, the set becomes −conj(X) in the case of the second quadrant, becomes conj(X) in the case of the third quadrant, and becomes −(X) in the case of the fourth quadrant. 
     That is, once the first quadrant is defined, the other quadrants may be expressed as follows: 
       1 Quarter (first quadrant)= X    
       2 Quarter (second quadrant)=−conj( X )
 
       3 Quarter (third quadrant)=conj( X ) 
       4 Quarter (fourth quadrant)=− X  
 
     Specifically, when the non-uniform M-QAM is used, M number of constellation points may be defined as z={z 0 , z 1 , . . . , z M-1 }. In this case, when the constellation points existing in the first quadrant are defined as {x 0 , x 1 , x 2 , . . . , x M/4-1 }, z may be defined as follows: 
       from  z   0  to  z   M/4-1 =from  x   0  to  x   M/4    
       from  z   M/4  to  z   2×M/4-1 =−conj(from  x   0  to  x   M/4 )
 
       from  z   2×M/4  to  z   3×M/4-1 =conj(from  x   0  to  x   M/4 ) 
       from  z   3×m/4  to  z   4×m/4-1 =−(from  x   0  to  x   m/4 )
 
     Accordingly, the modulator  130  may map the bits [y 0 , . . . , y m-1 ] output from the demultiplexer (not shown) onto constellation points in the non-uniform constellation method by mapping the output bits onto z L  having an index of 
     
       
         
           
             L 
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   0 
                 
                 
                   m 
                   - 
                   1 
                 
               
               
                 
                   ( 
                   
                     
                       y 
                       1 
                     
                     × 
                     
                       2 
                       
                         m 
                         - 
                         1 
                       
                     
                   
                   ) 
                 
                 . 
               
             
           
         
       
     
     An example of the constellation defined according to the non-uniform constellation method may be expressed as in tables 26 to 30 presented below when the code rate is 5/15, 7/15, 9/15, 11/15, 13/15: 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 26 
               
               
                   
                   
               
               
                   
                 Input data cell y 
                 Constellation point z s   
               
               
                   
                   
               
             
            
               
                   
                 (00) 
                  (1 + 1i)/√{square root over (2)} 
               
               
                   
                 (01) 
                  (1 − 1i)/√{square root over (2)} 
               
               
                   
                 (10) 
                 (−1 + 1i)/√{square root over (2)} 
               
               
                   
                 (11) 
                 (−1 − 1i)/√{square root over (2)} 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 27 
               
               
                   
               
             
            
               
                 x/Shape 
                 R6/15 
                 R7/15 
                 R8/15 
                 R9/15 
               
               
                   
               
               
                 x0 
                 0.4530 + 0.2663i 
                 1.2103 + 0.5026i 
                 0.4819 + 0.2575i 
                 0.4909 + 1.2007i 
               
               
                 x1 
                 0.2663 + 0.4530i 
                 0.5014 + 1.2103i 
                 0.2575 + 0.4819i 
                 1.2007 + 0.4909i 
               
               
                 x2 
                 1.2092 + 0.5115i 
                 0.4634 + 0.2624i 
                 1.2068 + 0.4951i 
                 0.2476 + 0.5065i 
               
               
                 x3 
                 0.5115 + 1.2092i 
                 0.2624 + 0.4627i 
                 0.4951 + 1.2068i 
                 0.5053 + 0.2476i 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 x/Shape 
                 R10/15 
                 R11/15 
                 R12/15 
                 R13/15 
               
               
                   
               
               
                 x0 
                 0.2173 + 0.4189i 
                 0.9583 + 0.9547i 
                 0.2999 + 0.2999i 
                 0.9517 + 0.9511i 
               
               
                 x1 
                 0.6578 + 0.2571i 
                 0.9547 + 0.2909i 
                 0.9540 + 0.2999i 
                 0.9524 + 0.3061i 
               
               
                 x2 
                 0.4326 + 1.1445i 
                 0.2921 + 0.9583i 
                 0.2999 + 0.9540i 
                 0.3067 + 0.9524i 
               
               
                 x3 
                 1.2088 + 0.5659i 
                 0.2909 + 0.2927i 
                 0.9540 + 0.9540i 
                 0.3061 + 0.3067i 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 28 
               
               
                   
               
             
            
               
                 x/Shape 
                 R64_6/15 
                 R64_7/15 
                 R64_8/15 
                 R64_9/15 
               
               
                   
               
               
                 x0 
                 0.4387 + 1.6023i 
                 0.3352 + 0.6028i 
                 1.4827 + 0.2920i 
                 0.3547 + 0.6149i 
               
               
                 x1 
                 1.6023 + 0.4387i 
                 0.2077 + 0.6584i 
                 1.2563 + 0.8411i 
                 0.1581 + 0.6842i 
               
               
                 x2 
                 0.8753 + 1.0881i 
                 0.1711 + 0.3028i 
                 1.0211 + 0.2174i 
                 0.1567 + 0.2749i 
               
               
                 x3 
                 1.0881 + 0.8753i 
                 0.1556 + 0.3035i 
                 0.8798 + 0.5702i 
                 0.1336 + 0.2700i 
               
               
                 x4 
                 0.2202 + 0.9238i 
                 0.6028 + 0.3345i 
                 0.2920 + 1.4827i 
                 0.6177 + 0.4030i 
               
               
                 x5 
                 0.2019 + 0.7818i 
                 0.6577 + 0.2084i 
                 0.8410 + 1.2563i 
                 0.7262 + 0.1756i 
               
               
                 x6 
                 0.3049 + 0.8454i 
                 0.3021 + 0.1711i 
                 0.2174 + 1.0211i 
                 0.3568 + 0.1756i 
               
               
                 x7 
                 0.2653 + 0.7540i 
                 0.3028 + 0.1556i 
                 0.5702 + 0.8798i 
                 0.3771 + 0.1336i 
               
               
                 x8 
                 0.7818 + 0.2019i 
                 0.5556 + 0.8922i 
                 0.3040 + 0.1475i 
                 0.5639 + 0.8864i 
               
               
                 x9 
                 0.9238 + 0.2202i 
                 0.2352 + 1.0190i 
                 0.3028 + 0.1691i 
                 0.1980 + 1.0277i 
               
               
                 x10 
                 0.7540 + 0.2653i 
                 0.8450 + 1.2619i 
                 0.6855 + 0.1871i 
                 0.8199 + 1.2515i 
               
               
                 x11 
                 0.8454 + 0.3049i 
                 0.2922 + 1.4894i 
                 0.6126 + 0.3563i 
                 0.2854 + 1.4691i 
               
               
                 x12 
                 0.2675 + 0.2479i 
                 0.8929 + 0.5549i 
                 0.1475 + 0.3040i 
                 0.8654 + 0.6058i 
               
               
                 x13 
                 0.2479 + 0.2675i 
                 1.0197 + 0.2359i 
                 0.1691 + 0.3028i 
                 1.0382 + 0.2141i 
               
               
                 x14 
                 0.2890 + 0.2701i 
                 1.2626 + 0.8457i 
                 0.1871 + 0.6855i 
                 1.2362 + 0.8416i 
               
               
                 x15 
                 0.2701 + 0.2890i 
                 1.4894 + 0.2922i 
                 0.3563 + 0.6126i 
                 1.4663 + 0.2973i 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 x/Shape 
                 R64_10/15 
                 R64_11/15 
                 R64_12/15 
                 E64_13/15 
               
               
                   
               
               
                 x0 
                 1.4388 + 0.2878i 
                 0.3317 + 0.6970i 
                 1.0854 + 0.5394i 
                 0.4108 + 0.7473i 
               
               
                 x1 
                 1.2150 + 0.8133i 
                 0.1386 + 0.8824i 
                 0.7353 + 0.4623i 
                 0.1343 + 0.5338i 
               
               
                 x2 
                 1.0386 + 0.2219i 
                 0.1323 + 0.4437i 
                 1.0474 + 0.1695i 
                 0.1570 + 0.9240i 
               
               
                 x3 
                 0.8494 + 0.6145i 
                 0.1015 + 0.1372i 
                 0.7243 + 0.1504i 
                 0.1230 + 0.1605i 
               
               
                 x4 
                 0.2931 + 1.4656i 
                 0.5682 + 0.4500i 
                 1.0693 + 0.9408i 
                 0.6285 + 0.4617i 
               
               
                 x5 
                 0.8230 + 1.2278i 
                 0.6739 + 0.1435i 
                 0.7092 + 0.8073i 
                 0.3648 + 0.3966i 
               
               
                 x6 
                 0.2069 + 1.0649i 
                 0.3597 + 0.3401i 
                 1.4261 + 0.2216i 
                 0.6907 + 0.1541i 
               
               
                 x7 
                 0.5677 + 0.8971i 
                 0.3660 + 0.1204i 
                 0.6106 + 1.1783i 
                 0.3994 + 0.1308i 
               
               
                 x8 
                 0.4119 + 0.1177i 
                 0.6004 + 0.8922i 
                 0.1392 + 0.4078i 
                 0.7268 + 0.8208i 
               
               
                 x9 
                 0.3998 + 0.2516i 
                 0.2120 + 1.2253i 
                 0.4262 + 0.4205i 
                 1.0463 + 0.9495i 
               
               
                 x10 
                 0.7442 + 0.1559i 
                 0.9594 + 1.0714i 
                 0.1407 + 0.1336i 
                 0.1866 + 1.2733i 
               
               
                 x11 
                 0.5954 + 0.4328i 
                 0.5829 + 1.3995i 
                 0.4265 + 0.1388i 
                 0.5507 + 1.1793i 
               
               
                 x12 
                 0.1166 + 0.1678i 
                 0.8439 + 0.5675i 
                 0.1388 + 0.7057i 
                 0.9283 + 0.5140i 
               
               
                 x13 
                 0.1582 + 0.3325i 
                 0.9769 + 0.1959i 
                 0.4197 + 0.7206i 
                 1.2648 + 0.5826i 
               
               
                 x14 
                 0.1355 + 0.7408i 
                 1.2239 + 0.6760i 
                 0.1682 + 1.0316i 
                 0.9976 + 0.1718i 
               
               
                 x15 
                 0.3227 + 0.6200i 
                 1.3653 + 0.2323i 
                 0.2287 + 1.3914i 
                 1.3412 + 0.1944i 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 29 
               
               
                   
               
             
            
               
                 x/Shape 
                 R64_6/15 
                 R64_7/15 
                 R64_8/15 
                 R64_9/15 
               
               
                   
               
               
                 x0 
                 0.4387 + 1.6023i 
                 0.3352 + 0.6028i 
                 1.4827 + 0.2920i 
                 0.3547 + 0.6149i 
               
               
                 x1 
                 1.6023 + 0.4387i 
                 0.2077 + 0.6584i 
                 1.2563 + 0.8411i 
                 0.1581 + 0.6842i 
               
               
                 x2 
                 0.8753 + 1.0881i 
                 0.1711 + 0.3028i 
                 1.0211 + 0.2174i 
                 0.1567 + 0.2749i 
               
               
                 x3 
                 1.0881 + 0.8753i 
                 0.1556 + 0.3035i 
                 0.8798 + 0.5702i 
                 0.1336 + 0.2700i 
               
               
                 x4 
                 0.2202 + 0.9238i 
                 0.6028 + 0.3345i 
                 0.2920 + 1.4827i 
                 0.6177 + 0.4030i 
               
               
                 x5 
                 0.2019 + 0.7818i 
                 0.6577 + 0.2084i 
                 0.8410 + 1.2563i 
                 0.7262 + 0.1756i 
               
               
                 x6 
                 0.3049 + 0.8454i 
                 0.3021 + 0.1711i 
                 0.2174 + 1.0211i 
                 0.3568 + 0.1756i 
               
               
                 x7 
                 0.2653 + 0.7540i 
                 0.3028 + 0.1556i 
                 0.5702 + 0.8798i 
                 0.3771 + 0.1336i 
               
               
                 x8 
                 0.7818 + 0.2019i 
                 0.5556 + 0.8922i 
                 0.3040 + 0.1475i 
                 0.5639 + 0.8864i 
               
               
                 x9 
                 0.9238 + 0.2202i 
                 0.2352 + 1.0190i 
                 0.3028 + 0.1691i 
                 0.1980 + 1.0277i 
               
               
                 x10 
                 0.7540 + 0.2653i 
                 0.8450 + 1.2619i 
                 0.6855 + 0.1871i 
                 0.8199 + 1.2515i 
               
               
                 x11 
                 0.8454 + 0.3049i 
                 0.2922 + 1.4894i 
                 0.6126 + 0.3563i 
                 0.2854 + 1.4691i 
               
               
                 x12 
                 0.2675 + 0.2479i 
                 0.8929 + 0.5549i 
                 0.1475 + 0.3040i 
                 0.8654 + 0.6058i 
               
               
                 x13 
                 0.2479 + 0.2675i 
                 1.0197 + 0.2359i 
                 0.1691 + 0.3028i 
                 1.0382 + 0.2141i 
               
               
                 x14 
                 0.2890 + 0.2701i 
                 1.2626 + 0.8457i 
                 0.1871 + 0.6855i 
                 1.2362 + 0.8416i 
               
               
                 x15 
                 0.2701 + 0.2890i 
                 1.4894 + 0.2922i 
                 0.3563 + 0.6126i 
                 1.4663 + 0.2973i 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 x/Shape 
                 R64_10/15 
                 R64_11/15 
                 R64_12/15 
                 E64_13/15 
               
               
                   
               
               
                 x0 
                 1.4388 + 0.2878i 
                 0.3317 + 0.6970i 
                 1.0854 + 0.5394i 
                 0.4108 + 0.7473i 
               
               
                 x1 
                 1.2150 + 0.8133i 
                 0.1386 + 0.8824i 
                 0.7353 + 0.4623i 
                 0.1343 + 0.5338i 
               
               
                 x2 
                 1.0386 + 0.2219i 
                 0.1323 + 0.4437i 
                 1.0474 + 0.1695i 
                 0.1570 + 0.9240i 
               
               
                 x3 
                 0.8494 + 0.6145i 
                 0.1015 + 0.1372i 
                 0.7243 + 0.1504i 
                 0.1230 + 0.1605i 
               
               
                 x4 
                 0.2931 + 1.4656i 
                 0.5682 + 0.4500i 
                 1.0693 + 0.9408i 
                 0.6285 + 0.4617i 
               
               
                 x5 
                 0.8230 + 1.2278i 
                 0.6739 + 0.1435i 
                 0.7092 + 0.8073i 
                 0.3648 + 0.3966i 
               
               
                 x6 
                 0.2069 + 1.0649i 
                 0.3597 + 0.3401i 
                 1.4261 + 0.2216i 
                 0.6907 + 0.1541i 
               
               
                 x7 
                 0.5677 + 0.8971i 
                 0.3660 + 0.1204i 
                 0.6106 + 1.1783i 
                 0.3994 + 0.1308i 
               
               
                 x8 
                 0.4119 + 0.1177i 
                 0.6004 + 0.8922i 
                 0.1392 + 0.4078i 
                 0.7268 + 0.8208i 
               
               
                 x9 
                 0.3998 + 0.2516i 
                 0.2120 + 1.2253i 
                 0.4262 + 0.4205i 
                 1.0463 + 0.9495i 
               
               
                 x10 
                 0.7442 + 0.1559i 
                 0.9594 + 1.0714i 
                 0.1407 + 0.1336i 
                 0.1866 + 1.2733i 
               
               
                 x11 
                 0.5954 + 0.4328i 
                 0.5829 + 1.3995i 
                 0.4265 + 0.1388i 
                 0.5507 + 1.1793i 
               
               
                 x12 
                 0.1166 + 0.1678i 
                 0.8439 + 0.5675i 
                 0.1388 + 0.7057i 
                 0.9283 + 0.5140i 
               
               
                 x13 
                 0.1582 + 0.3325i 
                 0.9769 + 0.1959i 
                 0.4197 + 0.7206i 
                 1.2648 + 0.5826i 
               
               
                 x14 
                 0.1355 + 0.7408i 
                 1.2239 + 0.6760i 
                 0.1682 + 1.0316i 
                 0.9976 + 0.1718i 
               
               
                 x15 
                 0.3227 + 0.6200i 
                 1.3653 + 0.2323i 
                 0.2287 + 1.3914i 
                 1.3412 + 0.1944i 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 30 
               
               
                   
               
             
            
               
                 x/Shape 
                 R6/15 
                 R7/15 
                 R8/15 
                 R9/15 
               
               
                   
               
               
                 x0 
                 0.6800 + 1.6926i 
                 1.2905 + 1.3099i 
                 1.0804 + 1.3788i 
                 1.3231 + 1.1506i 
               
               
                 x1 
                 0.3911 + 1.3645i 
                 1.0504 + 0.9577i 
                 1.0487 + 0.9862i 
                 0.9851 + 1.2311i 
               
               
                 x2 
                 0.2191 + 1.7524i 
                 1.5329 + 0.8935i 
                 1.6464 + 0.7428i 
                 1.1439 + 0.8974i 
               
               
                 x3 
                 0.2274 + 1.4208i 
                 1.1577 + 0.8116i 
                 1.3245 + 0.9414i 
                 0.9343 + 0.9271i 
               
               
                 x4 
                 0.8678 + 1.2487i 
                 1.7881 + 0.2509i 
                 0.7198 + 1.2427i 
                 1.5398 + 0.7962i 
               
               
                 x5 
                 0.7275 + 1.1667i 
                 1.4275 + 0.1400i 
                 0.8106 + 1.0040i 
                 0.9092 + 0.5599i 
               
               
                 x6 
                 0.8747 + 1.0470i 
                 1.4784 + 0.5201i 
                 0.5595 + 1.0317i 
                 1.2222 + 0.6574i 
               
               
                 x7 
                 0.7930 + 1.0406i 
                 1.3408 + 0.4346i 
                 0.6118 + 0.9722i 
                 0.9579 + 0.6373i 
               
               
                 x8 
                 0.2098 + 0.9768i 
                 0.7837 + 0.5867i 
                 1.6768 + 0.2002i 
                 0.7748 + 1.5867i 
               
               
                 x9 
                 0.2241 + 1.0454i 
                 0.8250 + 0.6455i 
                 0.9997 + 0.6844i 
                 0.6876 + 1.2489i 
               
               
                 x10 
                 0.1858 + 0.9878i 
                 0.8256 + 0.5601i 
                 1.4212 + 0.4769i 
                 0.5992 + 0.9208i 
               
               
                 x11 
                 0.1901 + 1.0659i 
                 0.8777 + 0.6110i 
                 1.1479 + 0.6312i 
                 0.6796 + 0.9743i 
               
               
                 x12 
                 0.5547 + 0.8312i 
                 1.0080 + 0.1843i 
                 0.6079 + 0.6566i 
                 0.5836 + 0.5879i 
               
               
                 x13 
                 0.5479 + 0.8651i 
                 1.0759 + 0.1721i 
                 0.7284 + 0.6957i 
                 0.6915 + 0.5769i 
               
               
                 x14 
                 0.6073 + 0.8182i 
                 1.0056 + 0.2758i 
                 0.5724 + 0.7031i 
                 0.5858 + 0.7058i 
               
               
                 x15 
                 0.5955 + 0.8420i 
                 1.0662 + 0.2964i 
                 0.6302 + 0.7259i 
                 0.6868 + 0.6793i 
               
               
                 x16 
                 1.4070 + 0.1790i 
                 0.8334 + 1.5554i 
                 0.1457 + 1.4010i 
                 1.6118 + 0.1497i 
               
               
                 x17 
                 1.7227 + 0.2900i 
                 0.8165 + 1.1092i 
                 0.1866 + 1.7346i 
                 0.9511 + 0.1140i 
               
               
                 x18 
                 1.3246 + 0.2562i 
                 0.6092 + 1.2729i 
                 0.1174 + 1.1035i 
                 1.2970 + 0.1234i 
               
               
                 x19 
                 1.3636 + 0.3654i 
                 0.6728 + 1.1456i 
                 0.1095 + 1.0132i 
                 1.0266 + 0.1191i 
               
               
                 x20 
                 1.3708 + 1.2834i 
                 0.3061 + 1.7469i 
                 0.4357 + 1.3636i 
                 1.5831 + 0.4496i 
               
               
                 x21 
                 1.6701 + 0.8403i 
                 0.1327 + 1.4056i 
                 0.5853 + 1.6820i 
                 0.9328 + 0.3586i 
               
               
                 x22 
                 1.1614 + 0.7909i 
                 0.3522 + 1.3414i 
                 0.3439 + 1.0689i 
                 1.2796 + 0.3894i 
               
               
                 x23 
                 1.2241 + 0.7367i 
                 0.2273 + 1.3081i 
                 0.3234 + 0.9962i 
                 1.0188 + 0.3447i 
               
               
                 x24 
                 0.9769 + 0.1863i 
                 0.5007 + 0.8098i 
                 0.1092 + 0.6174i 
                 0.5940 + 0.1059i 
               
               
                 x25 
                 0.9452 + 0.2057i 
                 0.5528 + 0.8347i 
                 0.1074 + 0.6307i 
                 0.7215 + 0.1100i 
               
               
                 x26 
                 1.0100 + 0.2182i 
                 0.4843 + 0.8486i 
                 0.1109 + 0.6996i 
                 0.5863 + 0.1138i 
               
               
                 x27 
                 0.9795 + 0.2417i 
                 0.5304 + 0.8759i 
                 0.1076 + 0.7345i 
                 0.6909 + 0.1166i 
               
               
                 x28 
                 0.8241 + 0.4856i 
                 0.1715 + 0.9147i 
                 0.3291 + 0.6264i 
                 0.5843 + 0.3604i 
               
               
                 x29 
                 0.8232 + 0.4837i 
                 0.1540 + 0.9510i 
                 0.3126 + 0.6373i 
                 0.6970 + 0.3592i 
               
               
                 x30 
                 0.8799 + 0.5391i 
                 0.1964 + 0.9438i 
                 0.3392 + 0.6999i 
                 0.5808 + 0.3250i 
               
               
                 x31 
                 0.8796 + 0.5356i 
                 0.1788 + 0.9832i 
                 0.3202 + 0.7282i 
                 0.6678 + 0.3290i 
               
               
                 x32 
                 0.1376 + 0.3342i 
                 0.3752 + 0.1667i 
                 0.9652 + 0.1066i 
                 0.1406 + 1.6182i 
               
               
                 x33 
                 0.1383 + 0.3292i 
                 0.3734 + 0.1667i 
                 0.9075 + 0.1666i 
                 0.1272 + 1.2984i 
               
               
                 x34 
                 0.1363 + 0.3322i 
                 0.3758 + 0.1661i 
                 0.9724 + 0.1171i 
                 0.1211 + 0.9644i 
               
               
                 x35 
                 0.1370 + 0.3273i 
                 0.3746 + 0.1649i 
                 0.9186 + 0.1752i 
                 0.1220 + 1.0393i 
               
               
                 x36 
                 0.1655 + 0.3265i 
                 0.4013 + 0.1230i 
                 0.6342 + 0.1372i 
                 0.1124 + 0.6101i 
               
               
                 x37 
                 0.1656 + 0.3227i 
                 0.4001 + 0.1230i 
                 0.6550 + 0.1495i 
                 0.1177 + 0.6041i 
               
               
                 x38 
                 0.1634 + 0.3246i 
                 0.4037 + 0.1230i 
                 0.6290 + 0.1393i 
                 0.1136 + 0.7455i 
               
               
                 x39 
                 0.1636 + 0.3208i 
                 0.4019 + 0.1218i 
                 0.6494 + 0.1504i 
                 0.1185 + 0.7160i 
               
               
                 x40 
                 0.1779 + 0.6841i 
                 0.6025 + 0.3934i 
                 1.3127 + 0.1240i 
                 0.4324 + 1.5679i 
               
               
                 x41 
                 0.1828 + 0.6845i 
                 0.5946 + 0.3928i 
                 0.9572 + 0.4344i 
                 0.3984 + 1.2825i 
               
               
                 x42 
                 0.1745 + 0.6828i 
                 0.6116 + 0.3879i 
                 1.2403 + 0.2631i 
                 0.3766 + 0.9534i 
               
               
                 x43 
                 0.1793 + 0.6829i 
                 0.6019 + 0.3837i 
                 1.0254 + 0.4130i 
                 0.3668 + 1.0301i 
               
               
                 x44 
                 0.3547 + 0.6009i 
                 0.7377 + 0.1618i 
                 0.6096 + 0.4214i 
                 0.3667 + 0.5995i 
               
               
                 x45 
                 0.3593 + 0.6011i 
                 0.7298 + 0.1582i 
                 0.6773 + 0.4284i 
                 0.3328 + 0.5960i 
               
               
                 x46 
                 0.3576 + 0.5990i 
                 0.7274 + 0.1782i 
                 0.5995 + 0.4102i 
                 0.3687 + 0.7194i 
               
               
                 x47 
                 0.3624 + 0.5994i 
                 0.7165 + 0.1746i 
                 0.6531 + 0.4101i 
                 0.3373 + 0.6964i 
               
               
                 x48 
                 0.2697 + 0.1443i 
                 0.1509 + 0.2425i 
                 0.1250 + 0.1153i 
                 0.1065 + 0.1146i 
               
               
                 x49 
                 0.2704 + 0.1433i 
                 0.1503 + 0.2400i 
                 0.1252 + 0.1158i 
                 0.1145 + 0.1108i 
               
               
                 x50 
                 0.2644 + 0.1442i 
                 0.1515 + 0.2437i 
                 0.1245 + 0.1152i 
                 0.1053 + 0.1274i 
               
               
                 x51 
                 0.2650 + 0.1432i 
                 0.1503 + 0.2425i 
                 0.1247 + 0.1156i 
                 0.1134 + 0.1236i 
               
               
                 x52 
                 0.2763 + 0.1638i 
                 0.1285 + 0.2388i 
                 0.3768 + 0.1244i 
                 0.1111 + 0.3821i 
               
               
                 x53 
                 0.2768 + 0.1626i 
                 0.1279 + 0.2419i 
                 0.3707 + 0.1237i 
                 0.1186 + 0.3867i 
               
               
                 x54 
                 0.2715 + 0.1630i 
                 0.1279 + 0.2431i 
                 0.3779 + 0.1260i 
                 0.1080 + 0.3431i 
               
               
                 x55 
                 0.2719 + 0.1618i 
                 0.1279 + 0.2406i 
                 0.3717 + 0.1252i 
                 0.1177 + 0.3459i 
               
               
                 x56 
                 0.6488 + 0.1696i 
                 0.3394 + 0.5764i 
                 0.1161 + 0.3693i 
                 0.3644 + 0.1080i 
               
               
                 x57 
                 0.6462 + 0.1706i 
                 0.3364 + 0.5722i 
                 0.1157 + 0.3645i 
                 0.3262 + 0.1104i 
               
               
                 x58 
                 0.6456 + 0.1745i 
                 0.3328 + 0.5758i 
                 0.1176 + 0.3469i 
                 0.3681 + 0.1173i 
               
               
                 x59 
                 0.6431 + 0.1753i 
                 0.3303 + 0.5698i 
                 0.1171 + 0.3424i 
                 0.3289 + 0.1196i 
               
               
                 x60 
                 0.5854 + 0.3186i 
                 0.1491 + 0.6316i 
                 0.3530 + 0.3899i 
                 0.3665 + 0.3758i 
               
               
                 x61 
                 0.5862 + 0.3167i 
                 0.1461 + 0.6280i 
                 0.3422 + 0.3808i 
                 0.3310 + 0.3795i 
               
               
                 x62 
                 0.5864 + 0.3275i 
                 0.1509 + 0.6280i 
                 0.3614 + 0.3755i 
                 0.3672 + 0.3353i 
               
               
                 x63 
                 0.5873 + 0.3254i 
                 0.1473 + 0.6225i 
                 0.3509 + 0.3656i 
                 0.3336 + 0.3402i 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 x/Shape 
                 R10/15 
                 R11/15 
                 R12/15 
                 R13/15 
               
               
                   
               
               
                 x0 
                 1.6097 + 0.1548i 
                 0.3105 + 0.3382i 
                 1.1014 + 1.1670i 
                 0.3556 + 0.3497i 
               
               
                 x1 
                 1.5549 + 0.4605i 
                 0.4342 + 0.3360i 
                 0.8557 + 1.2421i 
                 0.3579 + 0.4945i 
               
               
                 x2 
                 1.3226 + 0.1290i 
                 0.3149 + 0.4829i 
                 1.2957 + 0.8039i 
                 0.5049 + 0.3571i 
               
               
                 x3 
                 1.2772 + 0.3829i 
                 0.4400 + 0.4807i 
                 1.0881 + 0.8956i 
                 0.5056 + 0.5063i 
               
               
                 x4 
                 1.2753 + 1.0242i 
                 0.1811 + 0.3375i 
                 0.5795 + 1.2110i 
                 0.2123 + 0.3497i 
               
               
                 x5 
                 1.4434 + 0.7540i 
                 0.0633 + 0.3404i 
                 0.6637 + 1.4215i 
                 0.2116 + 0.4900i 
               
               
                 x6 
                 1.0491 + 0.8476i 
                 0.1818 + 0.4851i 
                 0.6930 + 1.0082i 
                 0.0713 + 0.3489i 
               
               
                 x7 
                 1.1861 + 0.6253i 
                 0.0633 + 0.4815i 
                 0.8849 + 0.9647i 
                 0.0690 + 0.4960i 
               
               
                 x8 
                 0.9326 + 0.0970i 
                 0.3084 + 0.1971i 
                 1.2063 + 0.5115i 
                 0.3527 + 0.2086i 
               
               
                 x9 
                 0.8962 + 0.2804i 
                 0.4356 + 0.1993i 
                 1.0059 + 0.4952i 
                 0.3497 + 0.0713i 
               
               
                 x10 
                 1.1044 + 0.1102i 
                 0.3098 + 0.0676i 
                 1.4171 + 0.5901i 
                 0.4960 + 0.2123i 
               
               
                 x11 
                 1.0648 + 0.3267i 
                 0.4342 + 0.0691i 
                 1.0466 + 0.6935i 
                 0.4974 + 0.0698i 
               
               
                 x12 
                 0.7325 + 0.6071i 
                 0.1775 + 0.1985i 
                 0.6639 + 0.6286i 
                 0.2086 + 0.2079i 
               
               
                 x13 
                 0.8260 + 0.4559i 
                 0.0640 + 0.1978i 
                 0.8353 + 0.5851i 
                 0.2094 + 0.0690i 
               
               
                 x14 
                 0.8744 + 0.7153i 
                 0.1775 + 0.0676i 
                 0.6879 + 0.8022i 
                 0.0676 + 0.2079i 
               
               
                 x15 
                 0.9882 + 0.5300i 
                 0.0647 + 0.0669i 
                 0.8634 + 0.7622i 
                 0.0698 + 0.0683i 
               
               
                 x16 
                 0.1646 + 1.6407i 
                 0.7455 + 0.3411i 
                 0.1213 + 1.4366i 
                 0.3586 + 0.7959i 
               
               
                 x17 
                 0.4867 + 1.5743i 
                 0.5811 + 0.3396i 
                 0.1077 + 1.2098i 
                 0.3571 + 0.6392i 
               
               
                 x18 
                 0.1363 + 1.3579i 
                 0.7556 + 0.4669i 
                 0.0651 + 0.9801i 
                 0.5034 + 0.8271i 
               
               
                 x19 
                 0.4023 + 1.3026i 
                 0.5862 + 0.4756i 
                 0.2009 + 1.0115i 
                 0.5063 + 0.6600i 
               
               
                 x20 
                 1.0542 + 1.2584i 
                 0.9556 + 0.3280i 
                 0.3764 + 1.4264i 
                 0.2146 + 0.7862i 
               
               
                 x21 
                 0.7875 + 1.4450i 
                 1.1767 + 0.3091i 
                 0.3237 + 1.2130i 
                 0.2109 + 0.6340i 
               
               
                 x22 
                 0.8687 + 1.0407i 
                 0.9673 + 0.4720i 
                 0.5205 + 0.9814i 
                 0.0713 + 0.8093i 
               
               
                 x23 
                 0.6502 + 1.1951i 
                 1.2051 + 0.5135i 
                 0.3615 + 1.0163i 
                 0.0698 + 0.6467i 
               
               
                 x24 
                 0.0982 + 0.9745i 
                 0.7367 + 0.2015i 
                 0.0715 + 0.6596i 
                 0.2799 + 1.0862i 
               
               
                 x25 
                 0.2842 + 0.9344i 
                 0.5811 + 0.2015i 
                 0.2116 + 0.6597i 
                 0.2806 + 1.2755i 
               
               
                 x26 
                 0.1142 + 1.1448i 
                 0.7316 + 0.0669i 
                 0.0729 + 0.8131i 
                 0.4328 + 0.9904i 
               
               
                 x27 
                 0.3385 + 1.0973i 
                 0.5782 + 0.0669i 
                 0.2158 + 0.8246i 
                 0.4551 + 1.1812i 
               
               
                 x28 
                 0.6062 + 0.7465i 
                 0.9062 + 0.1971i 
                 0.5036 + 0.6467i 
                 0.2309 + 0.9414i 
               
               
                 x29 
                 0.4607 + 0.8538i 
                 1.2829 + 0.1185i 
                 0.3526 + 0.6572i 
                 0.1077 + 1.3891i 
               
               
                 x30 
                 0.7263 + 0.8764i 
                 0.9156 + 0.0735i 
                 0.5185 + 0.8086i 
                 0.0772 + 0.9852i 
               
               
                 x31 
                 0.5450 + 1.0067i 
                 1.1011 + 0.0735i 
                 0.3593 + 0.8245i 
                 0.0802 + 1.1753i 
               
               
                 x32 
                 0.2655 + 0.0746i 
                 0.3244 + 0.8044i 
                 1.2545 + 0.1010i 
                 0.8301 + 0.3727i 
               
               
                 x33 
                 0.2664 + 0.0759i 
                 0.4589 + 0.8218i 
                 1.0676 + 0.0956i 
                 0.8256 + 0.5256i 
               
               
                 x34 
                 0.4571 + 0.0852i 
                 0.3207 + 0.6415i 
                 1.4782 + 0.1167i 
                 0.6593 + 0.3668i 
               
               
                 x35 
                 0.4516 + 0.1062i 
                 0.4509 + 0.6371i 
                 0.8981 + 0.0882i 
                 0.6623 + 0.5182i 
               
               
                 x36 
                 0.2559 + 0.1790i 
                 0.1920 + 0.8196i 
                 0.5518 + 0.0690i 
                 1.0186 + 0.3645i 
               
               
                 x37 
                 0.2586 + 0.1772i 
                 0.0633 + 0.8167i 
                 0.6903 + 0.0552i 
                 1.0001 + 0.5242i 
               
               
                 x38 
                 0.3592 + 0.2811i 
                 0.1811 + 0.6371i 
                 0.5742 + 0.1987i 
                 1.1857 + 0.2725i 
               
               
                 x39 
                 0.3728 + 0.2654i 
                 0.0640 + 0.6415i 
                 0.7374 + 0.1564i 
                 1.3928 + 0.3408i 
               
               
                 x40 
                 0.7706 + 0.0922i 
                 0.3331 + 1.0669i 
                 1.2378 + 0.3049i 
                 0.8011 + 0.2227i 
               
               
                 x41 
                 0.7407 + 0.2260i 
                 0.4655 + 1.0087i 
                 1.0518 + 0.3032i 
                 0.7981 + 0.0735i 
               
               
                 x42 
                 0.6180 + 0.0927i 
                 0.3433 + 1.2865i 
                 1.4584 + 0.3511i 
                 0.6459 + 0.2198i 
               
               
                 x43 
                 0.6019 + 0.1658i 
                 0.5004 + 1.5062i 
                 0.9107 + 0.2603i 
                 0.6430 + 0.07i3i 
               
               
                 x44 
                 0.6007 + 0.4980i 
                 0.1971 + 1.0051i 
                 0.6321 + 0.4729i 
                 0.9681 + 0.2205i 
               
               
                 x45 
                 0.6673 + 0.3928i 
                 0.0735 + 1.0298i 
                 0.7880 + 0.4392i 
                 0.9615 + 0.0735i 
               
               
                 x46 
                 0.4786 + 0.3935i 
                 0.1498 + 1.5018i 
                 0.6045 + 0.3274i 
                 1.3327 + 0.1039i 
               
               
                 x47 
                 0.5176 + 0.3391i 
                 0.0865 + 1.2553i 
                 0.7629 + 0.2965i 
                 1.1359 + 0.0809i 
               
               
                 x48 
                 0.0757 + 0.1003i 
                 0.7811 + 0.8080i 
                 0.0596 + 0.0739i 
                 0.8382 + 0.8709i 
               
               
                 x49 
                 0.0753 + 0.1004i 
                 0.6167 + 0.8153i 
                 0.1767 + 0.0731i 
                 0.8145 + 0.6934i 
               
               
                 x50 
                 0.0777 + 0.4788i 
                 0.7636 + 0.6255i 
                 0.0612 + 0.2198i 
                 0.6645 + 0.8486i 
               
               
                 x51 
                 0.0867 + 0.4754i 
                 0.6000 + 0.6327i 
                 0.1815 + 0.2192i 
                 0.6600 + 0.6786i 
               
               
                 x52 
                 0.1023 + 0.2243i 
                 0.9898 + 0.7680i 
                 0.4218 + 0.0715i 
                 1.1612 + 0.6949i 
               
               
                 x53 
                 0.1010 + 0.2242i 
                 1.5855 + 0.1498i 
                 0.2978 + 0.0725i 
                 0.9785 + 0.6942i 
               
               
                 x54 
                 0.1950 + 0.3919i 
                 0.9476 + 0.6175i 
                 0.4337 + 0.2115i 
                 1.3698 + 0.6259i 
               
               
                 x55 
                 0.1881 + 0.3969i 
                 1.4625 + 0.4015i 
                 0.3057 + 0.2167i 
                 1.2183 + 0.4841i 
               
               
                 x56 
                 0.0930 + 0.8122i 
                 0.8276 + 1.0225i 
                 0.0667 + 0.5124i 
                 0.7989 + 1.0498i 
               
               
                 x57 
                 0.2215 + 0.7840i 
                 0.6313 + 1.0364i 
                 0.2008 + 0.5095i 
                 0.4395 + 1.4203i 
               
               
                 x58 
                 0.0937 + 0.6514i 
                 0.8815 + 1.2865i 
                 0.0625 + 0.3658i 
                 0.6118 + 1.0246i 
               
               
                 x59 
                 0.1540 + 0.6366i 
                 0.6342 + 1.2705i 
                 0.1899 + 0.3642i 
                 0.6303 + 1.2421i 
               
               
                 x60 
                 0.4810 + 0.6306i 
                 1.0422 + 0.9593i 
                 0.4818 + 0.4946i 
                 1.0550 + 0.8924i 
               
               
                 x61 
                 0.3856 + 0.7037i 
                 1.2749 + 0.8538i 
                 0.3380 + 0.5050i 
                 0.8612 + 1.2800i 
               
               
                 x62 
                 0.3527 + 0.5230i 
                 1.1556 + 1.1847i 
                 0.4571 + 0.3499i 
                 1.2696 + 0.8969i 
               
               
                 x63 
                 0.3100 + 0.5559i 
                 1.4771 + 0.6742i 
                 0.3216 + 0.3599i 
                 1.0342 + 1.1181i 
               
               
                   
               
            
           
         
       
     
     Table 26 indicates non-uniform QPSK, table 27 indicates non-uniform 16-QAM, Tables 28 and 29 indicate non-uniform 64-QAM, and table 30 indicates non-uniform 256-QAM. 
     Referring to these tables, the constellation point of the first quadrant may be defined with reference to tables 26 to 30, and the constellation points in the other three quadrants may be defined in the above-described method. 
     However, this is merely an example and the modulator  130  may map the output bits outputted from the demultiplexer (not shown) onto the constellation points in various methods. 
     The interleaving is performed in the above-described method for the following reasons. 
     Specifically, when the LDPC codeword bits are mapped onto the modulation symbol, the bits may have different reliability (that is, receiving performance or receiving probability) according to where the bits are mapped onto in the modulation symbol. The LDPC codeword bits may have different codeword characteristics according to the configuration of a parity check matrix. That is, the LDPC codeword bits may have different codeword characteristics according to the number of 1 existing in the column of the parity check matrix, that is, the column degree. 
     Accordingly, the interleaver  120  may interleave to map the LDPC codeword bits having a specific codeword characteristic onto specific bits in the modulation symbol by considering both the codeword characteristics of the LDPC codeword bits and the reliability of the bits constituting the modulation symbol. 
     For example, when the LDPC codeword formed of bit groups X 0  to X 179  is group-interleaved based on Equation 21 and Table 11, the group interleaver  122  may output the bit groups in the order of X 55 , X 146 , X 83 , . . . , X 132 , X 135 . 
     In this case, when the modulation method is 16-QAM, the number of columns of the block interleaver  124  is four (4) and each column may be formed of 16200 rows. 
     Accordingly, from among the 180 groups constituting the LDPC codeword, 45 bit groups (X 55 , X 146 , X 83 , X 52 , X 62 , X 176 , X 160 , X 68 , X 53 , X 56 , X 81 , X 97 , X 79 , X 113 , X 163 , X 61 , X 58 , X 69 , X 133 , X 108 , X 66 , X 71 , X 86 , X 144 , X 57 , X 67 , X 116 , X 59 , X 70 , X 156 , X 172 , X 65 , X 149 , X 155 , X 82 , X 138 , X 136 , X 141 , X 111 , X 96 , X 170 , X 90 , X 140 , X 64 , X 159 ) may be inputted to the first column of the block interleaver  124 , 45 bit groups (X 15 , X 14 , X 37 , X 54 , X 44 , X 63 , X 43 , X 18 , X 47 , X 7 , X 25 , X 34 , X 29 , X 30 , X 26 , X 39 , X 16 , X 41 , X 45 , X 36 , X 0 , X 23 , X 32 , X 28 , X 27 , X 38 , X 48 , X 33 , X 22 , X 49 , X 51 , X 60 , X 46 , X 21 , X 4 , X 3 , X 20 , X 13 , X 50 , X 35 , X 24 , X 40 , X 17 , X 42 , X 6 ) may be inputted to the second column of the block interleaver  124 , 45 bit groups (X 112 , X 93 , X 127 , X 101 , X 94 , X 115 , X 105 , X 31 , X 19 , X 177 , X 74 , X 10 , X 145 , X 162 , X 102 , X 120 , X 126 , X 95 , X 73 , X 152 , X 129 , X 174 , X 125 , X 72 , X 128 , X 78 , X 171 , X 8 , X 142 , X 178 , X 154 , X 85 , X 107 , X 75 , X 12 , X 9 , X 151 , X 77 , X 117 , X 109 , X 80 , X 106 , X 134 , X 98 , X 1 ) may be inputted to the third column of the block interleaver  124 , and 45 bit groups (X 122 , X 173 , X 161 , X 150 , X 110 , X 175 , X 166 , X 131 , X 119 , X 103 , X 139 , X 148 , X 157 , X 114 , X 147 , X 87 , X 158 , X 121 , X 164 , X 104 , X 89 , X 179 , X 123 , X 118 , X 99 , X 88 , X 11 , X 92 , X 165 , X 84 , X 168 , X 124 , X 169 , X 2 , X 130 , X 167 , X 153 , X 137 , X 143 , X 91 , X 100 , X 5 , X 76 , X 132 , X 135 ) may be inputted to the fourth column of the block interleaver  124 . 
     In addition, the block interleaver  124  may output the bits inputted to the Pt row to the last row of each column serially, and the bits outputted from the block interleaver  124  may be inputted to the modulator  130  serially. In this case, the demultiplexer (not shown) may be omitted or the bits may be outputted serially without changing the order of bits inputted to the demultiplexer (not shown). Accordingly, the bits included in each of the bit groups X 55 , X 15 , X 112 , and X 122  may constitute the modulation symbol. 
     When the modulation method is 64-QAM, the number of columns of the block interleaver  124  is six (6) and each column may be formed of 10800 rows. 
     Accordingly, from among the 180 groups constituting the LDPC codeword, 30 bit groups (X 55 , X 146 , X 83 , X 52 , X 62 , X 176 , X 160 , X 68 , X 53 , X 56 , X 81 , X 97 , X 79 , X 113 , X 163 , X 61 , X 58 , X 69 , X 133 , X 108 , X 66 , X 71 , X 86 , X 144 , X 57 , X 67 , X 116 , X 59 , X 70 , X 156 ) may be inputted to the first column of the block interleaver  124 , 30 bit groups (X 172 , X 65 , X 149 , X 155 , X 82 , X 138 , X 136 , X 141 , X 111 , X 96 , X 170 , X 90 , X 140 , X 64 , X 159 , X 15 , X 14 , X 37 , X 54 , X 44 , X 63 , X 43 , X 18 , X 47 , X 7 , X 25 , X 34 , X 29 , X 30 , X 26 ) may be inputted to the second column of the block interleaver  124 , 30 bit groups (X 39 , X 16 , X 41 , X 45 , X 36 , X 0 , X 23 , X 32 , X 28 , X 27 , X 38 , X 48 , X 33 , X 22 , X 49 , X 51 , X 60 , X 46 , X 21 , X 4 , X 3 , X 20 , X 13 , X 50 , X 35 , X 24 , X 40 , X 17 , X 42 , X 6 ) may be inputted to the third column of the block interleaver  124 , 30 bit groups (X 112 , X 93 , X 127 , X 101 , X 94 , X 115 , X 105 , X 31 , X 19 , X 177 , X 74 , X 10 , X 145 , X 162 , X 102 , X 120 , X 126 , X 95 , X 73 , X 152 , X 129 , X 174 , X 125 , X 72 , X 128 , X 78 , X 171 , X 8 , X 142 , X 178 ) may be inputted to the fourth column of the block interleaver  124 , 30 bit groups (X 154 , X 85 , X 107 , X 75 , X 12 , X 9 , X 151 , X 77 , X 117 , X 109 , X 80 , X 106 , X 134 , X 98 , X 1 , X 122 , X 173 , X 161 , X 150 , X 110 , X 175 , X 166 , X 131 , X 119 , X 103 , X 139 , X 148 , X 157 , X 114 , X 147 ) may be inputted to the fifth column of the block interleaver  124 , and 30 bit groups (X 87 , X 158 , X 121 , X 164 , X 104 , X 89 , X 179 , X 123 , X 118 , X 99 , X 88 , X 11 , X 92 , X 165 , X 84 , X 168 , X 124 , X 169 , X 2 , X 130 , X 167 , X 153 , X 137 , X 143 , X 91 , X 100 , X 5 , X 76 , X 132 , X 135 ) may be inputted to the sixth column of the block interleaver  124 . 
     In addition, the block interleaver  124  may output the bits inputted to the Pt row to the last row of each column serially, and the bits outputted from the block interleaver  124  may be inputted to the modulator  130  serially. In this case, the demultiplexer (not shown) may be omitted or the bits may be outputted serially without changing the order of bits inputted to the demultiplexer (not shown). Accordingly, the bits included in each of the bit groups X 55 , X 172 , X 39 , X 112 , X 154 , and X 87  may constitute the modulation symbol. 
     As described above, since a specific bit is mapped onto a specific bit in a modulation symbol through interleaving, a receiver side can achieve high receiving performance and high decoding performance. 
     That is, when LDPC codeword bits of high decoding performance are mapped onto high reliability bits from among bits of each modulation symbol, the receiver side may show high decoding performance, but there is a problem that the LDPC codeword bits of the high decoding performance may not be received. In addition, when the LDPC codeword bits of high decoding performance are mapped onto low reliability bits from among the bits of the modulation symbol, initial receiving performance is excellent, and thus, overall performance is also excellent. However, when many bits showing poor decoding performance are received, error propagation may occur. 
     Accordingly, when LDPC codeword bits are mapped onto modulation symbols, an LDPC codeword bit having a specific codeword characteristic is mapped onto a specific bit of a modulation symbol by considering both codeword characteristics of the LDPC codeword bits and reliability of the bits of the modulation symbol, and is transmitted to the receiver side. Accordingly, the receiver side can achieve high receiving performance and decoding performance. 
     Hereinafter, a method for determining π(j), which is a parameter used for group interleaving, according to various exemplary embodiments, will be explained. 
     According to an exemplary embodiment, when the length of the LDPC codeword is 64800, the size of the bit group is determined to be 360 and thus 180 bit groups exist. In addition, there may be 180! possible interleaving patterns (Herein, factorial means A!=A×(A−1)× . . . ×2×1) regarding an integer A. 
     In this case, since a reliability level between the bits constituting a modulation symbol may be the same according to a modulationorder, many number of interleaving patterns may be regarded as the same interleaving operation when theoretical performance is considered. For example, when an MSB bit of the X-axis (or rear part-axis) and an MSB bit the Y-axis(or imaginary part-axis) of a certain modulation symbol have the same theoretical reliability, the same theoretical performance can be achieved regardless of the way how specific bits are interleaved to be mapped onto the two MSB bits. 
     However, such a theoretical prediction may become incorrect as a real channel environment is established. For example, in the case of the QPSK modulation method, two bits of a symbol in a part of a symmetric channel like an additive white Gaussian noise (AWGN) channel theoretically have the same reliability. Therefore, there should be no difference in the performance theoretically when any interleaving method is used. However, in a real channel environment, the performance may be different depending on the interleaving method. In the case of a well-known Rayleigh channel which is not a real channel, the performance of QPSK greatly depends on the interleaving method and thus the performance can be predicted somewhat only by the reliability between bits of a symbol according to a modulation method. However, there should be a limit to predicting the performance. 
     In addition, since code performance by interleaving may be greatly changed according to a channel which evaluates performance, channels should be always considered to drive an interleaving pattern. For example, a good interleaving pattern in the AWGN channel may be not good in the Rayleigh channel. If a channel environment where a given system is used is closer to the Rayleigh channel, an interleaving pattern which is better in the Rayleigh channel than in the AWGN channel may be selected. 
     As such, not only a specific channel environment but also various channel environments considered in a system should be considered in order to derive a good interleaving pattern. In addition, since there is a limit to predicting real performance only by theoretical performance prediction, the performance should be evaluated by directly conducting computation experiments and then the interleaving pattern should be finally determined. 
     However, since there are so many number of possible interleaving patterns to be applied (for example, 180!), reducing the number of interleaving patterns used to predict and test performance is an important factor in designing a high performance interleaver. 
     Therefore, the interleaver is designed through the following steps according to an exemplary embodiment. 
     1) Channels C 1 , C 2 , . . . , C k  to be considered by a system are determined. 
     2) A certain interleaver pattern is generated. 
     3) A theoretical performance value is predicted by applying the interleaver generated in step 2) to each of the channels determined in step 1). There are various methods for predicting a theoretical performance value, but a well-known noise threshold determining method like density evolution analysis is used according to an exemplary embodiment. The noise threshold recited herein refers to a value that can be expressed by a minimum necessary signal-to-noise ratio (SNR) capable of error-free transmission on the assumption that a cycle-free characteristic is satisfied when the length of a code is infinite and the code is expressed by the Tanner graph. The density evolution analysis may be implemented in various ways, but is not the subject matter of the inventive concept and thus a detailed description thereof is omitted. 
     4) When noise thresholds for the channels are expressed as TH 1 [i], TH 2 [i], . . . , TH k [i] for the i-th generated interleaver, a final determination threshold value may be defined as follows: 
         TH [ i ]= W   1   ×TH   1 [ i ] M+W   2   ×TH   2 [ i ]+ . . . + W   k   ×TH   k [ i ],         where W 1 +W 2 + . . . +W k =1,W 1 ,W 2 , . . . , W k &gt;0       
     Here, W 1 , W 2 , . . . , W k  are adjusted according to importance of the channels. That is, W 1 , W 2 , . . . , W k  are adjusted to a larger value in a more important channel and W 1 , W 2 , . . . , W k  are adjusted to a smaller value in a less important channel (for example, if the weight values of AWGN and Rayleigh channels are W 1  and W 2 , respectively, W 1  may be set to 0.25 and W 2  may be set to 0.75 when one of the channels is determined to be more important.). 
     5) B number of interleaver patterns are selected in an ascending order of TH[i] values from among the tested interleaver patterns and are directly tested by conducting performance computation experiments. An FER level for the test is determined as 10{circumflex over ( )}−3 (for example, B=100). 
     6) D number of best interleaver patterns are selected from among the B number of interleaver patterns tested in step 5) (for example, D=5). 
     In general, an interleaver pattern which has a great SNR gain in the area of FER=10{circumflex over ( )}−3 may be selected as a good performance interleaver in step of 5). However, according to an exemplary embodiment, as shown in  FIG.  16   , performance of FER required in the system based on the result of real computation experiments for the area of FER=10{circumflex over ( )}−3 may be predicted through extrapolation, and then an interleaver pattern having good performance in comparison with the expected performance in the FER required in the system may be determined as a good interleaver pattern. According to an exemplary embodiment, the extrapolation based on a linear function may be applied. However, various extrapolation methods may be applied.  FIG.  16    illustrates an example of performance extrapolation predicted by the result of computation experiments. 
     7) The D number of interleaver patterns selected in step 6) are tested by conducting performance computation experiments in each channel. Herein, the FER level for testing is selected as FER required in the system (for example, FER=10{circumflex over ( )}− 6 ) 
     8) When an error floor is not observed after the computation experiments, an interleaving pattern having the greatest SNR gain is determined as a final interleaving pattern. 
       FIG.  17    is a view schematically showing a process of determining B number of interleaver patterns in the steps 2), 3), 4), and 5) of the above-described method for determining the interleaving pattern in the case of AWGN and Rayleigh channels for example. 
     Referring to  FIG.  17   , necessary variables i, j, and etc. are initialized in operation S 1701 , and a noise threshold for the AWGN channel THIN and a noise threshold for the Rayleigh channel TH2[i] are calculated in operation S 1702 . Then, a final determination noise threshold TH[i] defined in step 4) is calculated in operation S 1703 , and is compared with a previously calculated final determination noise threshold TH[i−1] in operation S 1704 . When the final determination noise threshold TH[i] is smaller than the previously calculated final determination noise threshold TH[i−1], TH_S[i] is replaced with the TH[i] and is sotred in operation S 1706 . Next, i, j values increase by 1 in operation S 1707  and this process is repeated until the i value exceeds A which is pre-defined in operation S 1708 . In this case, A is the total number of interleaver patterns to be tested in steps 2), 3), 4), and 5) and A is typically determined to be greater than or equal to 10000. When all operations described above are completed, interleaver patterns corresponding to TH_S[0], TH_S[1], . . . , TH_S[B−1] which are stored in a descending order of final noise thresholds values in operation S 1709 . 
     The transmitting apparatus  100  may transmit the signal mapped onto the constellation to a receiving apparatus (for example,  1200  of  FIG.  18   ). For example, the transmitting apparatus  100  may map the signal mapped onto the constellation onto an Orthogonal Frequency Division Multiplexing (OFDM) frame using OFDM, and may transmit the signal to the receiving apparatus  1200  through an allocated channel. 
       FIG.  18    is a block diagram to illustrate a configuration of a receiving apparatus according to an exemplary embodiment. Referring to  FIG.  18   , the receiving apparatus  1200  includes a demodulator  1210 , a multiplexer  1220 , a deinterleaver  1230  and a decoder  1240 . 
     The demodulator  1210  receives and demodulates a signal transmitted from the transmitting apparatus  100 . Specifically, the demodulator  1210  generates a value corresponding to an LDPC codeword by demodulating the received signal, and outputs the value to the multiplexer  1220 . In this case, the demodulator  1210  may use a demodulation method corresponding to a modulation method used in the transmitting apparatus  100 . To do so, the transmitting apparatus  100  may transmit information regarding the modulation method to the receiving apparatus  1200 , or the transmitting apparatus  100  may perform modulation using a pre-defined modulation method between the transmitting apparatus  100  and the receiving apparatus  1200 . 
     The value corresponding to the LDPC codeword may be expressed as a channel value for the received signal. There are various methods for determining the channel value, and for example, a method for determining a Log Likelihood Ratio (LLR) value may be the method for determining the channel value. 
     The LLR value is a log value for a ratio of the probability that a bit transmitted from the transmitting apparatus  100  is 0 and the probability that the bit is 1. In addition, the LLR value may be a bit value which is determined by a hard decision, or may be a representative value which is determined according to a section to which the probability that the bit transmitted from the transmitting apparatus  100  is 0 or 1 belongs. 
     The multiplexer  1220  multiplexes the output value of the demodulator  1210  and outputs the value to the deinterleaver  1230 . 
     Specifically, the multiplexer  1220  is an element corresponding to a demultiplexer (not shown) provided in the transmitting apparatus  100 , and performs an operation corresponding to the demultiplexer (not shown). That is, the multiplexer  1220  performs an inverse operation of the operation of the demultiplexer (not shown), and performs cell-to-bit conversion with respect to the output value of the demodulator  1210  and outputs the LLR value in the unit of bit. However, when the demultiplexer (not shown) is omitted from the transmitting apparatus  100 , the multiplexer  1220  may be omitted from the receiving apparatus  1200 . 
     The information regarding whether the demultiplexing operation is performed or not may be provided by the transmitting apparatus  100 , or may be pre-defined between the transmitting apparatus  100  and the receiving apparatus  1200 . 
     The deinterleaver  1230  deinterleaves the output value of the multiplexer  1220  and outputs the values to the decoder  1240 . 
     Specifically, the deinterleaver  1230  is an element corresponding to the interleaver  120  of the transmitting apparatus  100  and performs an operation corresponding to the interleaver  120 . That is, the deinterleaver  1230  deinterleaves the LLR value by performing the interleaving operation of the interleaver  120  inversely. 
     To do so, the deinterleaver  1230  may include a block deinterleaver  1231 , a group twist deinterleaver  1232 , a group deinterleaver  1233 , and a parity deinterleaver  1234  as shown in  FIG.  18   . 
     The block deinterleaver  1231  deinterleaves the output of the multiplexer  1220  and outputs the value to the group twist deinterleaver  1232 . 
     Specifically, the block deinterleaver  1231  is an element corresponding to the block interleaver  124  provided in the transmitting apparatus  100  and performs the interleaving operation of the block interleaver  124  inversely. 
     That is, the block deinterleaver  1231  deinterleaves by writing the LLR value output from the multiplexer  1220  in each row in the row direction and reading each column of the plurality of rows in which the LLR value is written in the column direction by using at least one row formed of the plurality of columns. 
     In this case, when the block interleaver  124  interleaves by dividing the column into two parts, the block deinterleaver  1231  may deinterleave by dividing the row into two parts. 
     In addition, when the block interleaver  124  writes and reads in and from the bit group that does not belong to the first part in the row direction, the block deinterleaver  1231  may deinterleave by writing and reading values corresponding to the group that does not belong to the first part in the row direction. 
     Hereinafter, the block deinterleaver  1231  will be explained with reference to  FIG.  20   . However, this is merely an example and the block deinterleaver  1231  may be implemented in other methods. 
     An input LLR v i  (0≤i≤N ldpc ) is written in a r i  row and a c i  column of the block deinterleaver  1231 . Herein, c i =(i mod N c ) and 
     
       
         
           
             
               
                 r 
                 i 
               
               = 
               
                 ⌊ 
                 
                   i 
                   
                     N 
                     c 
                   
                 
                 ⌋ 
               
             
             , 
           
         
       
     
     On the other hand, an output LLR q i (0≤i≤N c ×N r1 ) is read from a c i  column and a r i  row of the first part of the block deinterleaver  1231 . Herein, 
     
       
         
           
             
               
                 c 
                 i 
               
               = 
               
                 ⌊ 
                 
                   i 
                   
                     N 
                     
                       r 
                       ⁢ 
                       1 
                     
                   
                 
                 ⌋ 
               
             
             , 
           
         
       
     
     r i =(i mod N r1 ). 
     In addition, an output LLR q i (N c ×N r1 ≤i≤N ldpc ) is read from a c i  column and a r i  row of the second part. Herein, 
     
       
         
           
             
               
                 c 
                 i 
               
               = 
               
                 ⌊ 
                 
                   
                     ( 
                     
                       i 
                       - 
                       
                         
                           N 
                           c 
                         
                         × 
                         
                           N 
                           
                             r 
                             ⁢ 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   
                     N 
                     
                       r 
                       ⁢ 
                       2 
                     
                   
                 
                 ⌋ 
               
             
             , 
           
         
       
     
     r i =N r1 +{(i−N c ×N r1 ) mode N r2 }. 
     The group twist deinterleaver  1232  deinterleaves the output value of the block deinterleaver  1231  and outputs the value to the group deinterleaver  1233 . 
     Specifically, the group twist deinterleaver  1232  is an element corresponding to the group twist interleaver  123  provided in the transmitting apparatus  100 , and may perform the interleaving operation of the group twist interleaver  123  inversely. 
     That is, the group twist deinterleaver  1232  may rearrange the LLR values of the same bit group by changing the order of the LLR values existing in the same bit group. When the group twist operation is not performed in the transmitting apparatus  100 , the group twist deinterleaver  1232  may be omitted. 
     The group deinterleaver  1233  (or the group-wise deinterleaver) deinterleaves the output value of the group twist deinterleaver  1232  and outputs the value to the parity deinterleaver  1234 . 
     Specifically, the group deinterleaver  1233  is an element corresponding to the group interleaver  122  provided in the transmitting apparatus  100  and may perform the interleaving operation of the group interleaver  122  inversely. 
     That is, the group deinterleaver  1233  may rearrange the order of the plurality of bit groups in bit group wise. In this case, the group deinterleaver  1233  may rearrange the order of the plurality of bit groups in bit group wise by applying the interleaving method of Tables 11 to 22 inversely according to a length of the LDPC codeword, a modulation method and a code rate. 
     The parity deinterleaver  1234  performs parity deinterleaving with respect to the output value of the group deinterleaver  1233  and outputs the value to the decoder  1240 . 
     Specifically, the parity deinterleaver  1234  is an element corresponding to the parity interleaver  121  provided in the transmitting apparatus  100  and may perform the interleaving operation of the parity interleaver  121  inversely. That is, the parity deinterleaver  1234  may deinterleave the LLR values corresponding to the parity bits from among the LLR values output from the group deinterleaver  1233 . In this case, the parity deinterleaver  1234  may deinterleave the LLR value corresponding to the parity bits inversely to the parity interleaving method of Equation 18. 
     However, the parity deinterleaver  1234  may be omitted depending on the decoding method and embodiment of the decoder  1240 . 
     Although the deinterleaver  1230  of  FIG.  18    includes three (3) or four (4) elements as shown in  FIG.  19   , operations of the elements may be performed by a single element. For example, when bits each of which belongs to each of bit groups X a , X b , X c , X d  constitute a single modulation symbol, the deinterleaver  1230  may deinterleave these bits to locations corresponding to their bit groups based on the received single modulation symbol. 
     For example, when the code rate is 6/15 and the modulation method is 16-QAM, the group deinterleaver  1233  may perform deinterleaving based on table 11. 
     In this case, bits each of which belongs to each of bit groups X 55 , X 15 , X 112 , X 122  may constitute a single modulation symbol. Since one bit in each of the bit groups X 55 , X 15 , X 112 , X 122  constitutes a single modulation symbol, the deinterleaver  1230  may map bits onto decoding initial values corresponding to the bit groups X 55 , X 15 , X 112 , X 122  based on the received single modulation symbol. 
     The decoder  1240  may perform LDPC decoding by using the output value of the deinterleaver  1230 . To achieve this, the decoder  1240  may include an LDPC decoder (not shown) to perform the LDPC decoding. 
     Specifically, the decoder  1240  is an element corresponding to the encoder  110  of the transmitting apparatus  100  and may correct an error by performing the LDPC decoding by using the LLR value output from the deinterleaver  1230 . 
     For example, the decoder  1240  may perform the LDPC decoding in an iterative decoding method based on a sum-product algorithm. The sum-product algorithm is one example of a message passing algorithm, and the message passing algorithm refers to an algorithm which exchanges messages (e.g., LLR value) through an edge on a bipartite graph, calculates an output message from messages input to variable nodes or check nodes, and updates. 
     The decoder  1240  may use a parity check matrix when performing the LDPC decoding. In this case, the parity check matrix used in the decoding may have the same configuration as that of the parity check matrix used in the encoding of the encoder  110 , and this has been described above with reference to  FIGS.  2  to  4   . 
     In addition, information on the parity check matrix and information on the code rate, etc. which are used in the LDPC decoding may be pre-stored in the receiving apparatus  1200  or may be provided by the transmitting apparatus  100 . 
       FIG.  21    is a flowchart to illustrate an interleaving method of a transmitting apparatus according to an exemplary embodiment. 
     First, an LDPC codeword is generated by LDPC encoding based on a parity check matrix (S 1410 ), and the LDPC codeword is interleaved (S 1420 ). 
     Then, the interleaved LDPC codeword is mapped onto a modulation symbol (S 1430 ). In this case, a bit included in a predetermined bit group from among a plurality of bit groups constituting the LDPC codeword may be mapped onto a predetermined bit in the modulation symbol. 
     Each of the plurality of bit groups may be formed of M number of bits, and M may be a common divisor of N ldpc  and K ldpc  and may be determined to satisfy Q ldpc =(N ldpc −K ldpc )/M. Herein, Q ldpc  is a cyclic shift parameter value regarding columns in a column group of an information word submatrix of the parity check matrix, N ldpc  is a length of the LDPC codeword, and K ldpc  is a length of information word bits of the LDPC codeword. 
     Operation S 1420  may include interleaving parity bits of the LDPC codeword, dividing the parity-interleaved LDPC codeword by the plurality of bit groups and rearranging the order of the plurality of bit groups in bit group wise, and interleaving the plurality of bit groups the order of which is rearranged. 
     The order of the plurality of bit groups may be rearranged in bit group wise based on the above-described Equation 21 presented above. 
     As described above, π(j) in Equation 21 may be determined based on at least one of a length of the LDPC codeword, a modulation method, and a code rate. 
     For example, when the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 6/15, π(j) may be defined as in table 11. 
     In addition, when the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 10/15, π(j) may be defined as in table 14. 
     In addition, when the LDPC codeword has a length of 64800, the modulation method is 16-QAM, and the code rate is 12/15, π(j) may be defined as in table 15. 
     In addition, when the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 6/15, π(j) may be defined as in table 17. 
     In addition, when the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 8/15, π(j) may be defined as in table 18. 
     In addition, when the LDPC codeword has a length of 64800, the modulation method is 64-QAM, and the code rate is 12/15, π(j) may be defined as in table 21. 
     The interleaving the plurality of bit groups may include: writing the plurality of bit groups in each of a plurality of columns in bit group wise in a column direction, and reading each row of the plurality of columns in which the plurality of bit groups are written in bit group wise in a row direction. 
     In addition, the interleaving the plurality of bit groups may include: serially write, in the plurality of columns, at least some bit group which is writable in the plurality of columns in bit group wise from among the plurality of bit groups, and then dividing and writing the other bit groups in an area which remains after the at least some bit group is written in the plurality of columns in bit group wise. 
     A non-transitory computer readable medium, which stores a program for performing the interleaving methods according to various exemplary embodiments in sequence, may be provided. 
     The non-transitory computer readable medium refers to a medium that stores data semi-permanently rather than storing data for a very short time, such as a register, a cache, and a memory, and is readable by an apparatus. Specifically, the above-described various applications or programs may be stored in a non-transitory computer readable medium such as a compact disc (CD), a digital versatile disk (DVD), a hard disk, a Blu-ray disk, a universal serial bus (USB), a memory card, and a read only memory (ROM), and may be provided. 
     At least one of the components, elements or units represented by a block as illustrated in  FIGS.  1 ,  5 ,  15 ,  18  and  19    may be embodied as various numbers of hardware, software and/or firmware structures that execute respective functions described above, according to an exemplary embodiment. For example, at least one of these components, elements or units may use a direct circuit structure, such as a memory, processing, logic, a look-up table, etc. that may execute the respective functions through controls of one or more microprocessors or other control apparatuses. Also, at least one of these components, elements or units may be specifically embodied by a module, a program, or a part of code, which contains one or more executable instructions for performing specified logic functions. Also, at least one of these components, elements or units may further include a processor such as a central processing unit (CPU) that performs the respective functions, a microprocessor, or the like. Further, although a bus is not illustrated in the above block diagrams, communication between the components, elements or units may be performed through the bus. Functional aspects of the above exemplary embodiments may be implemented in algorithms that execute on one or more processors. Furthermore, the components, elements or units represented by a block or processing steps may employ any number of related art techniques for electronics configuration, signal processing and/or control, data processing and the like. 
     The foregoing exemplary embodiments and advantages are merely exemplary and are not to be construed as limiting the present inventive concept. The exemplary embodiments can be readily applied to other types of apparatuses. Also, the description of the exemplary embodiments is intended to be illustrative, and not to limit the scope of the inventive concept, and many alternatives, modifications, and variations will be apparent to those skilled in the art.