# Patent

## Title
Transmitting apparatus and signal processing method thereof

## Abstract

A transmitting apparatus and a receiving apparatus are provided. The transmitting apparatus includes an encoder configured to generate a low density parity check (LDPC) codeword by performing LDPC encoding, an interleaver configured to interleave the LDPC codeword, and a modulator configured to modulate the interleaved LDPC codeword according to a modulation method to generate a modulation symbol. The interleaver performs interleaving by dividing the LDPC codeword into a plurality of groups, rearranging an order of the plurality of groups in group units, and dividing the plurality of rearranged groups based on a modulation order according to the modulation method.

## Background
CROSS-REFERENCE TO RELATED APPLICATION(S)
This application claims the benefit under 35 U.S.C. §119 from U.S. Provisional Application No. 61/897,460 field on Oct. 30, 2013, in the United States Patent and Trademark Office, and Korean Patent Application No. 10-2014-0149342, filed Oct. 30, 2014, in the Korean Intellectual Property Office, the disclosure of which are incorporated herein by reference in their entirety.


BACKGROUND
1. Technical Field
Apparatuses and methods consistent with exemplary embodiments relate to a transmitting apparatus and a signal processing method thereof, and more particularly, to a transmitting apparatus which processes data and transmits the data, and a signal processing method thereof.
2. Description of the Related Art
In a communication/broadcasting system, link performance may greatly deteriorate due to various noises of channels, a fading phenomenon, and an inter-symbol interference (ISI). Therefore, in order to implement high digital communication/broadcasting systems requiring high data throughput and reliability, such as next-generation mobile communication, digital broadcasting, and portable Internet, there is a demand for a method for overcoming the noise, fading, and inter-symbol interference. To overcome the noise, etc., research on an error-correction code has been actively conducted in recent years as a method for effectively restoring distorted information and enhancing reliability of communication.
The Low Density Parity Check (LDPC) code which was first introduced by Gallager in the 1960s has been forgotten for a long time due to its difficulty and complexity in realizing by the level of technology at that time. However, as the turbo code which was suggested by Berrou, Glavieux, Thitimajshima in 1993 showed performance equivalent to the channel capacity of Shannon, the performance and characteristics of the turbo code were actively interpreted and many researches on channel encoding based on iterative decoding and graph were conducted. This leaded the re-research on the LDPC code in the late 1990's and it turned out that decoding by applying iterative decoding based on a sum-product algorithm on a Tanner graph corresponding to the LDPC code resulted in the performance equivalent to the channel capacity of Shannon.
When the LDPC code is transmitted by using a high order modulation scheme, performance depends on how codeword bits are mapped onto high order modulation bits. Therefore, there is a need for a method for mapping LDPC codeword bits onto high order modulation bits to obtain an LDPC code of good performance.
SUMMARY
One or more exemplary embodiments may overcome the above disadvantages and other disadvantages not described above. However, it is understood that one or more exemplary embodiment are not required to overcome the disadvantages described above, and may not overcome any of the problems described above.
One or more exemplary embodiments provide a transmitting apparatus which can map a bit included in a predetermined group from among a plurality of groups of a Low Density Parity Check (LDPC) codeword onto a predetermined bit of a modulation symbol, and transmit the bit, and a signal processing method thereof.
According to an exemplary embodiment, there is provided a transmitting apparatus including: an encoder configured to generate a Low Density Parity Check (LDPC) codeword by performing LDPC encoding, an interleaver configured to interleave the LDPC codeword, and a modulator configured to modulate the interleaved LDPC codeword according to a modulation method to generate a modulation symbol, wherein the interleaver includes a group interleaver configured to group the LDPC codeword into a plurality of bit groups and rearrange an order of the plurality of bit groups in group units and a block interleaver configured to be formed of a plurality of columns each comprising a plurality of rows and classify and interleave the plurality of rearranged bit groups based on a modulation order which is determined according to the modulation method, and wherein the block interleaver performs interleaving by dividing each of the plurality of columns into a first part and a second part, sequentially writing the plurality of bit groups in a plurality of columns constituting the first part, dividing bits constituting the other bit groups into bit groups each consisting of predetermined number of bits based on the number of the plurality of columns, and sequentially writing the divided bit groups in a plurality of columns constituting the second part.
The number of the plurality of columns may have the same value as a modulation order according to the modulation method. In addition, each of the plurality of columns may be formed of rows corresponding to a value obtained by dividing the number of bits constituting the LDPC codeword by the number of the plurality of columns.
The first part may be formed of rows as many as the number of bits included in at least a part of bit group which is writable in bit group units in each of the plurality of columns among a plurality of bit groups constituting the LDPC codeword according to the number of the plurality of columns, the number of the bit groups, and the number of bits constituting each bit group, in each of the plurality of columns. In addition, the second part may be formed of rows as many as rows excluding the number of bits included in at least a part of bit group which is writable in bit group units in each of the plurality of columns of the rows constituting each of the plurality of columns, in each of the plurality of columns.
The number of rows of the second part may have the same value as a quotient obtained by dividing the number of bits included in all bit groups excluding the bit group corresponding to the first part by the number of columns constituting the block interleaver.
The block interleaver may sequentially write bits included in the at least a part of bit group which is writable in bit group units in each of the plurality of columns constituting the first part, divide bits included in the other bit groups excluding at least a part of bit group from a plurality of bit groups based on the number of columns, and sequentially write the divided bits in each of the plurality of columns constituting the second part.
The block interleaver may perform interleaving by dividing the bits included in the other bit groups by the number of the plurality of columns, writing each of the divided bits in each of the plurality of columns constituting the second part in a column direction, and reading the plurality of columns constituting the first part and the second part in a row direction.
In response to the modulation method being QPSK, 16-QAM, 64-QAM, 256-QAMN, 1024-QAM, and 4096-QAM, the modulation order may be 2, 4, 6, 8, 10, and 12.
According to an exemplary embodiment, there is provided a method for processing a signal of a transmitting apparatus, the method including: generating a Low Density Parity Check (LDPC) codeword by performing LDPC encoding, interleaving the LDPC codeword, and modulating the interleaved LDPC codeword according to a modulation method to generate a modulation symbol, wherein the performing interleaving includes grouping the LDPC codeword into a plurality of bit groups and rearranging an order of the plurality of bit groups in group units and classifying and interleaving the plurality of rearranged bit groups based on a modulation order which is determined according to the modulation method, by using a plurality of columns each comprising a plurality of rows, and wherein the classifying and interleaving the plurality of rearranged bit groups includes dividing each of the plurality of columns into a first part and a second part, sequentially writing the plurality of bit groups in a plurality of columns constituting the first part, dividing bits constituting the other bit groups into bit groups each consisting of predetermined number of bits based on the number of the plurality of columns, and sequentially writing the divided bit groups in a plurality of columns constituting the second part.
The number of the plurality of columns may have the same value as a modulation order according to the modulation method. In addition, each of the plurality of columns may be formed of rows corresponding to a value obtained by dividing the number of bits constituting the LDPC codeword by the number of the plurality of columns.
The first part may be formed of rows as many as the number of bits included in at least a part of bit group which is writable in bit group units in each of the plurality of columns among a plurality of bit groups constituting the LDPC codeword according to the number of the plurality of columns, the number of the bit groups, and the number of bits constituting each bit group in each of the plurality of columns. In addition, the second part may be formed of rows as many as rows excluding the number of bits included in at least a part of bit group which is writable in bit group units in each of the plurality of columns of the rows constituting each of the plurality of columns, in each of the plurality of columns.
The number of rows of the second part may have the same value as a quotient obtained by dividing the number of bits included in all bit groups excluding the bit group corresponding to the first part by the number of columns.
The performing interleaving may include sequentially writing the bits included in the at least a part of bit group which is writable in bit group units in each of the plurality of columns constituting the first part, dividing the bits included in the other bit group excluding at least a part of bit group from a plurality of bit groups based on the number of columns, and sequentially writing the divided bits in each of the plurality of columns constituting the second part.
The performing interleaving may include dividing bits included in the other bit groups by the number of the plurality of columns, writing each of the divided bits in each of the plurality of columns constituting the second part in a column direction, and reading the plurality of columns constituting the first part and the second part in a row direction.
In response to the modulation method being QPSK, 16-QAM, 64-QAM, 256-QAMN, 1024-QAM, and 4096-QAM, the modulation order may be 2, 4, 6, 8, 10, and 12.



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 and 3 are views to illustrate a configuration of a parity check matrix according to exemplary embodiments;
FIG. 4 is a block diagram to illustrate a configuration of an interleaver according to an exemplary embodiment;
FIGS. 5 to 7 are views illustrating a method for processing an LDPC codeword on a group basis according to exemplary embodiments;
FIGS. 8 to 11 are views to illustrate a configuration of a block interleaver and an interleaving method according to exemplary embodiments;
FIGS. 12 and 13 are views to illustrate an operation of a demultiplexer according to exemplary embodiments;
FIG. 14 is a view to illustrate an example of a uniform constellation modulation method according to an exemplary embodiment;
FIGS. 15 to 19 are views to illustrate an example of a non-uniform constellation modulation method according to exemplary embodiments;
FIG. 20 is a block diagram to illustrate a configuration of an interleaver according to another exemplary embodiment;
FIGS. 21 to 23 are views to illustrate a configuration of a block-row interleaver and an interleaving method according to exemplary embodiments;
FIG. 24 is a block diagram to illustrate a configuration of a receiving apparatus according to an exemplary embodiment;
FIGS. 25 and 27 are block diagrams to illustrate a configuration of a deinterleaver according to exemplary embodiments;
FIG. 26 is a view to illustrate a block deinterleaver according to an exemplary embodiment; and
FIG. 28 is a flowchart to illustrate a signal processing 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 a first 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. The encoder 110 may include an LDPC encoder (not shown) to perform the LDPC encoding.
Specifically, the encoder 110 LDPC-encodes input bits to information word bits to generate the LDPC codeword which is formed of the information word bits and parity bits (that is, LDPC parity bits). Here, since an LDPC code for the LDPC encoding 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 Nldpc number of bits, and includes Kldpc number of information word bits and Nparity=Nldpc−Kldpc number of parity bits.
In this case, the encoder 110 may generate the LDPC codeword by performing the LDPC encoding based on a parity check matrix. That is, since the LDPC encoding is a process for generating an LDPC codeword to satisfy H·CT=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 separate 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 (DVS-C2), Digital Video Broadcasting-Satellite-Second Generation (DVS-S2), Digital Video Broadcasting-Second Generation Terrestrial (DVS-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 configuration of a parity check matrix will be explained in detail with reference to FIGS. 2 and 3.
First, referring to FIG. 2, a parity check matrix 200 is formed of an information word submatrix 210 corresponding to information word bits, and a parity submatrix 220 corresponding to parity bits. In the parity check matrix 200, elements other than elements with 1 have 0.
The information word submatrix 210 includes Kldpc number of columns and the parity submatrix 220 includes Nparity=Nldpc−Kldpc 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, Nparity=Nldpc−Kldpc.
In addition, in the parity check matrix 200, Nldpc is a length of an LDPC codeword, Kldpc is a length of information word bits, and Nparity=Nldpc−Kldpc 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 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 Kldpc number of columns (that is, 0th column to (Kldpc−1)th column), and follows the following rules:
First, M number of columns from among Kldpc number of columns of the information word submatrix 210 belong to the same group, and Kldpc number of columns is divided into Kldpc/M number of column groups. In each column group, a column is cyclic-shifted from an immediately previous column by Qldpc or Qldpc number of bits.
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 Qldpc 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. M and Qldpc are integers and are determined to satisfy Qldpc=(Nldpc−Kldpc)/M. In this case, Kldpc/M is also an integer. M and Qldpc may have various values according to a length of the LDPC codeword and a code rate.
For example, when M=360 and the length of the LDPC codeword, Nldpc, is 64800, Qldpc may be defined as in table 1 presented below, and, when M=360 and the length Nldpc of the LDPC codeword is 16200, Qldpc may be defined as in table 2 presented below.












TABLE 1







Code Rate
NIdpc
M
QIdpc





















 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
NIdpc
M
QIdpc





















 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 0th column of the ith column group (i=0, 1, . . . , Kldpc/M−1) is Di (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 0th column of the ith column group is Ri,0(0), Ri,0(1), . . . , Ri,0(Di−1), an index Ri,j(k) of a row where kth weight-1 is located in the jth column in the ith column group (that is, an index of a row where kth 1 is located in the jth column in the ith column group) is determined by following Equation 1:

Ri,j(k)=Ri,(j−1)(k)+Qldpc mod(Nldpc−Kldpc)  (1)

where k=0, 1, 2, . . . Di−1; i=0, 1, . . . , Kldpc/M−1; and j=1, 2, . . . , M−1.

Equation 1 can be expressed as following Equation 2:

Ri,j(k)={Ri,0(k)+(j mod M)×Qldpc} mod(Nldpc−Kldpc)  (2)

where k=0, 1, 2, . . . Di−1; i=0, 1, . . . , Kldpc/M−1; and j=1, 2, . . . , M−1.

In the above equations, Ri,j(k) is an index of a row where kth weight-1 is located in the jth column in the ith column group, Nldpc is a length of an LDPC codeword, Kldpc is a length of information word bits, Di is a degree of columns belonging to the ith column group, M is the number of columns belonging to a single column group, and Qldpc is a size by which each column in the column group is cyclic-shifted.
As a result, referring to these equations, when only Ri,0(k) is known, the index Ri,j(k) of the row where the kth weight-1 is located in the jth column in the ith column group can be known. Therefore, when the index value of the row where the kth weight-1 is located in the first column of each column group is stored, a position of column and row where weight-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 ith column group have the same degree Di. 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 Nldpc is 30, Kldpc is 15, and Qldpc is 3, position information of the row where weight-1 is located in the 0th 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”.

R1,0(1)=1, R1,0(2)=2, R1,0(3)=8, R1,0(4)=10,

R2,0(1)=0, R2,0(2)=9, R2,0(3)=13,

R3,0(1)=0, R3,0(2)=14.

where Ri,j(k) is an index of a row where kth weight-1 is located in the jth column in the ith column group.

The weight-1 position sequence like Equation 3 which expresses an index of a row where 1 is located in the 0th 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 weight-1, that is, the value 1, in the parity check matrix, and the ith weight-1 position sequence is expressed by indexes of rows where weight-1 is located in the 0th column belonging to the ith 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 27 presented below, based on the above descriptions.
Specifically, Tables 4 to 27 show indexes of rows where 1 is located in the 0th column of the ith 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 0th column of each of the plurality of column groups may be defined by Tables 4 to 27.
Herein, the indexes of the rows where 1 is located in the 0th column of the ith column group mean “addresses of parity bit accumulators”. The “addresses of parity bit accumulators” have the same meaning as defined in the DVS-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 Nldpc of the LDPC codeword is 16200, the code rate R is 5/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 4 presented below:








TABLE 4





i
Index of row where 1 is located in the 0th column of the ith column group
















0
245 449 491 980 1064 1194 1277 1671 2026 3186 4399 4900 5283 



5413 5558 6570 7492 7768 7837 7984 8306 8483 8685 9357 9642



10045 10179 10261 10338 10412


1
1318 1584 1682 1860 1954 2000 2062 3387 3441 3879 3931 4240 



4302 4446 4603 5117 5588 5675 5793 5955 6097 6221 6449 6616



7218 7394 9535 9896 10009 10763


2
105 472 785 911 1168 1450 2550 2851 3277 3624 4128 4460 4572 



4669 4783 5102 5133 5199 5905 6647 7028 7086 7703 8121 8217



9149 9304 9476 9736 9884


3
1217 5338 5737 8334


4
855 994 2979 9443


5
7506 7811 9212 9982


6
848 3313 3380 3990


7
2095 4113 4620 9946


8
1488 2396 6130 7483


9
1002 2241 7067 10418


10
2008 3199 7215 7502


11
1161 7705 8194 8534


12
2316 4803 8649 9359


13
125 1880 3177


14
1441 8033 9072









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 6/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 5 presented below:








TABLE 5





i
Index of row where 1 is located in the 0th column of the ith column group
















0
13 88 136 188 398 794 855 918 954 1950 2762 2837 2847 



4209 4342 5092 5334 5498 5731 5837 6150 6942 7127



7402 7936 8235 8307 8600 9001 9419 9442 9710


1
619 792 1002 1148 1528 1533 1925 2207 2766 3021 3267 



3593 3947 4832 4873 5109 5488 5882 6079 6276 6499



6584 6738 6795 7550 7723 7786 8732 9060 9270 9401


2
499 717 1551 1791 2535 3135 3582 3813 4047 4309 5126 



5186 5219 5716 5977 6236 6406 6586 6591 7085 7199



7485 7726 7878 8027 8066 8425 8802 9309 9464 9553 9671


3
658 4058 7824 8512


4
3245 4743 8117 9369


5
465 6559 8112 9461


6
975 2368 4444 6095


7
4218 5993 9182 9473


8
9 3822 5306 5320


9
4 8311 9571 9669


10
13 8122 8949 9656


11
3353 4449 5829 8053


12
7885 9118 9674


13
7575 9591 9670


14
431 8123 9271


15
4228 7587 9270


16
8847 9146 9556


17
11 5213 7763









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 7/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 6 presented below:








TABLE 6





i
Index of row where 1 is located in the 0th column of the ith column group
















0
432 655 893 942 1285 1427 1738 2199 2441 2565 2932 



3201 4144 4419 4678 4963 5423 5922 6433 6564 6656



7478 7514 7892


1
220 453 690 826 1116 1425 1488 1901 3119 3182 3568 



3800 3953 4071 4782 5038 5555 6836 6871 7131 7609



7850 8317 8443


2
300 454 497 930 1757 2145 2314 2372 2467 2819 3191 



3256 3699 3984 4538 4965 5461 5742 5912 6135 6649 



7636 8078 8455


3
24 65 565 609 990 1319 1394 1465 1918 1976 2463 2987 



3330 3677 4195 4240 4947 5372 6453 6950 7066 8412



8500 8599


4
1373 4668 5324 7777


5
189 3930 5766 6877


6
3 2961 4207 5747


7
1108 4768 6743 7106


8
1282 2274 2750 6204


9
2279 2587 2737 6344


10
2889 3164 7275 8040


11
133 2734 5081 8386


12
437 3203 7121


13
4280 7128 8490


14
619 4563 6206


15
2799 6814 6991


16
244 4212 5925


17
1719 7657 8554


18
53 1895 6685


19
584 5420 6856


20
2958 5834 8103









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 8/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 7, 8 or 9 presented below:








TABLE 7





i
Index of row where 1 is located in the 0th column of the ith column group







0
32 384 430 591 1296 1976 1999 2137 2175 3638 4214 4304 4486 4662 



4999 5174 5700 6969 7115 7138 7189


 1
1788 1881 1910 2724 4504 4928 4973 5616 5686 5718 5846 6523 6893 



6994 7074 7100 7277 7399 7476 7480 7537


 2
2791 2824 2927 4196 4298 4800 4948 5361 5401 5688 5818 5862 5969 



6029 6244 6645 6962 7203 7302 7454 7534


 3
574 1461 1826 2056 2069 2387 2794 3349 3366 4951 5826 5834 5903 



6640 6762 6786 6859 7043 7418 7431 7554


 4
14 178 675 823 890 930 1209 1311 2898 4339 4600 5203 6485 6549 



6970 7208 7218 7298 7454 7457 7462


 5
4075 4188 7313 7553


 6
5145 6018 7148 7507


 7
3198 4858 6983 7033


 8
3170 5126 5625 6901


 9
2839 6093 7071 7450


10
11 3735 5413


11
2497 5400 7238


12
2067 5172 5714


13
1889 7173 7329


14
1795 2773 3499


15
2695 2944 6735


16
3221 4625 5897


17
1690 6122 6816


18
5013 6839 7358


19
1601 6849 7415


20
2180 7389 7543


21
2121 6838 7054


22
1948 3109 5046


23
272 1015 7464

















TABLE 8





i
Index of row where 1 is located in the 0th column of the ith column group







 0
5 519 825 1871 2098 2478 2659 2820 3200 3294 3650 3804 3949 



4426 4460 4503 4568 4590 4949 5219 5662 5738 5905



5911 6160 6404 6637 6708 6737 6814 7263 7412


 1
81 391 1272 1633 2062 2882 3443 3503 3535 3908 4033 4163 4490 



4929 5262 5399 5576 5768 5910 6331 6430 6844 



6867 7201 7274 7290 7343 7350 7378 7387 7440 7554


 2
105 975 3421 3480 4120 4444 5957 5971 6119 6617 6761 6810 7067 7353


 3
6 138 485 1444 1512 2615 2990 3109 5604 6435 6513 6632 6704 7507


 4
20 858 1051 2539 3049 5162 5308 6158 6391 6604 6744 7071 7195 7238


 5
1140 5838 6203 6748


 6
6282 6466 6481 6638


 7
2346 2592 5436 7487


 8
2219 3897 5896 7528


 9
2897 6028 7018


10
1285 1863 5324


11
3075 6005 6466


12
5 6020 7551


13
2121 3751 7507


14
4027 5488 7542


15
2 6012 7011


16
3823 5531 5687


17
1379 2262 5297


18
1882 7498 7551


19
3749 4806 7227


20
2 2074 6898


21
17 616 7482


22
9 6823 7480


23
5195 5880 7559

















TABLE 9





i
Index of row where 1 is located in the 0th column of the ith column group







 0
6 243 617 697 1380 1504 1864 1874 1883 2075 2122 2439 2489 3076 



3715 3719 3824 4028 4807 5006 5196 5532 5688 5881 6216 



6899 7000 7118 7284 7412 7417 7523


 1
0 6 17 20 105 1279 2443 2523 2800 3458 3684 4257 4799 4819 5499 



5665 5810 5927 6169 6536 6617 6669 7069 7127 7132 7158



7164 7230 7320 7393 7396 7465


 2
2 6 12 15 2033 2125 3352 3382 5931 7024 7143 7358 7391 7504


 3
5 17 1725 1932 3277 4781 4888 6025 6374 7001 7139 7510 7524 7548


 4
4 19 101 1493 4111 4163 4599 6517 6604 6948 6963 7008 7280 7319


 5
8 28 2289 5025


 6
5505 5693 6844 7552


 7
9 3441 7424 7533


 8
917 1816 3540 4552


 9
256 6362 6868


10
2125 3144 5576


11
3443 5553 7201


12
2219 3897 4541


13
6331 6481 7224


14
7 1444 5568


15
81 3125 3345


16
778 2726 7316


17
3512 6462 7259


18
768 3751 6028


19
4665 7130 7452


20
2375 6814 7450


21
7073 7209 7483


22
2592 6466 7018


23
3716 5838 7547









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 9/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 10 presented below:








TABLE 10





i
Index of row where 1 is located in the 0th column of the ith column group







 0
350 462 1291 1383 1821 2235 2493 3328 3353 3772 3872 



3923 4259 4426 4542 4972 5347 6217 6246 6332 6386


 1
177 869 1214 1253 1398 1482 1737 2014 2161 2331 3108 



3297 3438 4388 4430 4456 4522 4783 5273 6037 6395


 2
347 501 658 966 1622 1659 1934 2117 2527 3168 3231 3379 



3427 3739 4218 4497 4894 5000 5167 5728 5975


 3
319 398 599 1143 1796 3198 3521 3886 4139 4453 4556 4636 



4688 4753 4986 5199 5224 5496 5698 5724 6123


 4
162 257 304 524 945 1695 1855 2527 2780 2902 2958 



3439 3484 4224 4769 4928 5156 5303 5971 6358 6477


 5
807 1695 2941 4276


 6
2652 2857 4660 6358


 7
329 2100 2412 3632


 8
1151 1231 3872 4869


 9
1561 3565 5138 5303


10
407 794 1455


11
3438 5683 5749


12
1504 1985 3563


13
440 5021 6321


14
194 3645 5923


15
1217 1462 6422


16
1212 4715 5973


17
4098 5100 5642


18
5512 5857 6226


19
2583 5506 5933


20
784 1801 4890


21
4734 4479 4875


22
938 5081 5377


23
127 4125 4704


24
1244 2178 3352


25
3659 6350 6465


26
1686 3464 4336









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 10/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 11, 12, or 13 presented below:








TABLE 11





i
Index of row where 1 is located in the 0th column of the ith column group







 0
76 545 1005 1029 1390 1970 2525 2971 3448 3845 4088 4114 4163 4373 4640 4705 4970 5094


 1
14 463 600 1676 2239 2319 2326 2815 2887 4278 4457 4493 4597 4918 4989 5038 5261 5384


 2
451 632 829 1006 1530 1723 2205 2587 2801 3041 3849 4382 4595 4727 5006 5156 5224 5286


 3
211 265 1293 1777 1926 2214 2909 2957 3178 3278 3771 4547 4563 4737 4879 5068 5232 5344


 4
6 2901 3925 5384


 5
2858 4152 5006 5202


 6
9 1232 2063 2768


 7
7 11 2781 3871


 8
12 2161 2820 4078


 9
3 3510 4668 5323


10
253 411 3215 5241


11
3919 4789 5040 5302


12
12 5113 5256 5352


13
9 1461 4004 5241


14
1688 3585 4480 5394


15
8 2127 3469 4360


16
2827 4049 5084 5379


17
1770 3331 5315 5386


18
1885 2817 4900 5088


19
2568 3854 4660


20
1604 3565 5373


21
2317 4636 5156


22
2480 2816 4094


23
14 4518 4826


24
127 1192 3872


25
93 2282 3663


26
2962 5085 5314


27
2078 4277 5089


28
9 5280 5292


29
50 2847 4742

















TABLE 12





i
Index of row where 1 is located in the 0th column of the ith column group







 0
446 449 544 788 992 1389 1800 1933 2461 2975 3186 3442 3733 3773 4076 4308 4323 4605 4882 5034 5080 5135 5146 5269 5307


 1
25 113 139 147 307 1066 1078 1572 1773 1957 2143 2609 2642 2901 3371 3414 3935 4141 4165 4271 4520 4754 4971 5160 5179


 2
341 424 1373 1559 1953 2577 2721 3257 3706 4025 4273 4689 4995 5005


 3
442 465 1892 2274 2292 2999 3156 3308 3883 4084 4316 4636 4743 5200


 4
22 1809 2406 3332 3359 3430 3466 4610 4638 5224 5280 5288 5337 5381


 5
29 1203 1444 1720 1836 2138 2902 3601 3642 4138 4269 4457 4965 5315


 6
1138 2493 3852 4802


 7
3050 5361 5396


 8
278 399 4810


 9
1200 3577 4904


10
1705 2811 3448


11
2180 4242 5336


12
4539 5069 5363


13
3318 3645 4427


14
2902 5134 5176


15
5123 5130 5229


16
47 4474 5356


17
2399 3981 5067


18
2377 2465 5080


19
2413 2471 5328


20
2502 4911 5329


21
4770 5139 5356


22
3263 4000 4022


23
648 2015 4867


24
311 2309 4063


25
1284 3246 3740


26
7 1080 3820


27
1261 2408 4608


28
3838 4076 4842


29
2294 4592 5254

















TABLE 13





i
Index at row where 1 is located in the 0th column of the ith column group







 0
352 747 894 1437 1688 1807 1883 2119 2159 3321 3400 3543 3588 3770 3821 4384 4470 4884 5012 5036 5084 5101 



5271 5281 5353


 1
505 915 1156 1269 1518 1650 2153 2256 2344 2465 2509 2867 2875 3007 3254 3519 3687 4331 4439 4532 4940 5011 



5076 5113 5367


 2
268 346 650 919 1260 4389 4653 4721 4838 5054 5157 5162 5275 5362


 3
220 236 828 1590 1792 3259 3647 4276 4281 4325 4963 4974 5003 5037


 4
381 737 1099 1409 2364 2955 3228 3341 3473 3985 4257 4730 5173 5242


 5
88 771 1640 1737 1803 2408 2575 2974 3167 3464 3780 4501 4901 5047


 6
749 1502 2201 3189


 7
2873 3245 3427


 8
2158 2605 3165


 9
1 3438 3606


10
10 3019 5221


11
371 2901 2923


12
9 3935 4683


13
1937 3502 3735


14
507 3128 4994


15
25 3854 4550


16
1178 4737 5366


17
2 223 5304


18
1146 5175 5197


19
1816 2313 3649


20
740 1951 3844


21
1320 3703 4791


22
1754 2905 4058


23
7 917 5277


24
3048 3954 5396


25
4804 4824 5105


26
2812 3895 5226


27
0 5318 5358


28
1483 2324 4826


29
2266 4752 5387









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 11/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 14 presented below:








TABLE 14





i
Index of row where 1 is located in the 0th column of the ith column group







 0
108 297 703 742 1345 1443 1495 1628 1812 2341 2559 2669 2810 2877 3442 3690 3755 3904 4264


 1
180 211 477 788 824 1090 1272 1578 1685 1948 2050 2195 2233 2546 2757 2946 3147 3299 3544


 2
627 741 1135 1157 1226 1333 1378 1427 1454 1696 1757 1772 2099 2208 2592 3354 3580 4066 4242


 3
9 795 959 989 1006 1032 1135 1209 1382 1484 1703 1855 1985 2043 2629 2845 3136 3450 3742


 4
230 413 801 829 1108 1170 1291 1759 1793 1827 1976 2000 2423 2466 2917 3010 3600 3782 4143


 5
56 142 236 381 1050 1141 1372 1627 1985 2247 2340 3023 3434 3519 3957 4013 4142 4164 4279


 6
298 1211 2548 3643


 7
73 1070 1614 1748


 8
1439 2141 3614


 9
284 1594 2626


10
607 660 855


11
1195 2037 2753


12
49 1198 2562


13
296 1145 3540


14
1516 2315 2382


15
154 722 4016


16
759 2375 3825


17
162 194 1749


18
2335 2422 2632


19
6 1172 2583


20
726 1325 1428


21
985 2708 2769


22
255 2801 3181


23
2979 3720 4090


24
208 1428 4094


25
199 3743 3757


26
1229 2059 4282


27
458 1100 1387


28
1199 2481 3284


29
1161 1467 4060


30
959 3014 4144


31
2666 3960 4125


32
2809 3834 4318









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 12/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 15 or 16 presented below:








TABLE 15






Index of row where 1 is located in the 0th column of the ith 


i
column group







 0
3 394 1014 1214 1361 1477 1534 1660 1856 2745 2987 2991 3124 



3155


 1
59 136 528 781 803 928 1293 1489 1944 2041 2200 2613 2690 



2847


 2
155 245 311 621 1114 1269 1281 1783 1995 2047 2672 2803 2885 



3014


 3
79 870 974 1326 1449 1531 2077 2317 2467 2627 2811 3083 3101 



3132


 4
4 582 660 902 1048 1482 1697 1744 1928 2628 2699 2728 3045 



3104


 5
175 395 429 1027 1061 1068 1154 1168 1175 2147 2359 2376 



2613 2682


 6
1388 2241 3118 3148


 7
143 506 2067 3148


 8
1594 2217 2705


 9
398 988 2551


10
1149 2588 2654


11
678 2844 3115


12
1508 1547 1954


13
1199 1267 1710


14
2589 3163 3207


15
1 2583 2974


16
2766 2897 3166


17
929 1823 2742


18
1113 3007 3239


19
1753 2478 3127


20
0 509 1811


21
1672 2646 2984


22
965 1462 3230


23
3 1077 2917


24
1183 1316 1662


25
968 1593 3239


26
64 1996 2226


27
1442 2058 3181


28
513 973 1058


29
1263 3185 3229


30
681 1394 3017


31
419 2853 3217


32
3 2404 3175


33
2417 2792 2854


34
1879 2940 3235


35
647 1704 3060

















TABLE 16





i
Index of row where 1 is located in the 0th column of the ith column group







 0
69 170 650 1107 1190 1250 1309 1486 1612 1625 2091 2416 2580 2673 2921 2995 3175 3234


 1
299 652 680 732 1197 1394 1779 1848 1885 2206 2266 2286 2706 2795 3206 3229


 2
107 133 351 640 805 1136 1175 1479 1817 2068 2139 2586 2809 2855 2862 2930


 3
75 458 508 546 584 624 875 1948 2363 2471 2574 2715 3008 3052 3070 3166


 4
0 7 897 1664 1981 2172 2268 2272 2364 2873 2902 3016 3020 3121 3203 3236


 5
121 399 550 1157 1216 1326 1789 1838 1888 2160 2537 2745 2949 3001 3020 3152


 6
1497 2022 2726 2871


 7
872 2320 2504 3234


 8
851 1684 3210 3217


 9
1807 2918 3178


10
671 1203 2343


11
405 490 3212


12
1 1474 3235


13
527 1224 2139


14
3 1997 2072


15
833 2366 3183


16
385 1309 3196


17
1343 2691 3153


18
1815 2048 2394


19
812 2055 2926


20
166 826 2807


21
1 493 2961


22
2218 3032 3153


23
2099 2885 3228


24
1214 2677 3216


25
2292 2422 2835


26
574 2138 3053


27
576 1409 1912


28
354 1631 3142


29
3211 3228 3239


30
1335 2938 3184


31
729 995 1520


32
537 3115 323


33
4 2631 3231


34
1130 2851 3030


35
1136 2728 3203









In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate R is 13/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 17 presented below:








TABLE 17





i
Index of row where 1 is located in the 0th column of the ith column group







 0
37 144 161 199 220 496 510 589 731 808 834 965 1249 1264 1311 1377 1460 1520 1598 1707 1958 2055 2099 2154


 1
20 27 165 462 546 583 742 796 1095 1110 1129 1145 1169 1190 1254 1363 1383 1463 1718 1835 1870 1879 2108 2135


 2
288 362 463 505 638 691 745 861 1006 1083 1124 1175 1247 1275 1337 1353 1378 1506 1588 1632 1720 1868 1980 2135


 3
405 464 478 511 566 574 641 766 785 802 836 996 1128 1239 1247 1449 1491 1537 1616 1643 1668 1950 1975 2149


 4
86 192 245 357 363 374 700 713 852 903 992 1174 1245 1277 1342 1369 1381 1417 1463 1712 1900 1962 2053 2118


 5
101 327 378 550


 6
186 723 1318 1550


 7
118 277 504 1835


 8
199 407 1776 1965


 9
387 1253 1328 1975


10
62 144 1163 2017


11
100 475 572 2136


12
431 865 1568 2055


13
283 640 981 1172


14
220 1038 1903 2147


15
483 1318 1358 2118


16
92 961 1709 1810


17
112 403 1485 2042


18
431 1110 1130 1365


19
587 1005 1206 1588


20
704 1113 1943


21
375 1487 2100


22
1507 1950 2110


23
962 1613 2038


24
554 1295 1501


25
488 784 1446


26
871 1935 1964


27
54 1475 1504


28
1579 1617 2074


29
1856 1967 2131


30
330 1582 2107


31
40 1056 1809


32
1310 1353 1410


33
232 554 1939


34
168 641 1099


35
333 437 1556


36
153 622 745


37
719 931 1188


38
237 638 1607









In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 6/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 18 presented below: In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 6/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 18 presented below:








TABLE 18





i
Index of row where 1 is located in the 0th column of the ith column group







 0
2949 5639 7367 8468 8922 9552 11216 12483 12692 13305 14026 15331 16463 17169 17210 18200 18744 19729 21099 



21976 22751 23405 25903 27283 27785 28232 30140 31378 31517 32596 33276 34715 37150 38321 39030 41119 41822


 1
497 1675 2751 6204 6502 8092 9462 10174 11130 13320 15232 16384 19154 19161 19289 22598 23437 25056 27490 29258



29606 30519 30950 32033 33244 34263 34664 35613 36427 37214 39295 39721 40604 41339 42048 42956 42993


 2
81 2015 3625 3922 5312 5478 10562 12344 15258 17199 18144 18734 20426 20680 20892 23405 23775 26987 27051 27928



30165 30931 31468 31734 32911 33246 34674 35490 36528 37198 40207 40753 40978 41099 42308 42368 43163


 3
60 83 4850 12379 13152 15708 18322 18837 19306 19707 20498 20515 21581 25442 26973 28529 31811 33646 33932 34951



36620 38616 38999 39044 39113 40059 41349 41555 41862 42402 42498 42585 42675 42993 43024 43055 43096


 4
0 4117 4725 7284 8569 9958 12270 13621 15234 16376 16601 19689 21366 23666 23974 24076 24394 27950 30679 31287 



35577 36892 38152 38720 33876 39185 39252 39340 39775 40987 41909 41943 42437 42961 42973 43031 43097


 5
967 2629 3433 4645 4982 6055 9235 9343 12533 16491 21527 24963 25960 28150 28991 29257 30381 31084 31236 31335



32838 34483 35276 36410 37071 37122 37300 37462 39998 40146 40154 41508 41728 42263 42446 42610 42800


 6
20 183 4422 4776 8224 8345 9567 10020 12621 15894 16523 17333 20169 20579 23118 24244 27052 27722 29664 30390 



35672 35676 36997 37033 37413 37601 38892 39128 39289 40096 40772 41443 42048 42224 42467 42745 43011


 7
84 106 3077 11179 14909 18393 18487 19607 22021 22690 22803 25095 26576 27221 27921 28711 29634 29800 35182 



35277 35908 36592 37177 37183 37282 38588 40818 40959 41614 41645 41731 41848 42076 42133 42686 42960 43081


 8
11 69 118 1909 9297 9815 11631 13409 14586 15614 16965 23438 23714 25346 26766 27507 28079 30062 31032 31645 



32897 34419 37279 37875 33926 39755 41098 41630 41738 41745 41950 42210 42263 42404 42787 43073 43114


 9
847 4197 18971 21073 21632 21804 28372 29779 31957


10
887 2607 17001 22853 32823 42335 42766 42963 43068


11
10043 11204 16610 34509 38254 39794 40346 40904 41837


12
3140 6150 19328 27432 29211 30253 34813 35467 41585


13
43 7561 25050 27629 27970 36235 39071 41760 42680


14
4457 11073 22626 25705 26101 27653 37929 40444 43140


15
9356 22954 24346 26334 29985 38991 39405 39467 42161


16
5208 6811 9103 9459 9821 17992 20176 23613 25909


17
8028 8500 18269 26693 34977 39987 41007 41908 43134


18
9949 18141 19765 28691 35617 41501 42181 42613 42795


19
7088 18754 22437 24859 25006 25260 25986 40105 42107


20
1103 13160 38316


21
10019 14598 19503


22
9430 10336 25320


23
16076 21513 43031


24
16559 17352 42859


25
762 9254 27313


26
3248 31582 40854


27
39929 41844 42505


28
16862 37978 42989


29
1163 27452 40918


30
29919 41247 42965


31
5613 17649 33421


32
31620 37112 41081


33
2444 9823 40886


34
18347 24355 29735


35
17445 20377 23490


36
12214 30796 42127


37
2806 10061 31670


38
18024 36307 42997


39
1786 10898 40868 


40
91 12816 34474


41
14181 32766 42963


42
14002 20589 43180


43
4615 35058 43192


44
109 10827 40754


45
1782 7637 41511


46
39185 42681 42708


47
2790 37933 43108


48
2024 25595 35385


49
3205 35128 36500


50
653 12319 21362


51
158 6617 34314


52
520 42957 43092


53
37614 41770 43179


54 
16081 22755 40856


55
18163 19831 21768


56
3375 29411 43010


57
17033 22596 42767


58 
11786 19137 31011


59 
52 9964 19729









In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 6/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 19 presented below:








TABLE 19





i
Index of row where 1 is located in the 0th column of the ith column group







 0
1606 3402 4961 6751 7132 11516 12300 14282 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 25850 36129 37186


 7
111 1307 1628 2041 2524 5358 7988 8191 10322 11905 12919 14127 15515 15711 17061 19024 21195 22902 23727 



24401 24608 25228 27338 35398 37794 38196


 8
961 3035 7174 7948 1335 13607 14971 1819 18339 18665 18875 19142 20615 21136 21309 21758 23366 24745 25849 



25982 27583 300003 31118 32106 36469 36583 37920


 9
2990 3549 4273 4808 5707 6021 6509 7456 8240 10044 12262 122660 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 36329 



27723 28186 30434 31956 33971 34372 36764 38123


12
520 2562 2794 3528 3860 4402 5675 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 16667 27934 30021


27
8500 25016 33043 38070


28
7374 10207 16189 35811


29
611 18480 20064 38261


30
25416 27352 36089 28469


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 23446 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 31058


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 29195 37653









In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 7/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 20 or 21 presented below:








TABLE 20





i
Index of row where 1 is located in the 0th column of the ith column group







 0
13 127 927 930 1606 2348 3361 3704 5194 6327 7843 8081 8615 12199 13947 15317 15774 16289 16687 17122 20468 



21057 21853 22414 23829 23885 25452 28072 28699 28947 30289 31672 32470


 1
36 53 60 86 93 407 3975 4478 5884 6578 7599 7613 7696 9573 11010 11183 11233 13750 17182 17860 20181 23974 



24195 25089 25787 25892 26121 30880 32989 33383 33626 34153 34520


 2
27 875 2693 3435 3682 6195 6227 6711 7629 8005 9081 11052 11190 11443 14832 17431 17756 17998 18254 18632 22234



22880 23562 23647 27092 29035 29620 30336 33492 33906 33960 34337 34474


 3
10 722 1241 3558 5490 5508 6420 7128 12386 12847 12942 15305 15592 16799 18033 19134 20713 20870 21589 26380 



27538 27577 27971 29744 32344 32347 32673 32892 33018 33674 33811 34253 34511


 4
6 24 72 2552 3171 5179 11519 12484 13096 13282 15226 18193 19995 25166 25303 25693 26821 29193 30666 31952 



33137 33187 33190 33319 33653 33950 34062 34255 34292 34365 34433 34443 34527


 5
1 12 26 29 85 1532 3870 6763 7533 7630 8022 8857 11667 11919 14987 16133 20999 21830 23522 24160 27671 28451 



30618 31556 31894 33436 33543 34146 34197 34313 34437 34480 34550


 6
13 44 2482 5068 8153 13233 13728 14548 17278 20027 21273 22112 22376 24799 29175


 7
26 50 8325 8891 12816 15672 15933 24049 30372 31245 33194 33238 33934 34093 34547


 8
1412 6334 7945 8866 10886 14521 17224 23693 25160 29267 31337 31893 32346 33195 33687


 9
27 47 14505 14786 18416 19963 23250 23475 27275 27921 28090 33985 34371 37374 34512


10
16 31 4924 7028 10240 12380 13479 16405 20197 27989 28084 32440 33996 34090 34435


11
17 57 95 6786 7427 7548 10452 13714 25632 30647 33054 34195 34237 34304 34447


12
4 62 331 10220 10518 10575 18401 19286 28718 30521 30698 31329 31848 32614 34343


13
42 79 4682 4747 7335 11487 17405 18089 19470 22547 33433 34373 34471 34519 34540


14
27 65 4911 10752 14803 24122 24531 25322 29130 30081 31280 32050 32693 34435 34508


15
24 29 2107 2152 5271 11032 14001 14902 21705 23126 31276 33946 34382 34380 34469


16
16 62 72 7470 14839 15299 15894 17716 18068 24959 25024 33343 34186 34398 34429


17
37 56 70 2089 10016 11316 14652 15665 17202 19804 19847 30498 33938 34126 34391


18
68 963 2099 9596 17606 19249 21839 27437 29901 30714 33060 33456 34347 34498 34527


19
6 69 1845 2504 7189 8603 10379 11421 13742 15757 16857 20642 28039 32833 34270


20
2235 15032 31823


21
4737 33978 34504


22
2 20263 30373


23
923 18929 25743


24
4587 22945 28380


25
22094 26147 34544


26
5177 20758 26476


27
8938 17291 27352


28
5286 2417 29331


29
71 16442 32683


30
81 22810 28015


31
14112 14419 29708


32
4156 7522 23358


33
12850 20777 28294


34
14692 31178 34238


35
3447 12356 21997


36
6098 15443 33447


37
5947 11648 21719


38
72 8695 18421


39
2173 18978 27232


40
13656 18222 19869


41
49 24684 33849


42
84 13870 18354


43
54 10089 10516


44
8035 18741 23775


45
7553 13539 25652


46
9116 26724 27525


47
22960 24382 26185


48
17384 24749 26726


49
12197 18965 32473


50
95 23126 26909


51
19327 31338 34320


52
9843 34130 34381


53
4031 9940 22329


54
58 31795 34468


55
103 17411 25220


56
26 4338 24625


57
9758 34395 34531


58
2186 17077 27646


59
9156 19462 34059


60
6 59 29352


61
16316 29453 34128


62
16244 32865 34517


63
918 22159 29265


64
13612 19465 20671


65
1 8261 8849


66
11214 28864 32696


67
11513 27595 34479


68
11895 21430 34524


69
82 5535 10552


70
66 15799 26966


71
20555 21816 32855


72
3772 27923 33492


73
12837 15856 21575


74
2 16865 34413


75
2682 2702 21630


76
10 22173 34016


77
9740 23216 33800


78
61 33792 33839


79
3961 29314 33446


80
11337 16620 20008


81
18461 25285 34267


82
46 117 8394


83
12291 25671 34505

















TABLE 21





i
Index of row where 1 is located in the 0th column of the ith column group







 0
7 15 26 69 1439 3712 5756 5792 5911 8456 10579 19462 19782 21709 23214 25142 26040 30206 30475 31211 31427 32105



32989 33082 33502 34116 34241 34288 34292 34318 34373 34390 34465


 1
83 1159 2271 6500 6807 7823 10344 10700 13367 14162 14242 14352 15015 17301 18952 20811 24974 25795 27868 28081



33077 33204 33262 33350 33516 33677 33680 33930 34090 34250 34290 34377 34398


 2
25 2281 2995 3321 6006 7482 8428 11489 11601 14011 17409 26210 29945 30675 31101 31355 31421 31543 31697 32056 



32216 33282 33453 33487 33696 34044 34107 34213 34247 34261 34276 34467 34495


 3
0 43 87 2530 4485 4595 9951 11212 12270 12344 15566 21335 24699 26580 28518 28564 28812 29821 30418 31467 31871



32513 32597 33187 33402 33706 33838 33932 33977 34084 34283 34440 34473


 4
81 3344 5540 7711 13308 15400 15885 18265 18632 22209 23657 27736 29158 29701 29845 30409 30654 30855 31420 



31604 32519 32901 33267 33444 33525 33712 33878 34031 34172 34432 34496 34502 34541


 5
42 50 66 2501 4706 6715 6970 8637 9999 14555 22776 26479 27442 27984 28534 29587 31309 31783 31907 31927 31934 



32313 32369 32830 33364 33434 33553 33654 33725 33889 33962 34467 34482


 6
6534 7122 8723 13137 13183 15818 18307 19324 20017 26389 29326 31464 32678 33668 34217


 7
50 113 2119 5038 5581 6397 6550 10987 22308 25141 25943 29299 30186 33240 33399


 8
7262 8787 9246 10032 10505 13090 14587 14790 16374 19946 21129 25726 31033 33660 33675


 9
5004 5087 5291 7949 9477 11845 12698 14585 15239 17486 18100 18259 21409 21789 24280


10
28 82 3939 5007 6682 10312 12485 14384 21570 25512 26612 26854 30371 31114 32689


11
437 3055 9100 9517 12369 19030 19950 21328 24196 24236 25928 28458 30013 32181 33560


12
18 3590 4832 7053 8919 21149 24256 26543 27266 30747 31839 32671 33089 33571 34296


13
2678 4569 4667 6551 7639 10057 24276 24563 25818 26592 27879 28028 29444 29873 34017


14
72 77 2874 9092 10041 13669 20676 20778 25566 28470 28888 30338 31772 32143 33939


15
296 2196 7309 11901 14025 15733 16768 23587 25489 30936 31533 33749 34331 34431 34507


16
6 8144 12490 13275 14140 18706 20251 20644 21441 21938 23703 34190 34444 34463 34495


17
5108 14499 15734 19222 24695 25667 28359 28432 30411 30720 24161 34386 34465 34511 34522


18
61 89 3042 5524 12128 22505 22700 22919 24454 30526 33437 34114 34188 34490 34502


19
11 83 4668 4856 6361 11633 15342 16393 16958 26613 29136 309617 32559 34346 34504


20
3185 9728 25062


21
1643 5531 21573


22
2285 6088 24083


23
78 14678 19119


24
49 13705 33535


25
21192 32280 32781


26
10753 21469 22084


27
10082 11950 13889


28
7861 25107 29167


29
14051 24171 34430


30
706 894 8316


31
29693 30445 32281


32
10202 30964 3448


33
15815 32453 34463


34
4102 21608 24740


35
4472 29399 31435


36
1162 7118 23226


37
4791 33548 34096


38
1084 34099 34418


39
1765 20745 33714


40
1302 21300 33655


41
33 8736 16646


42
53 18671 19089


43
21 572 2028


44
3339 11506 16745


45
285 6111 12643


46
27 10336 11586


47
21046 32728 34538


48
22215 24195 34026


49
19975 26938 29374


50
16473 26777 34212


51
20 29260 32784


52
35 31645 32837


53
26132 34410 34495


54
12446 20649 26851


55
6796 10992 31061


56
0 46 8420


57
10 636 22885


58
7183 16342 18305


59
1 5604 28258


60
6071 18675 34489


61
16786 25023 33323


62
3573 5081 10925


62
5067 31761 34415


64
3735 33534 34522


65
85 32829 34518


66
6555 23368 34559


67
22083 29355 29390


68
6738 21110 34316


69
120 4192 11123


70
3313 4144 20824


71
27783 28550 31034


72
6597 8164 34427


73
18009 23474 32460


74
94 6342 12656


75
17 31962 34535


76
15091 24955 28545


77
15 3213 28298


78
26565 30236 34537


79
16832 20334 24628


80
4841 20669 26509


81
18055 23700 34534


82
23576 31496 34492


83
10699 13826 34440









In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 8/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 22 presented below:








TABLE 22





i
Index of row where 1 is located in the 0th column of the 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 18412 19058 22985 24257 26202 26596 27899


17
1361 2195 4146 6708 7158 7538 9138 9998 14862 15359 16076 18925 2140121573 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
12581 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 Nldpc of the LDPC codeword is 64800, the code rate R is 9/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 23 presented below:








TABLE 23





i
Index of row where 1 is located in the 0th column of the ith column group
















0
113 1557 3316 5680 6241 10407 13404 13947 14040 14353 15522 15698 16079 17363 19374 19543 20530 22833 24339


1
271 1361 6236 7006 7307 7333 12768 15441 15568 17923 18341 20321 21502 22023 23938 25351 25590 25876 25910


2
73 605 872 4008 6279 7653 10346 10799 12482 12935 13604 15909 16526 19782 20506 22804 23629 24859 25600


3
1445 1690 4304 4851 8919 9176 9252 13783 16076 16675 17274 18806 18882 20819 21958 22451 23869 23999 24177


4
1290 2337 5661 6371 8996 10102 10941 11360 12242 14918 16808 20571 23374 24046 25045 25060 25662 25783 25913


5
28 42 1926 3421 3503 8558 9453 10168 15820 17473 19571 19685 22790 2336 23367 23890 24061 25657 25680


6
0 1709 4041 4932 5968 7123 8430 9564 10596 11026 14761 19484 20762 20858 23803 24016 24795 25853 25863


7
29 1625 6500 6609 16831 18517 18568 18738 19387 20159 20544 21603 21941 24137 24269 24416 24803 25154 25395


8
55 66 871 3700 11426 13221 15001 16367 17601 18380 22796 23488 23938 25476 25635 25678 25807 25857 25872


9
1 19 5958 8548 8860 11489 16845 18450 18469 19496 20190 23173 25262 25566 25668 25679 25858 25888 25915


10
7520 7690 8855 9183 14654 16695 17121 17854 18083 18428 19633 20470 20736 21720 22335 23273 25083 25293 25403


11
48 58 410 1299 3786 10668 18523 18963 20864 22106 22308 23033 23107 23128 23990 24286 24409 24595 25802


12
12 51 3894 6539 8276 10885 11644 12777 13427 14039 15954 17078 19053 20537 22863 24521 25087 25463 25838


13
3509 8748 9581 11509 15884 16230 17583 19264 20900 21001 21310 22547 22756 22959 24768 24814 25594 25626 25880


14
21 29 69 1448 2386 4601 6626 6667 10242 13141 13852 14137 18640 19951 22449 23454 24431 25512 25814


15
18 53 7890 9934 10063 16728 19040 19809 20825 21522 21800 23582 24556 25031 25547 25562 25733 25789 25906


16
4096 4582 5766 5894 6517 10027 12182 13247 15207 17041 18958 20133 20503 22228 24332 24613 25689 25855 25883


17
0 25 819 5539 7076 7536 7695 9532 13668 15051 17683 19665 20253 21996 24136 24890 25758 25784 25807


18
34 40 44 4215 6076 7427 7965 8777 11017 15593 19542 22202 22973 23397 23423 24418 24873 25107 25644 


19
1595 6216 22850 25439


20
1562 15172 19517 22362


21
7508 12879 24324 24496


22
6298 15819 16757 18721


23
11173 15175 19966 21195


24
59 13505 16941 23793


25
2267 4830 12023 20587


26
8827 9278 13072 16664 


27
14419 17463 23398 25348


28
6112 16534 20423 22698


29
493 8914 21103 24799


30
6896 12761 13206 25873


31
2 1380 12322 21701


32
11600 21306 25753 25790


33
8421 13076 14271 15401


34
9630 14112 19017 20955


35
212 13932 21781 25824


36
5961 9110 16654 19636


37
58 5434 9936 12770


38
6575 11433 19798


39
2731 7338 20926


40
14253 18463 25404


41
21791 24805 25869


42
2 11646 15850


43
6075 8586 23819


44
18435 22093 24852


45
2103 2368 11704


46
10925 17402 18232


47
9062 25061 25674


48
18497 20853 23404


49
18606 19364 19551


50
7 1022 25543


51
6744 15481 25868


52
9081 17305 25164


53
8 23701 25883


54
9680 19955 22848


55
56 4564 19121


56
5595 15086 25892


57
3174 17127 23183


58
19397 19817 20275


59
12561 24571 25825


60
7111 9889 25865


61
19104 20189 21851


62
549 9686 25548


63
6586 20325 25906


64
3224 20710 21637


65
641 15215 25754


66
13484 23729 25818


67
2043 7493 24246


68
16860 25230 25768


69
22047 24200 24902


70
9391 18040 19499


71
7855 24336 25069


72
23834 25570 25852


73
1977 8800 25756


74
6671 21772 25859


75
3279 6710 24444


76
24099 25117 25820 


77
5553 12306 25915


78
48 11107 23907


79
10832 11974 25773


80
2223 17905 25484


81
16782 17135 20446 


82
475 2861 3457


83
16218 22449 24362


84
11716 22200 25897


85
8315 15009 22633


86
13 20480 25852


87
12352 18658 25687


88
3681 14794 23703


89
30 24531 25846


90
4103 22077 24107


91
23837 25622 25812


92
3627 13387 25839


93
908 5367 19388


94
0 6894 25795


95
20322 23546 25181


96
8178 25260 25437


97
2449 13244 22565


98
31 18928 22741


99
1312 5134 14838


100
6085 13937 24220


101
66 14633 25670


102
47 22512 25472


103
8867 24704 25279


104
6742 21623 22745


105
147 9948 24178


106
8522 24261 24307


107
19202 22406 24609









In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 10/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 24 presented below:








TABLE 24





i
Index of row where 1 is located in the 0th column of the 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 1622316370 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 13046 15384 18576 20349 21424 21587


9
985 1591 3248 3509 3706 3847 6174 6276 7864 9033 13618 15675 16646 18355 18843


10
975 3774 4083 5825 6166 7218 7633 9633 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 2825 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 3404 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 9685 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 4625 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
377 12035 14081


99
1336 6468 12328


100
5948 9145 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 Nldpc of the LDPC codeword is 64800, the code rate R is 11/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 25 presented below:








TABLE 25





i
Index of row where 1 is located in the 0th column of the ith column group
















0
696 989 1238 3091 3116 3738 4269 6406 7033 8048 9157 10254 12033 16456 16912


1
444 1488 6541 8626 10735 12447 13111 13706 14135 15195 15947 16453 16916 17137 17268


2
401 460 992 1145 1576 1678 2238 2320 4280 6770 10027 12486 15363 16714 17157


3
1161 3108 3727 4508 5092 5348 5582 7727 11793 12515 12917 13362 14247 16717 17205


4
542 1190 6883 7911 8349 8835 10489 11631 14195 15009 15454 15482 16632 17040 17063


5
17 487 776 880 5077 6172 9771 11446 12798 16016 16109 16171 17087 17132 17226


6
1337 3275 3462 4229 9246 10180 10845 10866 12250 13633 14482 16024 16812 17186 17241


7
15 980 2305 3674 5971 8224 11499 11752 11770 12897 14082 14836 15311 16391 17209


8
0 3926 5869 8696 9351 9391 11371 14052 14172 14636 14974 16619 16961 17033 17237


9
3033 5317 6501 8579 10698 12168 12966 14019 15392 15806 15991 16493 16690 17062 17090


10
981 1205 4400 6410 11003 13319 13405 14695 15846 16297 16492 16563 16616 16862 16953


11
1725 4276 8869 9588 14062 14486 15474 15548 16300 16432 17042 17050 17060 17175 17273


12
1807 5921 9960 10011 14305 14490 14872 15852 16054 16061 16306 16799 16833 17136 17262


13
2826 4752 6017 6540 7016 8201 14245 14419 14716 15983 16569 16652 17171 17179 17247


14
1662 2516 3345 5229 8086 9686 11456 12210 14595 15808 16011 16421 16825 17112 17195


15
2890 4821 5987 7226 8823 9869 12468 14694 15352 15805 16075 16462 17102 17251 17263


16
3751 3890 4382 5720 10281 10411 11350 12721 13121 14127 14980 15202 15335 16735 17123


17
26 30 2805 5457 6630 7188 7477 7556 11065 16608 16859 16909 16943 17030 17103


18
40 4524 5043 5566 9645 10204 10282 11696 13080 14837 15607 16274 17034 17225 17266


19
904 3157 6284 7151 7984 11712 12887 13767 15547 16099 16753 16829 17044 17250 17259


20
7 311 4876 8334 9249 11267 14072 14559 15003 15235 15686 16331 17177 17238 17253


21
4410 8066 8596 9631 10369 11249 12610 15769 16791 16960 17018 1 703 7 17062 17165 17204


22
24 8261 9691 10138 11607 12782 12786 13424 13933 15262 15795 16476 17084 17193 17220


23
88 11622 14705 15890


24
304 2026 2638 6018


25
1163 4268 11620 17232


26
9701 11785 14463 17260


27
4118 10952 12224 17006


28
3647 10823 11521 12060


29
1717 3753 9199 11642


30
2187 14280 17220


31
14787 16903 17061


32
381 3534 4294


33
3149 6947 8323


34
12562 16724 16881


35
7289 9997 15306


36
5615 13152 17260


37
5666 16926 17027


38
4190 7798 16831


39
4778 10629 17180


40
10001 13884 15453


41
6 2237 8203


42
7831 15144 15160


43
9186 17204 17243


44
9435 17168 17237


45
42 5701 17159


46
7812 14259 15715


47
39 4513 6658


48
38 9368 11273


49
1119 4785 17182


50
5620 16521 16729


51
16 6685 17242


52
210 3452 12383


53
466 14462 16250


54
10548 12633 13962


55
1452 6005 16453


56
22 4120 13684


57
5195 11563 16522


58
5518 16705 17201


59
12233 14552 15471


60
6067 13440 17248


61
8660 8967 17061


62
8673 12176 15051


63
5959 15767 16541


64
3244 12109 12414


65
16936 17122 17162


66
4868 8451 13183


67
3714 4451 16919


68
11313 13801 17132


69
17070 17191 17242


70
1911 11201 17186


71
14 17190 17254


72
11760 16008 16832


73
14543 17033 17278


74
16129 16765 17155


75
6891 15561 17007


76
12741 14744 17116


77
8992 16661 17277


78
1861 11130 16742


79
4822 13331 16192


80
13281 14027 14989


81
38 14887 17141


82
10698 13452 15674


83
4 2539 16877


84
857 17170 17249


85
11449 11906 12867


86
285 14118 16831


87
15191 17214 17242


88
39 728 16915


89
2469 12969 15579


90
16644 17151 17164


91
2592 8280 10448


92
9236 12431 17173


93
9064 16892 17233


94
4526 16146 17038


95
31 2116 16083


96
15837 16951 17031


97
5362 8382 16618


98
6137 13199 17221


99
2841 15068 17068


100
24 3620 17003


101
9880 15718 16764


102
1784 10240 17209


103
7731 10203 10846


104
3121 8723 16598


105
8563 15662 17088


106
13 1167 14676


107
29 13850 15963


108
3654 7553 8114


109
23 4362 14865


110
4434 14741 16688


111
8362 13901 17244


112
13687 16736 17232


113
46 4229 13394


114
13169 16383 16972


115
16031 16681 16952


116
3384 9894 12580


117
9841 14414 16165


118
5013 17099 17115


119
2130 8941 17266


120
6907 15428 17241


121
16 1860 17235


122
2151 16014 16643


123
14954 15958 17222


124
3969 8419 15116


125
31 15593 16984


126
11514 16605 17255









In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 12/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 26 presented below:








TABLE 26





i
Index of row where 1 is located in the 0th column of the 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 4524 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 5594 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 another example, when the length Nldpc of the LDPC codeword is 64800, the code rate R is 13/15, and M is 360, the indexes of the rows where 1 is located in the 0th column of the ith column group of the information word submatrix 210 are as shown in Table 27 presented below:








TABLE 27






Index of row where 1 is located in the 0th column 


i
of the ith column group
















0
142 2307 2598 2650 4028 4434 5781 5881 6016 6323 6681 6698 8125


1
2932 4928 5248 5256 5983 6773 6828 7789 8426 8494 8534 8539 8583


2
899 3295 3833 5399 6820 7400 7753 7890 8109 8451 8529 8564 8602


3
21 3060 4720 5429 5636 5927 6966 8110 8170 8247 8355 8365 8616


4
20 1745 2838 3799 4380 4418 4646 5059 7343 8161 8302 8456 8631


5
9 6274 6725 6792 7195 7333 8027 8186 8209 8273 8442 8548 8632


6
494 1365 2405 3799 5188 5291 7644 7926 8139 8458 8504 8594 8625


7
192 574 1179 4387 4695 5089 5831 7673 7789 8298 8301 8612 8632


8
11 20 1406 6111 6176 6256 6708 6834 7828 8232 8457 8495 8602


9
6 2654 3554 4483 4966 5866 6795 8069 8249 8301 8497 8509 8623


10
21 1144 2355 3124 6773 6805 6887 7742 7994 8358 8374 8580 8611


11
335 4473 4883 5528 6096 7543 7586 7921 8197 8319 8394 8489 8636


12
2919 4331 4419 4735 6366 6393 6844 7193 8165 8205 8544 8586 8617


13
12 19 742 930 3009 4330 6213 6224 7292 7430 7792 7922 8137


14
710 1439 1588 2434 3516 5239 6248 6827 8230 8448 8515 8581 8619


15
200 1075 1868 5581 7349 7642 7698 8037 8201 8210 8320 8391 8526


16
3 2501 4252 5256 5292 5567 6136 6321 6430 6486 7571 8521 8636


17
3062 4599 5885 6529 6616 7314 7319 7567 8024 8153 8302 8372 8598


18
105 381 1574 4351 5452 5603 5943 7467 7788 7933 8362 8513 8587


19
787 1857 3386 3659 6550 7131 7965 8015 8040 8312 8484 8525 8537


20
15 1118 4226 5197 5575 5761 6762 7038 8260 8338 8444 8512 8568


21
36 5216 5368 5616 6029 6591 8038 8067 8299 8351 8565 8578 8585


22
1 23 4300 4530 5426 5532 5817 6967 7124 7979 8022 8270 8437


23
629 2133 4828 5475 5875 5890 7194 8042 8345 8385 8518 8598 8612


24
11 1065 3782 4237 4993 7104 7863 7904 8104 8228 8321 8383 8565


25
2131 2274 3168 3215 3220 5597 6347 7812 8238 8354 8527 8557 8614


26
5600 6591 7491 7696


27
1766 8281 8626


28
1725 2280 5120


29
1650 3445 7652


30
4312 6911 8626


31
15 1013 5892


32
2263 2546 2979


33
1545 5873 7406


34
67 726 3697


35
2860 6443 8542


36
17 911 2820


37
1561 4580 6052


38
79 5269 7134


39
22 2410 2424


40
3501 5642 8627


41
808 6950 8571


42
4099 6389 7482


43
4023 5000 7833


44
5476 5765 7917


45
1008 3194 7207


46
20 495 5411


47
1703 8388 8635


48
6 4395 4921


49
200 2053 8206


50
1089 5126 5562


51
10 4193 7720


52
1967 2151 4608


53
22 738 3513


54
3385 5066 8152


55
440 1118 8537


56
3429 6058 7716


57
5213 7519 8382


58
5564 8365 8620


59
43 3219 8603


60
4 5409 5815


61
5 6376 7654


62
4091 5724 5953


63
5348 6754 8613


64
1634 6398 6632


65
72 2058 8605


66
3497 5811 7579


67
3846 6743 8559


68
15 5933 8629


69
2133 5859 7068


70
4151 4617 8566


71
2960 8270 8410


72
2059 3617 8210


73
544 1441 6895


74
4043 7482 8592


75
294 2180 8524


76
3058 8227 8373


77
364 5756 8617


78
5383 8555 8619


79
1704 2480 4181


80
7338 7929 7990


81
2615 3905 7981


82
4298 4548 8296


83
8262 8319 8630


84
892 1893 8028


85
5694 7237 8595


86
1487 5012 5810


87
4335 8593 8624


88
3509 4531 5273


89
10 22 830


90
4161 5208 6280


91
275 7063 8634


92
4 2725 3113


93
2279 7403 8174


94
1637 3328 3930


95
2810 4939 5624


96
3 1234 7687


97
2799 7740 8616


98
22 7701 8636


99
4302 7857 7993


100 
7477 7794 8592


101 
9 6111 8591


102 
5 8606 8628


103 
347 3497 4033


104 
1747 2613 8636


105 
1827 5600 7042


106 
580 1822 6842


107 
232 7134 7783


108 
4629 5000 7231


109 
951 2806 4947


110 
571 3474 8577


111 
2437 2496 7945


112 
23 5873 8162


113 
12 1168 7686


114 
8315 8540 8596


115 
1766 2506 4733


116 
929 1516 3338


117 
21 1216 6555


118 
782 1452 8617


119 
8 6083 6087


120 
667 3240 4583


121 
4030 4661 5790


122 
559 7122 8553


123 
3202 4388 4909


124 
2533 3673 8594


125 
1991 3954 6206


126 
6835 7900 7980


127 
189 5722 8573


128 
2680 4928 4998


129 
243 2579 7735


130 
4281 8132 8566


131 
7656 7671 8609


132 
1116 2291 4166


133 
21 388 8021


134 
6 1123 8369


135 
311 4918 8511


136 
0 3248 6290


137
13 6762 7172


138
4209 5632 7563


139
49 127 8074


140
581 1735 4075


141
0 2235 5470


142
2178 5820 6179


143
16 3575 6054


144
1095 4564 6458


145
9 1581 5953


146
2537 6469 8552


147
14 3874 4844


148
0 3269 3551


149
2114 7372 7926


150
1875 2388 4057


151
3232 4042 6663


152
9 401 583


153
13 4100 6584


154
2299 4190 4410


155
21 3670 4979









According to an exemplary embodiment, even when the order of numbers, i.e., indexes, in a sequence corresponding to the ith column group of the parity check matrix 200 as shown in the above-described Tables 4 to 27 is changed, the changed parity check matrix is a parity check matrix used for the same LDPC code. Therefore, a case in which the order of numbers in the sequence corresponding to the ith column group in Tables 4 to 27 is changed is covered by the inventive concept.
According to an exemplary embodiment, even when one sequence corresponding to one column group is changed and another sequence corresponding to another column group are changed to each other in Tables 4 to 27, cycle characteristics on a graph of the LDPC 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 27 is changed is also covered by the inventive concept.
In addition, even when a multiple of Qldpc is equally added to all numbers, i.e., indexes, corresponding to a certain column group in Tables 4 to 27, the cycle characteristics on the graph of the LDPC code or the algebraic characteristics such as degree distribution are not changed. Therefore, a result of equally adding a multiple of Qldpc to the sequences shown in Tables 4 to 27 is also covered by the inventive concept. However, it should be noted that, when the resulting value obtained by adding a multiple of Qldpc to a given sequence is greater than or equal to (Nldpc−Kldpc), a value obtained by applying a modulo operation for (Nldpc−Kldpc) to the resulting value should be applied instead.
Once positions of the rows where 1 exists in the 0th column of the ith column group of the information word submatrix 210 are defined as shown in Tables 4 to 27, 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 0th column are cyclic-shifted by Qldpc in the next column.
For example, in the case of Table 4, in the 0th column of the 0th column group of the information word submatrix 210, 1 exists in the 245th row, 449th row, 491st row, . . . .
In this case, since Qldpc=(Nldpc−Kldpc)/M=(16200−5400)/360=30, the indexes of the rows where 1 is located in the 1st column of the 0th column group may be 275(=245+30), 479(=449+30), 521(=491+30), . . . , and the indexes of the rows where 1 is located in the 2nd column of the 0th column group may be 305(=275+30), 509(=479+30), 551(=521+30).
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 Nldpc−Kldpc number of columns (that is, Kldpcth column to (Nlpdc−1)th column), and has a dual diagonal or staircase configuration. Accordingly, the degree of columns except the last column (that is, (Nldpc−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 27, and the parity submatrix 220 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.

Qldpc·i+j→M·j+i (0≦i<M,0≦j<Qldpc)  (4)

Kldpc+Qldpc·k+1→Kldpc+M·l+k (0≦k<M,0≦l<Qldpc)  (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 Xth row, i and j satisfying X=Qldpc×i+j are calculated and the Xth row is permutated by assigning the calculated i and j to M×j+i. For example, regarding the 7th row, i and j satisfying 7=2×i+j are 3 and 1, respectively. Therefore, the 7th row is permutated to the 13th 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 Nqc_column=Nldpc/M number of column blocks and Nqc_row=Nparity/M number of row blocks.
Hereinafter, the submatrix of M×M will be explained.
First, the (Nqc_column−1)th column block of the 0th 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 0th row and the (M−1)th column are all “0”, and, regarding 0≦i≦(M−2), the (i+1)th row of the ith column is “1” and the other values are “0”.
Second, regarding 0≦i≦(Nldpc−Kldpc)/M−1 in the parity submatrix 320, the ith row block of the (Kldpc/M+i)th column block is configured by a unit matrix IM×M 340. In addition, regarding 0≦i≦(Nldpc−Kldpc)/M−2, the (i+1)th row block of the (Kldpc/M+i)th column block is configured by a unit matrix IM×M 340.
Third, a block 350 constituting the information word submatrix 310 may have a cyclic-shifted format of a cyclic matrix P, Paij, or an added format of the cyclic-shifted matrix Paij 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 aij is 0, the cyclic matrix P, that is, P0 indicates a unit matrix IM×M, and when aij is ∞, P∞ is a zero matrix.
A submatrix existing where the ith row block and the jth column block intersect in the parity check matrix 300 of FIG. 3 may be Paij. 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 Nldpc=M×Nqc_column, and the total number of rows is Nparity=M×Nqc_row. That is, the parity check matrix 300 is formed of Nqc_column number of column blocks and Nqc_row number of row blocks.
Referring back to FIG. 1, the encoder 110 may perform the LDPC encoding by using various code rates such as 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 and 3.
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 bits to which the BCH parity bits are added into information word 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 group from among a plurality of groups constituting the LDPC codeword (that is, a plurality of bit 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.
Hereinafter, interleaving rules used in the interleaver 120 will be explained in detail according to cases.
Case in which a Block Interleaver is Used
According to an exemplary embodiment, the interleaver 120 may interleave the LDPC codeword in a method described below such that a bit included in a predetermined group from among a plurality of groups constituting the interleaved LDPC codeword is mapped onto a predetermined bit in a modulation symbol. A detailed description thereof is provided with reference to FIG. 4.
FIG. 4 is a block diagram to illustrate a configuration of an interleaver according to exemplary embodiment. Referring to FIG. 4, the interleaver 120 includes 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 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 8 presented below:

ui=ci for 0≦i<Kldpc, and

uKldpc+M·t+s=cKldpc+Qldpc·s+t for 0≦s<M, 0≦t<Qldpc  (8),

where 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, that is, the number of columns included in a column group (for example, M=360), and Qldpc 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=(c0, c1, . . . , cNldpc−1), and outputs U=(u0, u1, . . . , uNldpc−1).

When the LDPC codeword encoded based on the parity check matrix 200 of FIG. 2 is parity-interleaved based on Equations 8, the parity-interleaved LDPC codeword is the same as the LDPC codeword encoded by the parity check matrix 300 of FIG. 3. Accordingly, when the LDPC codeword is generated based on the parity check matrix 300 of FIG. 3, the parity interleaver 121 may be omitted.
The LDPC codeword parity-interleaved after having been encoded based on the parity check matrix 200 of FIG. 2, or the LDPC codeword encoded based on the parity check matrix having the format of FIG. 3 may be characterized in 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 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 ith LDPC codeword bit, ci (i=0, 1, . . . , Nldpc−1) and the ith column of the parity check matrix should be a “0” vector. Accordingly, the ith LDPC codeword bit may be regarded as corresponding to the ith column of the parity check matrix.
In the case of the parity check matrix 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.
In addition, in the case of the parity check matrix 300 of FIG. 3, since the information word submatrix 310 and the parity submatrix 320 of the parity check matrix 300 have the same characteristics on the basis of a column group including M number of columns due to the row and column permutation, the information word bits and the parity bits of the LDPC codeword encoded based on the parity check matrix 300 are formed of M number of continuous bits of the same codeword characteristics.
Herein, the row permutation does not influence the cycle characteristic or algebraic characteristic of the LDPC codeword such as a degree distribution, a minimum distance, etc. since the row permutation is just to rearrange the order of rows in the parity check matrix. In addition, since the column permutation is performed for the parity submatrix 320 to correspond to parity interleaving performed in the parity interleaver 121, the parity bits of the LDPC codeword encoded by the parity check matrix 300 of FIG. 3 are formed of M number of continuous bits like the parity bits of the LDPC codeword encoded by the parity check matrix 200 of FIG. 2.
Accordingly, the bits constituting an LDPC codeword may have the same characteristics on the basis of M number of continuous bits, according to the present exemplary embodiment.
The group interleaver 122 may divide the LDPC codeword into a plurality of groups and rearrange the order of the plurality of groups or may divide the parity-interleaved LDPC codeword into a plurality of groups and rearrange the order of the plurality of groups. That is, the group interleaver 122 interleaves the plurality of groups in group units.
To achieve this, the group interleaver 122 divides the parity-interleaved LDPC codeword into a plurality of groups by using Equation 9 or Equation 10 presented below.




  
    
      
        
          
            X
            j
          
          =
          
            {
            
              
                
                  
                    u
                    k
                  
                  ❘
                  j
                
                =
                
                  ⌊
                  
                    k
                    360
                  
                  ⌋
                
              
              ,
              
                0
                ≤
                k
                <
                
                  N
                  ldpc
                
              
            
            }
          
        
        ⁢
        
          

        
        ⁢
        
          
            for
            ⁢
            
              
            
            ⁢
            0
          
          ≤
          j
          <
          
            N
            group
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          
            X
            j
          
          =
          
            {
            
              
                
                  u
                  k
                
                ❘
                
                  
                    360
                    ×
                    j
                  
                  ≤
                  k
                  <
                  
                    360
                    ×
                    
                      (
                      
                        j
                        +
                        1
                      
                      )
                    
                  
                
              
              ,
              
                0
                ≤
                k
                <
                
                  N
                  ldpc
                
              
            
            }
          
        
        ⁢
        
          

        
        ⁢
        
          
            for
            ⁢
            
              
            
            ⁢
            0
          
          ≤
          j
          <
          
            N
            group
          
        
      
    
    
      
        (
        10
        )
      
    
  




where Ngroup is the total number of groups, Xj is the jth group, and uk is the kth 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 groups may be as shown in FIG. 5.
Referring to FIG. 5, the LDPC codeword is divided into the plurality of groups and each group is formed of M number of continuous bits. When M is 360, each of the plurality of groups may be formed of 360 bits. Accordingly, the 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, Kldpc number of information word bits are divided into (Kldpc/M) number of groups and Nldpc−Kldpc number of parity bits are divided into (Nldpc−Kldpc)/M number of groups. Accordingly, the LDPC codeword may be divided into (Nldpc/M) number of groups in total. For example, when M=360 and the length Nldpc of the LDPC codeword is 64800, the number of groups Ngroups is 180, and, when the length Nldpc of the LDPC codeword is 16200, the number of groups Ngroup is 45.
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 group is M. However, this is merely an example and the number of bits constituting each group is variable.
For example, the number of bits constituting each group may be an aliquot part of M. That is, the number of bits constituting each 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 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 groups such that the number of bits constituting each group is one of the aliquot parts of 360.
Hereinafter, the case in which the number of bits constituting a group is M will be explained for convenience of explanation.
Thereafter, the group interleaver 122 interleaves the LDPC codeword in group units. Specially, the group interleaver 122 may group the LDPC codeword into a plurality of groups and rearrange the plurality of groups in group units. That is, the group interleaver 122 changes positions of the plurality of groups constituting the LDPC codeword and rearranges the order of the plurality of groups constituting the LDPC codeword.
Herein, the group interleaver may rearrange an order of the plurality of groups in group units so that groups including bits mapped onto the same modulation symbol from among the plurality of groups are spaced a predetermined distance apart.
In this case, the group interleaver may rearrange the order of the plurality of groups in group units so that the groups including the bits mapped onto the same modulation symbol are spaced a predetermined distance apart, by considering the number of columns and rows constituring the block interleaver 124, the number of groups constituting the LDPC codeword, and the number of bits included in each group.
For doing this, the group interleaver 122 may rearrange the order of the plurality of groups by using Equation 11 presented below:

Yj=Xπ(j)(0≦j<Ngroup)  (11),

where Xj is the jth group before group interleaving, and Yj is the jth 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 code rate and a modulation method.

Accordingly, XΠ(j) is a Π(j)th group before group interleaving, and Equation 11 means that the pre-interleaving Π(j)th group is interleaved into the jth group.
According to an exemplary embodiment, an example of Π(j) may be defined as in Tables 28 to 35 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 group units 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 10/15 to generate an LDPC codeword of a length of 16200, 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 10/15 in tables 28 to 35 presented below.
For example, when the length of the LDPC codeword is 16200, the code rate is 10/15, and the modulation method is 16-Quadrature Amplitude Modulation (QAM), the group interleaver 122 may perform interleaving by using Π(j) defined as in Table 28.
An example of Π(j) is as follows:
For example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 28 presented below:









TABLE 28







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


10/15, 11/15,
7
17
33
31
26
10
32
41
28
8
24
42
20
9
35
43
22
12
38
3
5
14
37


12/15, 13/15
40
19
16
27
39
25
4
21
1
23
18
36
0
6
11
34
2
29
15
30
13
44









In the case of Table 28, Equation 11 may be expressed as Y0=XΠ(0)=X7, Y1=XΠ(1)=X17, Y2=XΠ(2)=X33, . . . , Y43=XΠ(43)=X13, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 7th group to the 0th group, the 17th group to the 1st group, the 33rd group to the 2nd group, . . . , the 13th group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 29 presented below:









TABLE 29







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


6/15, 7/15,
32
4
23
27
35
24
16
39
5
22
33
40
18
13
8
6
37
34
0
15
21
38
30


8/15, 9/15
26
14
17
10
31
25
28
12
1
29
9
41
3
20
19
36
11
7
2
42
43
44









In the case of Table 29, Equation 11 may be expressed as Y0=XΠ(0)=X32, Y1=XΠ(1)=X4, Y2=XΠ(2)=X23, . . . , Y43=XΠ(43)=X43, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 32nd group to the 0th group, the 4th group to the 1st group, the 23rd group to the 2nd group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 30 presented below:









TABLE 30







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


10/15, 11/15,
28
6
15
8
0
22
37
35
21
26
7
12
27
1
32
33
13
11
10
18
34
9
39


12/15, 13/15
38
24
17
29
25
5
16
30
2
4
19
23
14
20
3
31
36
40
41
42
43
44









In the case of Table 30, Equation 11 may be expressed as Y0=XΠ(0)=X28, Y1=XΠ(1)=X6, Y2=XΠ(2)=X15, . . . , Y43=XΠ(43)=X43, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 28th group to the 0th group, the 6th group to the 1st group, the 15th group to the 2nd group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 31 presented below:









TABLE 31







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


6/15, 7/15,
16
13
1
25
24
33
4
29
32
30
0
17
22
18
8
9
27
11
37
35
12
15
10


8/15, 9/15
20
5
6
36
38
2
26
14
7
19
3
21
23
31
34
28
39
40
41
42
43
44









In the case of Table 31, Equation 11 may be expressed as Y0=XΠ(0)=X16, Y1=XΠ(1)=X13, Y2=XΠ(2)=X1, . . . , Y43=XΠ(43)=X43, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 16th group to the 0th group, the 13rd group to the 1st group, the 1st group to the 2nd group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.

In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 32 presented below:








TABLE 32






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
48
4
15
97
108
76
1
174
61
0
59
71
120
175
187
114
65
98
101


7/15,
19
112
109
152
138
35
62
43
86
153
73
173
165
23
49
91
5
169
99


8/15,
77
149
26
36
25
56
156
155
110
80
58
42
40
103
159
83
127
111
63


9/15
89
11
52
144
142
133
154
44
96
93
66
122
123
79
141
51
21
17
45



126
150
3
168
41
106
124
64
147
78
8
118
113
39
69
140
14
131
82



134
55
33
50
84
28
105
6
145
7
27
132
92
115
164
74
10
68
102



67
30
151
18
148
129
53
100
22
107
16
170
143
121
38
57
95
90
172



81
158
171
32
119
37
24
130
136
161
75
29
9
47
60
162
146
137
157



70
104
31
34
166
128
117
125
2
13
85
88
135
116
12
163
20
48
87



94
139
54
72
160
176
177
178
179









In the case of Table 32, Equation 11 may be expressed as Y0=XΠ(0)=X48, Y1=XΠ(1)=X4, Y2=XΠ(2)=X15, . . . , Y178=XΠ(178)=X178, and Y179=XΠ(179)=X179. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 48th group to the 0th group, the 4th group to the 1st group, the 15th group to the 2nd group, . . . , the 178th group to the 178th group, and the 179th group to the 179th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 33 presented below:








TABLE 33






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
121
39
139
61
52
45
44
66
55
50
153
40
68
58
38
43
171
42
69


7/15,
166
64
148
46
47
130
117
179
57
63
67
126
144
112
48
135
175
60
71


8/15,
49
54
51
41
157
162
70
65
19
32
20
4
110
23
76
11
158
53
1


9/15
150
86
84
22
30
36
92
29
28
25
167
127
35
62
31
24
59
142
118



26
21
33
88
27
134
174
15
37
90
56
94
8
34
152
149
99
73
83



89
177
72
77
17
6
128
85
137
164
78
81
169
82
80
75
155
95
140



93
119
132
87
115
172
91
113
160
103
123
13
107
101
74
97
105
125
109



143
146
0
14
178
176
173
136
161
16
168
104
138
18
100
163
154
129
147



141
124
170
10
98
133
151
156
159
165
120
7
122
114
106
145
108
111
102



131
3
5
116
9
79
12
2
96









In the case of Table 33, Equation 11 may be expressed as Y0=XΠ(0)=X121, Y1=XΠ(1)=X39, Y2=XΠ(2)=X139, . . . , Y178=XΠ(178)=X2, and Y179=XΠ(179)=X96. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 121st group to the 0th group, the 39th group to the 1st group, the 139th group to the 2nd group, . . . , the 2nd group to the 178th group, and the 96th group to the 179th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 64-QAM, Π(j) may be defined as in Table 34 presented below:








TABLE 34






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
49
105
104
47
116
112
84
113
75
99
52
4
76
102
7
35
48
109
106


11/15,
42
86
51
31
143
41
33
134
30
62
92
69
59
118
70
45
24
91
87


12/15,
88
44
63
32
101
73
83
81
56
37
79
139
98
38
13
54
114
89
40


13/15
85
55
27
61
23
57
93
66
97
107
115
14
110
50
36
43
68
67
80



103
78
64
60
100
82
71
74
108
25
53
117
34
111
130
8
146
138
39



96
132
94
77
72
29
90
28
140
3
142
17
137
46
135
95
65
119
154



2
133
136
26
128
58
5
150
10
168
174
155
22
6
148
162
179
11
152



21
171
145
16
129
0
151
144
9
149
177
153
165
18
147
141
15
159
157



161
156
127
178
19
125
166
120
126
173
158
170
172
122
121
124
20
123
131



160
12
175
169
163
1
167
164
176









In the case of Table 34, Equation 11 may be expressed as Y0=XΠ(0)=X49, Y1=XΠ(1)=X105, Y2=XΠ(2)=X104, . . . , Y178=XΠ(178)=X164, and Y179=XΠ(179)=X176. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 49th group to the 0th group, the 105th group to the 1st group, the 104th group to the 2nd group, . . . , the 164th group to the 178th group, and the 176th group to the 179th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 35 presented below:








TABLE 35






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
45
37
31
51
47
53
43
55
59
49
24
39
0
2
33
41
57
35
46


11/15,
122
56
50
23
42
44
52
40
58
30
36
32
38
1
34
48
54
71
3


12/15,
79
73
63
147
77
65
26
81
67
83
61
145
69
75
5
85
84
86
82


13/15
70
78
6
68
149
74
72
62
4
66
64
76
154
60
80
87
97
133
103



143
95
99
13
89
11
128
91
93
7
9
137
101
141
152
8
90
102
10



104
139
98
100
12
132
14
136
94
88
142
96
92
116
106
146
127
120
15



114
150
112
153
29
123
134
27
130
108
110
125
28
105
115
25
111
117
107



151
126
131
113
109
135
148
124
129
158
121
160
119
140
174
168
170
18
20



162
178
155
22
176
16
172
164
157
166
163
179
118
161
173
17
159
175
138



156
169
171
167
144
177
21
19
165









In the case of Table 35, Equation 11 may be expressed as Y0=XΠ(0)=X45, Y1=XΠ(1)=X37, Y2=XΠ(2)=X31, . . . , Y178=XΠ(178)=X19, and Y179=XΠ(179)=X165. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 45th group to the 0th group, the 37th group to the 1st group, the 31st group to the 2nd group, . . . , the 19th group to the 178th group, and the 165th group to the 179th group.
As described above, the group interleaver 122 may rearrange the order of the plurality of groups by using Equation 11 and Tables 28 to 35.
Meanwhile, the group interleaver 122 may rearrange the order of the plurality of groups by using Equation 12 presented below:

Yπ(j)=Xj (0≦j<Ngroup)  (12)

where Xj is the jth group before group interleaving, and Yj is the jth 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 code rate and a modulation method.

Accordingly, XΠ(j) is a Π(j)th group before group interleaving, and Equation 12 means that the pre-interleaving Π(j)th group is interleaved into the jth group.
According to an exemplary embodiment, an example of Π(j) may be defined as in Tables 36 to 43 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 group units 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 10/15 to generate an LDPC codeword of a length of 16200, 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 10/15 in Tables 36 to 43 presented below.
For example, when the length of the LDPC codeword is 16200, the code rate is 10/15, and the modulation method is 16-Quadrature Amplitude Modulation (QAM), the group interleaver 122 may perform interleaving by using Π(j) defined as in Table 36.
An example of Π(j) is as follows:
For example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 36 presented below:









TABLE 36







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


10/15, 11/15,
35
31
39
19
29
20
36
0
9
13
5
37
17
43
21
41
25
1
33
24
12
30
16


12/15, 13/15
32
10
28
4
26
8
40
42
3
6
2
38
14
34
22
18
27
23
7
11
15
44









In the case of Table 36, Equation 12 may be expressed as X0=YΠ(0)=Y35, X1=YΠ(1)=Y31, X2=YΠ(2)=Y39, . . . , X43=YΠ(43)=Y15, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 35th group, the 1st group to the 31st group, the 2nd group to the 39th group, . . . , the 43rd group to the 15th group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 64-QAM, Π(j) may be defined as in Table 37 presented below:









TABLE 37







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


6/15, 7/15,
18
31
41
35
1
8
15
40
14
33
26
39
30
13
24
19
6
25
12
37
36
20
9


8/15, 9/15
2
5
28
23
3
29
32
22
27
0
10
17
4
38
16
21
7
11
34
42
43
44









In the case of Table 37, Equation 12 may be expressed as X0=YΠ(0)=Y18, X1=YΠ(1)=Y31, X2=YΠ(2)=Y41, . . . , X43=YΠ(43)=Y43, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 18th group, the 1st group to the 31st group, the 2nd group to the 41st group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 38 presented below:









TABLE 38







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


10/15, 11/15,
4
13
31
37
32
28
1
10
3
21
18
17
11
16
35
2
29
25
19
33
36
8
5


12/15, 13/15
34
24
27
9
12
0
26
30
38
14
15
20
7
39
6
23
22
40
41
42
43
44









In the case of Table 38, Equation 12 may be expressed as X0=YΠ(0)=Y4, X1=YΠ(0)=Y13, X2=YΠ(2)=Y31, . . . , X43=YΠ(43)=Y43, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 4th group, the 1st group to the 13th group, the 2nd group to the 31st group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 39 presented below:









TABLE 39







Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)







































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44


6/15, 7/15,
10
2
28
33
6
24
25
31
14
15
22
17
20
1
30
21
0
11
13
32
23
34
12


8/15, 9/15
35
4
3
29
16
38
7
9
36
8
5
37
19
26
18
27
39
40
41
42
43
44









In the case of Table 39, Equation 12 may be expressed as X0=YΠ(0)=Y10, X1=YΠ(1)=Y2, X2=YΠ(2)=Y28, . . . , X43=YΠ(43)=Y43, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 10th group, the 1st group to the 2nd group, the 2nd group to the 28th group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 40 presented below:








TABLE 40






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
9
6
160
78
1
35
102
104
86
145
111
58
166
161
92
2
124
74
117


7/15,
19
168
73
122
32
139
42
40
105
100
144
115
154
136
97
155
24
41
138


8/15,
128
89
50
80
49
26
64
75
169
146
0
33
98
72
59
120
173
96
43


9/15
129
48
10
147
8
25
56
83
16
67
114
112
90
152
11
174
29
110
143



5
38
85
70
47
133
94
53
99
162
27
170
163
57
131
34
107
66
171



130
65
3
17
37
121
18
113
51
153
101
81
123
4
21
46
55
20
88



15
108
165
158
87
137
12
127
68
69
82
159
76
54
157
119
140
93
106



62
95
164
141
150
23
172
91
71
61
126
60
103
149
84
118
39
77
116



22
28
63
45
44
151
134
52
175
142
148
167
109
31
156
14
79
36
125



135
132
30
7
13
179
178
177
176









In the case of Table 40, Equation 12 may be expressed as X0=YΠ(0)=Y9, X1=YΠ(1)=Y6, X2=YΠ(2)=Y160, . . . , X178=YΠ(178)=Y177, and X179=YΠ(179)=Y176. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 9th group, the 1st group to the 6th group, the 2nd group to the 160th group, . . . , the 178th group to the 177th group, and the 179th group to the 176th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 41 presented below:








TABLE 41






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
135
56
178
172
49
173
100
163
88
175
155
53
177
125
136
83
142
99
146


7/15,
46
48
77
60
51
72
66
76
80
65
64
61
71
47
78
89
69
62
84


8/15,
14
1
11
41
17
15
6
5
22
23
33
38
9
40
4
55
39
8
86


9/15
27
13
73
36
3
70
28
20
45
7
29
12
18
44
37
97
93
128
110



52
98
105
176
109
106
108
94
59
102
58
117
79
95
85
120
63
114
87



112
179
129
156
92
147
127
170
123
144
130
166
126
168
132
50
169
32
121



165
118
174
25
75
115
162
0
164
124
153
131
30
68
101
150
24
171
116



157
81
34
140
103
145
2
113
152
74
133
31
167
134
151
21
91
57
158



90
10
149
111
159
42
54
160
122
141
43
148
104
161
19
67
143
107
154



16
119
139
82
35
138
96
137
26









In the case of Table 41, Equation 12 may be expressed as X0=YΠ(0)=Y135, X1=YΠ(1)=Y56, X2=YΠ(2)=Y178, . . . , X178=YΠ(178)=Y137, and X179=YΠ(179)=Y26. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 135th group, the 1st group to the 56th group, the 2nd group to the 178th group, . . . , the 178th group to the 137th group, and the 179th group to the 26th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 64-QAM, Π(j) may be defined as in Table 42 presented below:








TABLE 42






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
138
176
114
104
11
120
127
14
91
141
122
131
172
52
68
149
136
106
146


11/15,
156
168
133
126
61
35
85
117
59
102
100
27
22
41
25
88
15
71
47


12/15,
51
94
56
24
19
72
39
34
108
3
16
0
70
21
10
86
53
58
46


13/15
62
119
31
79
60
28
40
78
111
64
74
73
30
33
82
99
43
83
8



12
98
77
48
75
45
81
44
6
57
20
37
38
55
101
36
29
63
97



110
95
65
50
9
80
42
13
76
2
1
18
66
84
17
69
89
5
7



54
67
4
87
32
112
159
166
165
169
167
157
160
154
118
137
90
170
96



115
26
109
116
107
93
49
103
148
105
23
140
135
92
147
128
142
121
139



132
144
113
125
153
151
162
150
171
152
129
175
178
145
158
177
123
174
163



134
164
161
124
173
179
143
155
130









In the case of Table 42, Equation 12 may be expressed as X0=YΠ(0)=Y138, X1=YΠ(1)=Y176, X2=YΠ(2)=Y114, . . . , X178=YΠ(178)=Y155, and X179=YΠ(179)=Y130. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 138th group, the 1st group to the 176th group, the 2nd group to the 114th group, . . . , the 178th group to the 155th group, and the 179th group to the 130th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 43 presented below:








TABLE 43






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
12
32
13
37
65
52
59
85
91
86
94
81
99
79
101
113
157
167
150


11/15,
178
151
177
155
22
10
129
44
121
126
118
28
2
30
14
33
17
29
1


12/15,
31
11
26
15
23
6
24
0
18
4
34
9
21
3
25
5
35
7
20


13/15
16
27
8
70
48
64
40
67
43
66
46
60
50
57
36
63
39
62
51



68
42
58
38
71
45
56
47
54
53
55
72
104
80
92
83
107
84
103



77
106
73
97
78
98
88
93
75
95
127
109
132
123
137
124
130
116
136



114
128
108
131
164
145
112
143
19
119
140
125
134
111
82
141
122
135
100



74
120
138
102
87
170
96
146
89
105
76
175
49
110
41
139
61
115
133



90
117
69
154
171
160
142
168
144
165
152
162
159
179
161
174
148
172
149



173
158
166
147
169
156
176
153
163









In the case of Table 43, Equation 12 may be expressed as X0=YΠ(0)=Y12, X1=YΠ(1)=Y32, X2=YΠ(2)=Y13, . . . , X178=YΠ(178)=Y153, and X179=YΠ(179)=Y163. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups by changing the 0th group to the 12th group, the 1st group to the 32nd group, the 2nd group to the 13th group, . . . , the 178th group to the 153rd group, and the 179th group to the 163rd group.
As described above, the group interleaver 122 may rearrange the order of the plurality of groups in group units by using Equation 12 and Tables 36 to 43.
On the other hand, since the order of the groups constituting the LDPC codeword is rearranged by the group interleaver 122, and then the 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 28 to 43 in relation to Π(j).
The LDPC codeword which is group-interleaved in the above-described method is illustrated in FIG. 6. Comparing the LDPC codeword of FIG. 6 and the LDPC codeword of FIG. 5 before group interleaving, it can be seen that the order of the plurality of groups constituting the LDPC codeword is rearranged.
That is, as shown in FIGS. 5 and 6, the groups of the LDPC codeword are arranged in order of group X0, group X1, . . . , group XNgroup-1 before being group-interleaved, and are arranged in an order of group Y0, group Y1, . . . , group YNgroup-1 after being group-interleaved. In this case, the order of arranging the groups by the group interleaving may be determined based on Tables 28 to 43.
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 group by changing the order of the bits in the same group.
In this case, the group twist interleaver 123 may rearrange the order of the bits in the same group by cyclic-shifting a predetermined number of bits from among the bits in the same group.
For example, as shown in FIG. 7, the group twist interleaver 123 may cyclic-shift bits included in the group Y1 to the right by 1 bit. In this case, the bits located in the 0th position, the 1st position, the 2nd position, . . . , the 358th position, and the 359th position in the group Y1 as shown in FIG. 7 are cyclic-shifted to the right by 1 bit. As a result, the bit located in the 359th position before being cyclic-shifted is located in the front of the group Y1 and the bits located in the 0th position, the 1st position, the 2nd position, . . . , the 358th 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 group by cyclic-shifting a different number of bits in each group.
For example, the group twist interleaver 123 may cyclic-shift the bits included in the group Y1 to the right by 1 bit, and may cyclic-shift the bits included in the group Y2 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 group and does not change the order of the groups. Therefore, the group twist interleaver 123 may be placed before the group interleaver 122.
The block interleaver 124 interleaves the plurality of groups the order of which has been rearranged. Specifically, the block interleaver 124 is formed of a plurality of columns each including a plurality of rows, and classify and interleave the plurality of rearranged groups based on a modulation order which is determined according to a modulation method.
In this case, the block interleaver 124 may interleave the plurality of groups the order of which has been rearranged by the group interleaver 122 in group units. Specifically, the block interleaver 124 may divide and interleave the plurality of rearranged groups based on a modulation order by using the first part and the second part.
Specially, the block interleaver 124 may perform interleaving by dividing each of a plurality of columns into a first part and a second part, sequentially writing a plurality of groups in group units in a plurality of columns constituting the first part, dividing bits constituting the other groups into groups consisting of predetermined number of bits (or, sub bit group) based on the number of the plurality of column, and sequentially writing the divided groups in a plurality of columns constituting the second part.
Herein, the number of groups which are interleaved in group units may be determined by at least one of the number of rows and columns constituting the block interleaver 124, the number of groups and the number of bits included in each group. In other words, the block interleaver 124 may determine the groups which are to be interleaved in group units considering at least one of the number of rows and columns constituting the block interleaver 124, the number of groups and the number of bits included in each group, interleave the corresponding groups in group units, and divide and interleave the remaining groups. For example, the block interleaver 124 may interleave at least a part of the plurality of groups in group units by using the first part and divide and interleave the remaining groups by using the second part.
Meanwhile, interleaving groups in group units means that the bits included in the same group are written in the same column. In other words, in case of groups which are interleaved in group units, the block interleaver 124 may not divide the bits included in the same groups and write the bits in the same column, and in case of groups which are not interleaved in group units, the block interleaver 124 may divide the bits included in the groups and write and interleave the bits in different columns.
Accordingly, the number of rows constituting the first part may be multiple of the number of bits (for example, 360) included in a group, and the number of rows constituting the second part may be less than the number of bits included in a group.
In addition, in case of all groups which are interleaved by the first part, the bits included in the same groups may be written in the same column of the first part and interleaved, and in case of at least one group which is interleaved by the second part, the bits may be divided and written in at least two columns constituting 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 same group and does not change the order of groups by interleaving. Accordingly, the order of the groups to be block-interleaved by the block interleaver 124, that is, the order of the groups to be input to the block interleaver 124, may be determined by the group interleaver 122. Specifically, the order of the groups to be block-interleaved by the block interleaver 124 may be determined by Π(j) defined in Tables 28 to 43.
As described above, the block interleaver 124 may be formed of a plurality of columns each including a plurality of rows, and may divide the plurality of columns into at least two parts and interleave an LDPC codeword.
For example, the block interleaver 124 may divide each of a plurality of columns into a first part and a second part and interleave a plurality of groups constituting an 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 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 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 groups constituting the LDPC codeword in group units without dividing each of the plurality of columns into parts.
Specifically, the block interleaver 124 may interleave by writing the plurality of groups of the LDPC codeword on each of the columns in group units in a column direction, and reading each row of the plurality of columns in which the plurality of groups are written in group units in a row direction.
In this case, the block interleaver 124 may interleave by writing bits included in a predetermined number of groups which corresponds to a quotient of the number of groups of the LDPC codeword divided 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 jth position after being interleaved by the group interleaver 122 will be referred to as group Yj.
For example, it is assumed that the block interleaver 124 is formed of C number of columns each including R1 number of rows. In addition, it is assumed that the LDPC codeword is formed of Ygroup number of groups and the number of groups Ygroup is a multiple of C.
In this case, since a quotient obtained by dividing the number of groups constituting the LDPC codeword, that is, Ygroup by the number of columns constituting the block interleaver 12, that is, C, is Ygroup/C, the block interleaver 124 may interleave by writing Ygroup/C number of 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. 8, the block interleaver 124 writes bits included in group Y0, group Y1, . . . , group Yp−1 in the 1st column from the 1st row to the R1th row, writes bits included in group Yp, group Yp+1, . . . , group Yq−1 in the 2nd column from the 1st row to the R1th row, . . . , and writes bits included in group Yz, Yz+1, . . . , group YNgroup-1 in the column C from the 1st row to the R1th 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 groups constituting the LDPC codeword in group units.
However, when the number of 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 interleave a part of the plurality of groups of the LDPC codeword in group units by dividing each column into 2 parts and divide and interleave the remaining groups. In this case, the bits included in the remaining groups, that is, the bits included in the groups which correspond to remainder obtained by dividing the number of groups constituting the LDPC codeword by the number of columns are not interleaved in group units, 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 a first part (part 1) and a second part (part 2) based on the number of columns of the block interleaver 124, the number of groups constituting the LDPC codeword, and the number of bits of each of the plurality of groups.
Here, each of the plurality of groups may be formed of 360 bits. In addition, the number of groups constituting the LDPC codeword is determined according to a length of the LDPC codeword and the number of bits included in each group. For example, when an LDPC codeword length of which is 16200 is divided in such a way that each group is formed of 360 bits, the LDPC codeword may be divided into 45 groups. When an LDPC codeword length of which is 64800 is divided in such a way that each group is formed of 360 bits, the LDPC codeword may be divided into 180 groups. In addition, 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 groups constituting the LDPC codeword, and the number of bits constituting each of the plurality of groups.
Specifically, in each of the plurality of columns, the first part may be formed of as many rows as the number of of bits included in at least one group which can be written in each column in group units from among the plurality of groups of the LDPC codeword, according to the number of columns constituting the block interleaver 124, the number of groups constituting the LDPC codeword, and the number of bits constituting each 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 groups which can be written in each of the plurality of columns in group units. 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 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 groups which are not written in the first part from among 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 groups which can be written in each column in group units, 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 groups, that is, as many rows as an integer multiple of M. However, since the number of codeword bits constituting each 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 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 constiting the first part and the second part in which the LDPC codeword is written in a row direction.
That is, the block interleaver may interleave by writing bits included in at least some groups which can be written in each of the plurality of columns in group units in each of the plurality of columns of the first part sequentially, dividing bits included in the other groups except the at least some 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 divide and interleave the other groups except the at least some groups from among the plurality of groups based on the number of columns constituting the block interleaver 124.
Specifically, the block interleaver 124 may perform interleaving by dividing the bits include in the other 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 in which the divided bits are written in a row direction.
That is, the block interleaver 124 may divide the bits included in the other groups except the groups written in the first part from among the plurality of groups of the LDPC codeword, that is, the bits included in the groups which correspond to the remainder obtained by dividing the number of groups constituting the LDCP codeword 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 R1 number of rows. In addition, it is assumed that the LDPC codeword is formed of Ygroup number of groups, the number of groups Ygroup is not a multiple of C, and A×C+1=Ygroup (A is an integer greater than 0). That is, it is assumed that, when the number of groups constituting the LDCP codeword is divided by the number of columns, the quotient is A and the remainder is 1.
In this case, as shown in FIGS. 9 and 10, the block interleaver 124 may divide each column into a first part including R1 number of rows and a second part including R2 number of rows. In this case, R1 may correspond to the number of bits included in groups which can be written in each column in group units, and R2 may be R1 subtracted from the number of rows of each column.
That is, in the above-described example, the number of groups which can be written in each column in group units 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 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 groups which can be written in each column in group units, that is, A number of groups, in the first part of each column in the column direction.
That is, as shown in FIGS. 9 and 10, the block interleaver 124 writes the bits included in each of group Y0, group Y1, . . . , group Yn−1 in the 1st to R1th rows of the first part of the 1st column, writes bits included in each of group Yn, group Yn+1, . . . , group Ym−1 in the 1st to R1th rows of the first part of the 2nd column, . . . , writes bits included in each of group Ye, group Ye+1, . . . , group YNgroup-2 in the 1st to R1th rows of the first part of the column C.
As described above, the block interleaver 124 writes the bits included in the groups which can be written in each column in group units in the first part of each column in group units.
That is, the bits included in each of group Y0, group Y1, . . . , group Yn−1 may not be divided and may be written in the first column, and the bits included in each of group Yn, group Yn+1, . . . , group Ym−1 may not be divided and may be written in the second column, and the bits included in each of group Ye, group Ye+1, . . . , group YNgroup-2 may not be divided and may be written in C column. As such, it can be seen that, in case of all groups which are interleaved by the first part, the bits included in the same group are written in the same column of the first part.
Thereafter, the block interleaver 124 divides bits included in the other groups except the groups written in the first part of each column from among the plurality of 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 groups except the 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=Ygroup, when the groups constituting the LDPC codeword are written in the first part sequentially, the last group YNgroup-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 group YNgroup-1 by C as shown in FIG. 9, and writes the divided bits (that is, the bits which correspond to the quotient obtained by dividing the bits included in the last group YNgroup-1 by C) in the second part of each column serially.
Herein, each of the bits divided based on the number of columns may be called a sub bit group. In this case, it may be seen that each of the sub bit groups is written in each column of the second part. That is, bits included in a bit group may be divided and form a sub bit group.
That is, the block interleaver 124 writes the bits in the 1st to R2th rows of the second part of the 1st column, writes the bits in the 1st to R2th rows of the second part of the 2nd column, . . . , etc., and writes the bits in the 1st to R2th 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. 9.
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. That is, in the above example, since the last group YNgroup-1 is formed of M bits, the bits included in the last group YNgroup-1 may be divided in M/C units and written in each column. In other words, bits included in the last group (YNgroup-1) may be divided M/C units, form a sub bit group in the divided M/C units, and each of the sub bit group may be written in each column of the second part.
Accordingly, it can be seen that, in case of at least one group which is interleaved by the second part, the bits included in at least one 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 parts in a 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. 10, the block interleaver 124 writes the bits from the 1st row of the second part in the 1st column to the 1st row of the second part in the column C, writes the bits from the 2nd row of the second part in the 1st column to the 2nd row of the second part in the column C, . . . , etc., and writes the bits from the R2th row of the second part in the 1st column to the R2th 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. 9 and 10, 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 a plurality of groups constituting the LDPC codeword in group units, and divide and interleave the remaining groups. That is, the block interleaver 124 may peform interleaving by writing the LDCP codeword constituting a predetermined number of groups from among a plurality of groups in a plurality of columns constituting the first part in group units, dividing and writing the LDPC codeword constituting the other groups in each column constituting the second part, and reading a plurality of columns constituting the first part and the second part in a row direction.
As described above, the block interleaver 124 may interleave the plurality of groups in the methods described above with reference to FIGS. 8 to 10.
In particular, in the case of FIG. 9, the bits included in the 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 group which does not belong to the first part is rearranged. Since the bits included in the 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 group which does not belong to the first part may not be interleaved as shown in FIG. 10. That is, since the block interleaver 124 writes and read the bits included in the group which does not belong to the first part on 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. 9 and 10, the last single group of the plurality of groups is written in the second part. However, this is merely an example. The number of groups written in the second part may vary according to the total number of groups of the LDPC codeword, the number of columns and rows, the number of transmission antennas, etc.
The block interleaver 124 may have a different configuration according to whether bits included in a same group are mapped onto a single bit of each modulation symbol or bits included in a same group are mapped onto two bits of each modulation symbol.
On the other hand, in the case of a transceiving system using a plurality of antennas, the number of columns constituting the block interleaver 124 may be determined by considering the number of bits constituting a modulation symbol and the number of used antennas simultaneously. For example, when bits included in a same group are mapped onto a single bit in a modulation symbol and two antennas are used, the block interleaver 124 may determine the number of columns to be two times the number of bits constituting the modulation symbol.
First, when bits included in the same group are mapped onto a single bit of each modulation symbol, the block interleaver 124 may have configurations as shown in Tables 44 and 45:










TABLE 44









Nldpc = 64800
















16
64
256
1024
4096



QPSK
QAM
QAM
QAM
QAM
QAM

















C
2
4
6
8
10
12


R1
32400
16200
10800
7920
6480
5400


R2
0
0
0
180
0
0



















TABLE 45









Nldpc = 16200
















16
64
256
1024
4096



QPSK
QAM
QAM
QAM
QAM
QAM



















C
2
4
6
8
10
12



R1
7920
3960
2520
1800
1440
1080



R2
180
90
180
225
180
270










Herein, C (or NC) is the number of columns of the block interleaver 124, R1 is the number of rows constituting the first part in each column, and R2 is the number of rows constituting the second part in each column.
Referring to Tables 44 and 45, the number of a plurality of columns has the same value as a modulation order according to a modulation method, and each of the plurality of columns is formed of columns which correspond to a value obtained by dividing the number of bits constituting the LDPC codeword by the number of the plurality of columns.
For example, when a length of the LDPC codeword is Nldpc=64800, and modulation is performed in a 16-QAM method, a modulation order is 4. Thus, the block interleaver 124 is formed of four columns, and each column is formed of rows of R1+R2=16200(=64800/4).
Meanwhile, referring to Tables 44 and 45, when the number of groups constituting an LDPC codeword is an integer multiple of the number of columns, the block interleaver 124 interleaves without dividing each column. Therefore, R1 corresponds to the number of rows constituting each column, and R2 is 0. In addition, when the number of 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 R1 number of rows, and the second part formed of R2 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 group are mapped onto a single bit of each modulation symbol as shown in Tables 44 and 45.
For example, when Nldpc=64800 and the modulation method is 16-QAM, the block interleaver 124 may use four (4) columns each including 16200 rows. In this case, a plurality of groups of an LDPC codeword are written in the four (4) columns in group units and 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 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 group written in the 1st column may be mapped onto the first bit of each modulation symbol.
On the other hand, when bits included in a same group are mapped onto two bits of each modulation symbol, the block interleaver 124 may have configurations as shown in Tables 46 and 47:








TABLE 46








Nldpc = 64800















16
64
256
1024
4096



QPSK
QAM
QAM
QAM
QAM
QAM
















C
1
2
3
4
5
6


R1
64800
32400
21600
16200
12960
10800


R2
0
0
0
0
0
0


















TABLE 47








Nldpc = 16200
















16
64
256
1024
4096



QPSK
QAM
QAM
QAM
QAM
QAM


















C
1
2
3
4
5
6



R1
16200
7920
5400
3960
3240
2520



R2
0
180
0
90
0
180









Herein, C (or NC) is the number of columns of the block interleaver 124, R1 is the number of rows constituting the first part in each column, and R2 is the number of rows constituting the second part in each column.
Referring to Tables 46 and 47, when the number of groups constituting an LDPC codeword is an integer multiple of the number of columns, the block interleaver 124 interleaves without dividing each column. Therefore, R1 corresponds to the number of rows constituting each column, and R2 is 0. In addition, when the number of 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 R1 number of rows, and the second part formed of R2 number of rows.
When the number of columns of the block interleaver 124 is half of the number of bits constituting a modulation symbol as shown in Tables 46 and 47, bits included in a same group are mapped onto two bits of each modulation symbol.
For example, when Nldpc=64800 and the modulation method is 16-QAM, the block interleaver 124 may use two (2) columns each including 32400 rows. In this case, a plurality of groups of an LDPC codeword are written in the two (2) columns in group units and bits written in the same row in each column are output serially. Since four (4) bits constitute a single modulation symbol in the modulation method of 16-QAM, bits output from two rows constitute a single modulation symbol. Accordingly, bits included in the same group, that is, bits output from a single column, may be mapped onto two bits of each modulation symbol. For example, bits included in a group written in the 1st column may be mapped onto bits existing in any two positions of each modulation symbol.
Referring to Tables 44 to 47, the total number of rows of the block interleaver 124, that is, R1+R2, is Nldpc/C.
In addition, the number of rows of the first part, R1, is an integer multiple of the number of bits included in each group, M (e.g., M=360), and may be expressed as └Ngroup/C┘×M, and the number of rows of the second part, R2, may be Nldpc/C−R1. Herein, └Ngroup/C┘ is the largest integer below Ngroup/C. Since R1 is an integer multiple of the number of bits included in each group, M, bits may be written in R1 in group units.
In addition, when the number of groups of an LDPC codeword is not a multiple of the number of columns, it can be seen from Tables 44 to 47 that the block interleaver 124 interleaves a plurality of groups of the LDPC codeword by dividing each column into two parts.
Specifically, the length of an 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 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 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 44. In this case, each group of the LDPC codeword is formed of 360 bits, and the LDPC codeword is formed of 64800/360(=180) groups.
When the modulation method is 16-QAM, the block interleaver 124 may use four (4) columns and each column may have 64800/4(=16200) rows.
In this case, since the number of groups of an LDPC codeword divided by the number of columns is 180/4(=45), bits can be written in each column in group units without dividing each column into two parts. That is, bits included in 45 groups which are the quotients obtained by dividing the number of groups constituting the LDPC codeword 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 use eight (8) columns and each column may have 64800/8(=8100) rows.
In this case, since the number of groups of an LDPC codeword divided by the number of columns is 180/8=22.5, the number of 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 group units.
In this case, since the bits should be written in the first part of each column in group units, the number of groups which can be written in the first part of each column in group units is 22 which are the quotients obtained by dividing the number of groups constituting the LDPC codeword by the number of columns, and accordingly, the first part of each column has 22×360(=7920) rows. Accordingly, 7920 bits included in 22 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 group which has not been written in the first part are divided and written in the second part of each column.
Specifically, since 22×8(=176) groups are written in the first part, the number of groups to be written in the second part is 180-176(=4) (for example, group Y176, group Y177, group Y178, and group Y179 from among group Y0, group Y1, group Y2, . . . , group Y178, and group Y179 constituting an LDPC codeword).
Accordingly, the block interleaver 124 may write the four (4) 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 group Y176 in the 1st row to the 180th row of the second part of the 1st column in the column direction, and may write the other 180 bits in the 1st row to the 180th row of the second part of the 2nd column in the column direction. In addition, the block interleaver 124 may write 180 bits of the 360 bits included in the group Y177 in the 1st row to the 180th row of the second part of the 3rd column in the column direction, and may write the other 180 bits in the 1st row to the 180th row of the second part of the 4th column in the column direction. In addition, the block interleaver 124 may write 180 bits of the 360 bits included in the group Y178 in the 1st row to the 180th row of the second part of the 5th column in the column direction, and may write the other 180 bits in the 1st row to the 180th row of the second part of the 6th column in the column direction. In addition, the block interleaver 124 may write 180 bits of the 360 bits included in the group Y179 in the 1st row to the 180th row of the second part of the 7th column in the column direction, and may write the other 180 bits in the 1st row to the 180th row of the second part of the 8th column in the column direction.
Accordingly, the bits included in the 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 of FIG. 4 according to an exemplary embodiment will be explained in detail with reference to FIG. 11.
In a group-interleaved LDPC codeword (v0, v1, . . . , VNldpc−1), Yj is continuously arranged like V={Y0, Y1, . . . , YNgroup−1}.
The LDPC codeword after group interleaving may be interleaved by the block interleaver 124 as shown in FIG. 11. 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 groups. In this case, in the first part, the bits constituting groups may be written in the same column, and in the second part, the bits constituting groups may be written in a plurality of columns.
In the block interleaver 124, the data bits vi from the group-wise interleaver 122 are written serially into the block interleaver column-wise starting in the first part 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 bits included in the same group in the first part may be mapped onto 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 method as in Table 48 presented below. The first part and the second part block interleaving configurations for each modulation format and code length are specified in Table 32. 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, Nr1 and the number of rows of the second part, Nr2, is equal to Nldpc/NC (herein, NC is the number of columns). In addition, since Nr1(=└Ngroup/NC┘×360) is a multiple of 360, so that multiple of bit groups are written into the first part of block interleaver.










TABLE 48








Rows in Part 1 Nr1
Rows in Part 2 Nr2














Nldpc = 
Nldpc = 
Nldpc = 
Nldpc = 
Columns


Modulation
64800
16200
64800
16200
Nc















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. 11, the input bit vi (0≦i<NC×Nr1) is written in ri row of ci column of the first part of the block interleaver 124. Herein, ci and ri are




  
    c
    i
  
  =
  
    ⌊
    
      i
      
        N
        
          r
          ⁢
          
            
          
          ⁢
          1
        
      
    
    ⌋
  




and ri=(i mod Nr1), respectively.

In addition, the input bit vi (NC×Nr1≦i<Nldpc) is written in an ri row of ci column of the second part of the block interleaver 124. Herein, ci and ri are




  
    c
    i
  
  =
  
    ⌊
    
      
        (
        
          i
          -
          
            
              N
              C
            
            ×
            
              N
              
                r
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
        )
      
      
        N
        
          r
          ⁢
          
            
          
          ⁢
          2
        
      
    
    ⌋
  




and ri=Nr1+{(i−NC×Nr1) mod Nr2}, respectively.

An output bit qj(0≦j<Nldpc) is read from cj column of rj row. Herein, rj and cj are




  
    r
    j
  
  =
  
    ⌊
    
      j
      
        N
        c
      
    
    ⌋
  




and cj=(j mod NC), respectively.

For example, when the length Nldpc of an LDPC codeword is 64800 and the modulation method is 256-QAM, an order of bits output from the block interleaver 124 may be (q0, q1, q2, . . . , q63357, q63358, q63359, q63360, q63361, . . . , q64799)=(v0, v7920, v15840, . . . , v47519, v55439, v63359, v63360, v63540, . . . , v64799). 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.
Meanwhile, in the above example, the number of columns constituting the block interleaver 124 may be the same value as a modulation degree or half the modulation degree, but this is only an example. The number of columns constituting the block interleaver 124 may be a multiple value of the modulation degree. In this case, the number of rows constituting each column may be the length of the LDPC codeword divided by the number of columns.
For example, in case that the modulation method is QPSK (that is, the modulation degree is 2), the number of columns may be 4 instead of 2. In this case, if the length Nldpc of the LDPC codeword is 16200, the number of rows constituting each column may be 4050(=16200/4).
Meanwhile, even when the number of columns is the multiple value of the modulation degree, the block interleaver 124 may perform interleaving using the same method as when the number of columns is the same value as the modulation degree of half the modulation degree, so detailed description thereof will not be provided.
In this case, the number of columns constituting the block interleaver 124 may have the same value as the modulation degree or the integer multiple of the modulation degree and thus, the number of the second part may be the same value as a quotient when the number of bits included in all bit groups excluding groups corresponding to the first part is divided by the modulation degree or the multiple of the modulation degree.
Referring back to FIG. 1, the modulator 130 modulates an interleaved LDPC codeword according to a modulation method to generate a modulation symbol. Specifically, the modulator 130 may demultiplex the interleaved LDPC codeword and modulate the demultiplexed LDPC codeword and map it onto a constellation, thereby generating a modulation symbol.
In this case, the modulator 130 may generate a modulation symbol using bits included in each of a plurality of groups.
In other words, as described above, the bits included in different groups are written in each column of the block interleaver 124, and the block interleaver 124 reads the bits written in each column in a 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 group.
For example, it is assumed that the modulation symbol consists of C bits (C refers to the number of bits). In this case, the bits which are read from each row of C 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 bits belong to C different 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. 12, the demultiplexer (not shown) receives the LDPC codeword Q=(q0, q1, q2, . . . ) output from the interleaver 120, outputs the received LDPC codeword bits to one of a plurality of substreams serially, converts the input LDPC codeword bits into cells, and outputs the cells.
Herein, the number of substreams, Nsubstreams, may be equal to the number of bits constituting a modulation symbol, ηmod, and the number of bits constituting the cell may be equal to Nldpc/ηmod. ηmod varying according to a modulation method and the number of cells generated according to the length Nldpc of the LDPC codeword are as in Table 49 presented below:










TABLE 49







Number of output
Number of output




data cells for Nldpc =
data cells for Nldpc =


Modulation mode
ηMOD
64800
16100


















QPSK
2
32400
8100


 16-QAM
4
16200
4050 


 64-QAM
6
10800
2700


 256-QAM
8
 8100
2025


1024-QAM
10
 6480
1620









Bits having the same index in each of the plurality of sub-streams may constitute a same cell. That is, in FIG. 12, each cell may be expressed as (y0,0, y1,0, . . . , yηMOD−1,0), (y0,1, y1,1, . . . , yηMOD−1,1).
The demultiplexer (not shown) may demultiplex input LDPC codeword bits in various methods. That is, the demultiplexer (not shown) may change an order of the LDPC codeword bits and output the bits to each of the plurality of substreams, or may output the bits to each of the plurality of streams serially without changing the order of the LDPC codeword bits. These operations may be determined according to the number of columns used for interleaving in the block interleaver 124.
Specifically, when the block interleaver 124 includes as many columns as half of the number of bits constituting a modulation symbol, the demultiplexer (not shown) may change the order of the input LDPC codeword bits and output the bits to each of the plurality of sub-streams. An example of a method for changing the order is illustrated in Table 50 presented below:









TABLE 50








Modulation format
QPSM














input bit
0
1



di mod Nsubstreams





output bit-number 
0
1













Modulation format
16 QAM
















input bit
0
1
2
3



di mod Nsubstreams







output bit-number 
0
2
1
3













Modulation format
64 QAM


















Input bit
0
1
2
3
4
5



di mod Nsubstreams









output bit-number 
0
3
1
4
2
5













Modulation format
256 QAM




















input bit
0
1
2
3
4
5
6
7



di mod Nsubstreams











output bit-number 
0
4
1
5
2
6
3
7













Modulation format
1024 QAM






















input bit
0
1
2
3
4
5
6
7
8
9



di mod Nsubstreams













output bit-number 
0
5
1
6
2
7
3
8
4
9











Modulation format
4096 QAM






















input bit
0
1
2
3
4
5
6
7
8
9
10
11


di mod Nsubstreams














output bit-number 
0
6
1
7
2
8
3
9
4
10
5
11









According to Table 50, when the modulation method is 16-QAM for example, the number of substreams is four (4) since the number of bits constituting the modulation symbol is four (4) in the case of 16-QAM. In this case, the demultiplexer (not shown) may output, from among the serially input bits, bits with an index i satisfying i mod 4=0 to the 0th substream, bits with an index i satisfying i mod 4=1 to the 2nd substream, bits with an index i satisfying i mode 4=2 to the 1st substream, and bits with an index i satisfying i mode 4=3 to the 3rd substream.
Accordingly, the LDPC codeword bits input to the demultiplexer (not shown), (q0, q1, q2, . . . ), may be output as cells like (y0,0, y1,0, y2,0, y3,0)=(q0, q2, q1, q3), (y0,1, y1,1, y2,1, y3,1)=(q4, q6, q5, q7), . . . .
When the block interleaver 124 includes the same number of columns as the number of bits constituting a modulation symbol, the demultiplexer (not shown) may output the input LDPC codeword bits to each of the plurality of streams serially without changing the order of the bits. That is, as shown in FIG. 13, the demultiplexer (not shown) may output the input LDPC codeword bits (q0, q1, q2, . . . ) to each of the substreams serially, and accordingly, each cell may be configured as (y0,0, y1,0, . . . , yηMOD−1,0)=(q0, q1, . . . , qηMOD−1), (y0,1, y1,1, . . . , yηMOD−1,1)=(qηMOD,qηMOD+1, . . . , q2×ηMOD−1), . . . .
In the above-described example, the demultiplexer (not shown) outputs the input LDPC codeword bits to each of the plurality of streams serially without changing the order of the bits. However, this is merely an example. That is, according to an exemplary embodiment, when the block interleaver 124 includes the same number of columns as the number of bits constituting a modulation symbol, the demultiplexer (not shown) may be omitted.
The modulator 130 may map the demultiplexed LDPC codeword onto modulation symbols. However, when the demultiplexer (not shown) is omitted as described above, the modulator 130 may map LDPC codeword bits output from the interleaver 120, that is, block-interleaved LDPC codeword bits, onto modulation symbols.
The modulator 130 may modulate bits (that is, cells) output from the demultiplexer (not shown) in various modulation methods such as QPSK, 16-QAM, 64-QAM, 256-QAM, 1024-QAM, 4096-QAM, etc. When the modulation method is QPSK, 16-QAM, 64-QAM, 256-QAM, 1024-QAM and 4096-QAM, the number of bits constituting a modulation symbol, ηMOD (that is, a modulation degree), 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 a modulation symbol, the modulator 130 may generate a modulation symbol by mapping each cell output from the demultiplexer (not shown) onto a constellation point serially. Herein, a modulation symbol corresponds to a constellation point on the constellation.
However, when the demultiplexer (not shown) is omitted, the modulator 130 may generate modulation symbols by grouping a predetermined number of bits from interleaved bits sequentially and mapping the predetermined number of bits onto constellation points. In this case, the modulator 130 may generate the modulation symbols by using ηMOD number of bits sequentially according to a modulation method.
The modulator 130 may modulate by mapping cells output from the demultiplexer (not shown) onto constellation points in a uniform constellation (UC) method.
The uniform constellation method refers to a method for mapping a modulation symbol onto a constellation point so that a real number component Re(zq) and an imaginary number component Im(zq) of a constellation point have symmetry and the modulation symbol is placed at equal intervals. Accordingly, at least two of modulation symbols mapped onto constellation points in the uniform constellation method may have the same demodulation performance.
Examples of the method for generating a modulation symbol in the uniform constellation method according to an exemplary embodiment are illustrated in Tables 51 to 58 presented below, and an example of a case of a uniform constellation 64-QAM is illustrated in FIG. 14.











TABLE 51









y0, q
1
0



Re(zq)
−1
1





















TABLE 52









Y1, q
1
0



Im(zq)
−1
1























TABLE 53









y0, q
1
1
0
0



y2, q
0
1
1
0



Re(zq)
−3
−1
1
3























TABLE 54









y1, q
1
1
0
0



y3, q
0
1
1
0



Im(zq)
−3
−1
1
3


























TABLE 55








y0, q
1
1
1
1
0
0
0
0



y2, q
0
0
1
1
1
1
0
0



y4, q
0
1
1
0
0
1
1
0



Re(zq)
−7
−5
−3
−1
1
3
5
7

























TABLE 56








y1, q
1
1
1
1
0
0
0
0



y3, q
0
0
1
1
1
1
0
0



y5, q
0
1
1
0
0
1
1
0



Im(zq)
−7
−5
−3
−1
1
3
5
7
































TABLE 57







y0, q
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0


y2, q
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0


y4, q
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0


y6, q
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0


Re(zq)
−15
−13
−11
−9
−7
−5
−3
−1
1
3
5
7
9
11
13
15
































TABLE 58







y1, q
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0


y3, q
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0


y5, q
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0


y7, q
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0


Im(zq)
−15
−13
−11
−9
−7
−5
−3
−1
1
3
5
7
9
11
13
15









Tables 51 and 52 are used for determining a real number component Re(zq) and an imaginary number component Im(zq) when the modulation is performed in a QPSK method, Tables 53 and 54 are used for determining a real number component Re(zq) and an imaginary number component Im(zq) when the modulation is performed in a 16-QAM method, Tables 55 and 56 are used for determining a real number component Re(zq) and an imaginary number component Im(zq) when the modulation is performed in a 64-QAM method, and Tables 57 and 58 are used for determining a real number component Re(zq) and an imaginary number component Im(zq) when the modulation is performed in a 256-QAM method.
Referring to Tables 51 to 58, performance (e.g., reliability) varies according to whether a plurality of bits constituting a modulation symbol correspond to most significant bits (MSBs) or least significant bits (LSBs).
For example, in the case of 16-QAM, from among four (4) bits constituting a modulation symbol, each of the first and second bits determines a sign of each of the real number component Re(zq) and the imaginary number component Im(zq) of a constellation point onto which a modulation symbol is mapped, and the third and fourth bits determine a size of the constellation point onto which the modulation symbol is mapped.
In this case, the first and second bits for determining the sign from among the four (4) bits constituting the modulation symbol have a higher reliability than the third and fourth bits for determining the size.
In another example, in the case of 64-QAM, from among six (6) bits constituting a modulation symbol, each of the first and second bits determines a sign of each of the real number component Re(zq) and the imaginary number component Im(zq) of a constellation point onto which the modulation symbol is mapped. In addition, the third to sixth bits determine a size of the constellation point onto which the modulation symbol is mapped. From among these bits, the third and fourth bits determine a relatively large size, and the fifth and sixth bits determine a relatively small size (for example, the third bit determines which of sizes (−7, −5) and (−3, −1) corresponds to the constellation point onto which the modulation symbol is mapped, and, when (−7, −5) is determined by the third bit, the fourth bit determines which of −7 and −5 corresponds to the size of the constellation point.).
In this case, the first and second bits for determining the sign from among the six bits constituting the modulation symbol have the highest reliability, and the third and fourth bits for determining the relatively large size has the higher reliability than the fifth and sixth bits for determining the relatively small size.
As described above, in the case of the uniform constellation method, the bits constituting a modulation symbol have different reliability according to mapping locations in the modulation symbol.
The modulator 130 may modulate by mapping cells output from the demultiplexer (not shown) onto constellation points in a non-uniform constellation (NUC) method.
Specifically, the modulator 130 may modulate bits output from the demultiplexer (not shown) in various modulation methods such as non-uniform 16-QAM, non-uniform 64-QAM, non-uniform 256-QAM, non-uniform 1024-QAM, non-uniform 4096-QAM, etc.
Hereinafter, a method for generating a modulation symbol by using the non-uniform constellation method according to an exemplary embodiment will be explained.
First, the non-uniform constellation method has the following characteristics:
In the non-uniform constellation method, the constellation points may not regularly be arranged unlike in the uniform constellation method. Accordingly, when the non-uniform constellation method is used, performance for a signal-to-noise ratio (SNR) less than a specific value can be improved and a high SNR gain can be obtained in comparison to the uniform constellation method.
In addition, the characteristics of the constellation may be determined by one or more parameters such as a distance between constellation points. Since the constellation points are regularly distributed in the uniform constellation, the number of parameters for specifying the uniform constellation method may be one (1). However, the number of parameters necessary for specifying the non-uniform constellation method is relatively larger and the number of parameters increases as the constellation (e.g., the number of constellation points) increases.
In the case of the non-uniform constellation method, an x-axis and a y-axis may be designed to be symmetric to each other or may be designed to be asymmetric to each other. When the x-axis and the y-axis are designed to be asymmetric to each other, improved performance can be guaranteed, but decoding complexity may increase.
Hereinafter, an example of a case in which the x-axis and the y-axis are designed to be asymmetric to each other will be explained. In this case, once a constellation point of the first quadrant is defined, locations of 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={z0, z1, . . . , zM−1}. In this case, when the constellation points existing in the first quadrant are defined as {x0, x1, x2, . . . , xM/4−1}, z may be defined as follows:

from z0 to zM/4−1=from x0 to xM/4 

from zM/4 to z2×M/4−1=−conj(from x0 to xM/4)

from z2×M/4 to z3×m/4−1=conj(from x0 to xM/4)

from z3×M/4 to z4×M/4−1=−(from x0 to xM/4)

Accordingly, the modulator 130 may map the bits [y0, . . . , ym−1] output from the demultiplexer (not shown) onto constellation points in the non-uniform constellation method by mapping the output bits onto zL having an index of




  L
  =
  
    
      ∑
      
        i
        =
        0
      
      
        m
        -
        1
      
    
    ⁢
    
      
    
    ⁢
    
      
        (
        
          
            y
            1
          
          ×
          
            2
            
              m
              -
              1
            
          
        
        )
      
      .
    
  




An example of the constellation of the non-uniform constellation method is illustrated in FIGS. 15 to 19.

An example of the method for modulating asymmetrically in the non-uniform constellation method in the modulator 130 is illustrated as in Tables 59 to 64 presented below. That is, according to an exemplary embodiment, modulation is performed in the non-uniform constellation method by defining constellation points existing in the first quadrant and defining constellations points existing in the other quadrants based on Tables 59 to 64.












TABLE 59







w/Shape
NUC_16_6/15
NUC_16_7/15
NUC_16_8/15
NUC_16_9/15
NUC_16_10/15





w0
0.4530 + 0.2663i
1.2103 + 0.5026i
0.4819 + 0.2575i
0.4909 + 1.2007i
0.2173 + 0.4189i


w1
0.2663 + 0.4530i
0.5014 + 1.2103i
0.2575 + 0.4819i
1.2007 + 0.4909i
0.6578 + 0.2571i


w2
1.2092 + 0.5115i
0.4634 + 0.2624i
1.2068 + 0.4951i
0.2476 + 0.5065i
0.4326 + 1.1445i


w3
0.5115 + 1.2092i
0.2624 + 0.4627i
0.4951 + 1.2068i
0.5053 + 0.2476i
1.2088 + 0.5659i















x/Shape
NUC_16_11/15
NUC_16_12/15
NUC_16_13/15






x0
0.9583 + 0.9547i
0.2999 + 0.2999i
0.9517 + 0.9511i



x1
0.9547 + 0.2909i
0.9540 + 0.2999i
0.9524 + 0.3061i



x2
0.2921 + 0.9583i
0.2999 + 0.9540i
0.3067 + 0.9524i



x3
0.2909 + 0.2927i
0.9540 + 0.9540i
0.3061 + 0.3067i





















TABLE 60







x/Shape
R64_6/15
R64_7/15
R64_8/15
R64_9/15
R64_10/15





x0
0.4387 + 1.6023i
0.3352 + 0.6028i
1.4827 + 0.2920i
0.3547 + 0.6149i
1.4388 + 0.2878i


x1
1.6023 + 0.4387i
0.2077 + 0.6584i
1.2563 + 0.8411i
0.1581 + 0.6842i
1.2150 + 0.8133i


x2
0.8753 + 1.0881i
0.1711 + 0.3028i
1.0211 + 0.2174i
0.1567 + 0.2749i
1.0386 + 0.2219i


x3
1.0881 + 0.8753i
0.1556 + 0.3035i
0.8798 + 0.5702i
0.1336 + 0.2700i
0.8494 + 0.6145i


x4
0.2202 + 0.9238i
0.6028 + 0.3345i
0.2920 + 1.4827i
0.6177 + 0.4030i
0.2931 + 1.4656i


x5
0.2019 + 0.7818i
0.6577 + 0.2084i
0.8410 + 1.2563i
0.7262 + 0.1756i
0.8230 + 1.2278i


x6
0.3049 + 0.8454i
0.3021 + 0.1711i
0.2174 + 1.0211i
0.3568 + 0.1756i
0.2069 + 1.0649i


x7
0.2653 + 0.7540i
0.3028 + 0.1556i
0.5702 + 0.8798i
0.3771 + 0.1336i
0.5677 + 0.8971i


x8
0.7818 + 0.2019i
0.5556 + 0.8922i
0.3040 + 0.1475i
0.5639 + 0.8864i
0.4119 + 0.1177i


x9
0.9238 + 0.2202i
0.2352 + 1.0190i
0.3028 + 0.1691i
0.1980 + 1.0277i
0.3998 + 0.2516i


x10
0.7540 + 0.2653i
0.8450 + 1.2619i
0.6855 + 0.1871i
0.8199 + 1.2515i
0.7442 + 0.1559i


x11
0.8454 + 0.3049i
0.2922 + 1.4894i
0.6126 + 0.3563i
0.2854 + 1.4691i
0.5954 + 0.4328i


x12
0.2675 + 0.2479i
0.8929 + 0.5549i
0.1475 + 0.3040i
0.8654 + 0.6058i
0.1166 + 0.1678i


x13
0.2479 + 0.2675i
1.0197 + 0.2359i
0.1691 + 0.3028i
1.0382 + 0.2141i
0.1582 + 0.3325i


x14
0.2890 + 0.2701i
1.2626 + 0.8457i
0.1871 + 0.6855i
1.2362 + 0.8416i
0.1355 + 0.7408i


x15
0.2701 + 0.2890i
1.4894 + 0.2922i
0.3563 + 0.6126i
1.4663 + 0.2973i
0.3227 + 0.6200i















x/Shape
R64_11/15
R64_12/15
R64_13/15






x0
0.3317 + 0.6970i
1.0854 + 0.5394i
0.4108 + 0.7473i



x1
0.1386 + 0.8824i
0.7353 + 0.4623i
0.1343 + 0.5338i



x2
0.1323 + 0.4437i
1.0474 + 0.1695i
0.1570 + 0.9240i



x3
0.1015 + 0.1372i
0.7243 + 0.1504i
0.1230 + 0.1605i



x4
0.5682 + 0.4500i
1.0693 + 0.9408i
0.6285 + 0.4617i



x5
0.6739 + 0.1435i
0.7092 + 0.8073i
0.3648 + 0.3966i



x6
0.3597 + 0.3401i
1.4261 + 0.2216i
0.6907 + 0.1541i



x7
0.3660 + 0.1204i
0.6106 + 1.1783i
0.3994 + 0.1308i



x8
0.6004 + 0.8922i
0.1392 + 0.4078i
0.7268 + 0.8208i



x9
0.2120 + 1.2253i
0.4262 + 0.4205i
1.0463 + 0.9495i



x10
0.9594 + 1.0714i
0.1407 + 0.1336i
0.1866 + 1.2733i



x11
0.5829 + 1.3995i
0.4265 + 0.1388i
0.5507 + 1.1793i



x12
0.8439 + 0.5675i
0.1388 + 0.7057i
0.9283 + 0.5140i



x13
0.9769 + 0.1959i
0.4197 + 0.7206i
1.2648 + 0.5826i



x14
1.2239 + 0.6760i
0.1682 + 1.0316i
0.9976 + 0.1718i



x15
1.3653 + 0.2323i
0.2287 + 1.3914i
1.3412 + 0.1944i





















TABLE 61







x/Shape
NUC_64_6/15
NUC_64_7/15
NUC_64_8/15
NUC_64_9/15
NUC_64_10/15





x0
0.4387 + 1.6023i
0.3352 + 0.6028i
1.4827 + 0.2920i
0.3547 + 0.6149i
1.4388 + 0.2878i


x1
1.6023 + 0.4387i
0.2077 + 0.6584i
1.2563 + 0.8411i
0.1581 + 0.6842i
1.2150 + 0.8133i


x2
0.8753 + 1.0881i
0.1711 + 0.3028i
1.0211 + 0.2174i
0.1567 + 0.2749i
1.0386 + 0.2219i


x3
1.0881 + 0.8753i
0.1556 + 0.3035i
0.8798 + 0.5702i
0.1336 + 0.2700i
0.8494 + 0.6145i


x4
0.2202 + 0.9238i
0.6028 + 0.3345i
0.2920 + 1.4827i
0.6177 + 0.4030i
0.2931 + 1.4656i


x5
0.2019 + 0.7818i
0.6577 + 0.2084i
0.8410 + 1.2563i
0.7262 + 0.1756i
0.8230 + 1.2278i


x6
0.3049 + 0.8454i
0.3021 + 0.1711i
0.2174 + 1.0211i
0.3568 + 0.1756i
0.2069 + 1.0649i


x7
0.2653 + 0.7540i
0.3028 + 0.1556i
0.5702 + 0.8798i
0.3771 + 0.1336i
0.5677 + 0.8971i


x8
0.7818 + 0.2019i
0.5556 + 0.8922i
0.3040 + 0.1475i
0.5639 + 0.8864i
0.4119 + 0.1177i


x9
0.9238 + 0.2202i
0.2352 + 1.0190i
0.3028 + 0.1691i
0.1980 + 1.0277i
0.3998 + 0.2516i


x10
0.7540 + 0.2653i
0.8450 + 1.2619i
0.6855 + 0.1871i
0.8199 + 1.2515i
0.7442 + 0.1559i


x11
0.8454 + 0.3049i
0.2922 + 1.4894i
0.6126 + 0.3563i
0.2854 + 1.4691i
0.5954 + 0.4328i


x12
0.2675 + 0.2479i
0.8929 + 0.5549i
0.1475 + 0.3040i
0.8654 + 0.6058i
0.1166 + 0.1678i


x13
0.2479 + 0.2675i
1.0197 + 0.2359i
0.1691 + 0.3028i
1.0382 + 0.2141i
0.1582 + 0.3325i


x14
0.2890 + 0.2701i
1.2626 + 0.8457i
0.1871 + 0.6855i
1.2362 + 0.8416i
0.1355 + 0.7408i


x15
0.2701 + 0.2890i
1.4894 + 0.2922i
0.3563 + 0.6126i
1.4663 + 0.2973i
0.3227 + 0.6200i















x/Shape
NUC_64_11/15
NUC_64_12/15
NUC_64_13/15






x0
0.3317 + 0.6970i
1.0854 + 0.5394i
0.8624 + 1.1715i



x1
0.1386 + 0.8824i
0.7353 + 0.4623i
1.1184 + 0.8462i



x2
0.1323 + 0.4437i
1.0474 + 0.1695i
0.2113 + 1.3843i



x3
0.1015 + 0.1372i
0.7243 + 0.1504i
0.7635 + 0.7707i



x4
0.5682 + 0.4500i
1.0693 + 0.9408i
1.1796 + 0.1661i



x5
0.6739 + 0.1435i
0.7092 + 0.8073i
1.0895 + 0.4882i



x6
0.3597 + 0.3401i
1.4261 + 0.2216i
0.8101 + 0.1492i



x7
0.3660 + 0.1204i
0.6106 + 1.1783i
0.7482 + 0.4477i



x8
0.6004 + 0.8922i
0.1392 + 0.4078i
0.1524 + 0.9943i



x9
0.2120 + 1.2253i
0.4262 + 0.4205i
0.1482 + 0.6877i



x10
0.9594 + 1.0714i
0.1407 + 0.1336i
0.4692 + 1.0853i



x11
0.5829 + 1.3995i
0.4265 + 0.1388i
0.4492 + 0.7353i



x12
0.8439 + 0.5675i
0.1388 + 0.7057i
0.1578 + 0.1319i



x13
0.9769 + 0.1959i
0.4197 + 0.7206i
0.1458 + 0.4025i



x14
1.2239 + 0.6760i
0.1682 + 1.0316i
0.4763 + 0.1407i



x15
1.3653 + 0.2323i
0.2287 + 1.3914i
0.4411 + 0.4267i




















TABLE 62







x/Shape
7/15
13/15









x0
0.1543 + 0.3088i
1.4293 + 0.2286i



x1
0.1719 + 0.3074i
0.6234 + 1.1799i



x2
0.2021 + 0.6601i
1.0719 + 0.9247i



x3
0.3396 + 0.6009i
0.6841 + 0.8071i



x4
0.3080 + 0.1543i
1.0440 + 0.1692i



x5
0.3069 + 0.1716i
0.7232 + 0.1541i



x6
0.6607 + 0.2018i
1.0639 + 0.5312i



x7
0.6011 + 0.3395i
0.7147 + 0.4706i



x8
0.2936 + 1.4847i
0.2128 + 1.4368i



x9
0.8412 + 1.2593i
0.1990 + 1.0577i



x10
0.2321 + 1.0247i
0.1176 + 0.6586i



x11
0.5629 + 0.8926i
0.3691 + 0.7533i



x12
1.4850 + 0.2935i
0.1457 + 0.1261i



x13
1.2599 + 0.8426i
0.4329 + 0.1380i



x14
1.0247 + 0.2320i
0.1424 + 0.3819i



x15
0.8925 + 0.5631i
0.4216 + 0.4265i






















TABLE 63







x/Shape
R6/15
R7/15
R8/15
R9/15
R10/15





x0
0.6800 + 1.6926i
1.2905 + 1.3099i
1.0804 + 1.3788i
1.3231 + 1.1506i
1.6097 + 0.1548i


x1
0.3911 + 1.3645i
1.0504 + 0.9577i
1.0487 + 0.9862i
0.9851 + 1.2311i
1.5549 + 0.4605i


x2
0.2191 + 1.7524i
1.5329 + 0.8935i
1.6464 + 0.7428i
1.1439 + 0.8974i
1.3226 + 0.1290i


x3
0.2274 + 1.4208i
1.1577 + 0.8116i
1.3245 + 0.9414i
0.9343 + 0.9271i
1.2772 + 0.3829i


x4
0.8678 + 1.2487i
1.7881 + 0.2509i
0.7198 + 1.2427i
1.5398 + 0.7962i
1.2753 + 1.0242i


x5
0.7275 + 1.1667i
1.4275 + 0.1400i
0.8106 + 1.0040i
0.9092 + 0.5599i
1.4434 + 0.7540i


x6
0.8747 + 1.0470i
1.4784 + 0.5201i
0.5595 + 1.0317i
1.2222 + 0.6574i
1.0491 + 0.8476i


x7
0.7930 + 1.0406i
1.3408 + 0.4346i
0.6118 + 0.9722i
0.9579 + 0.6373i
1.1861 + 0.6253i


x8
0.2098 + 0.9768i
0.7837 + 0.5867i
1.6768 + 0.2002i
0.7748 + 1.5867i
0.9326 + 0.0970i


x9
0.2241 + 1.0454i
0.8250 + 0.6455i
0.9997 + 0.6844i
0.6876 + 1.2489i
0.8962 + 0.2804i


x10
0.1858 + 0.9878i
0.8256 + 0.5601i
1.4212 + 0.4769i
0.5992 + 0.9208i
1.1044 + 0.1102i


x11
0.1901 + 1.0659i
0.8777 + 0.6110i
1.1479 + 0.6312i
0.6796 + 0.9743i
1.0648 + 0.3267i


x12
0.5547 + 0.8312i
1.0080 + 0.1843i
0.6079 + 0.6566i
0.5836 + 0.5879i
0.7325 + 0.6071i


x13
0.5479 + 0.8651i
1.0759 + 0.1721i
0.7284 + 0.6957i
0.6915 + 0.5769i
0.8260 + 0.4559i


x14
0.6073 + 0.8182i
1.0056 + 0.2758i
0.5724 + 0.7031i
0.5858 + 0.7058i
0.8744 + 0.7153i


x15
0.5955 + 0.8420i
1.0662 + 0.2964i
0.6302 + 0.7259i
0.6868 + 0.6793i
0.9882 + 0.5300i


x16
1.4070 + 0.1790i
0.8334 + 1.5554i
0.1457 + 1.4010i
1.6118 + 0.1497i
0.1646 + 1.6407i


x17
1.7227 + 0.2900i
0.8165 + 1.1092i
0.1866 + 1.7346i
0.9511 + 0.1140i
0.4867 + 1.5743i


x18
1.3246 + 0.2562i
0.6092 + 1.2729i
0.1174 + 1.1035i
1.2970 + 0.1234i
0.1363 + 1.3579i


x19
1.3636 + 0.3654i
0.6728 + 1.1456i
0.1095 + 1.0132i
1.0266 + 0.1191i
0.4023 + 1.3026i


x20
1.3708 + 1.2834i
0.3061 + 1.7469i
0.4357 + 1.3636i
1.5831 + 0.4496i
1.0542 + 1.2584i


x21
1.6701 + 0.8403i
0.1327 + 1.4056i
0.5853 + 1.6820i
0.9328 + 0.3586i
0.7875 + 1.4450i


x22
1.1614 + 0.7909i
0.3522 + 1.3414i
0.3439 + 1.0689i
1.2796 + 0.3894i
0.8687 + 1.0407i


x23
1.2241 + 0.7367i
0.2273 + 1.3081i
0.3234 + 0.9962i
1.0188 + 0.3447i
0.6502 + 1.1951i


x24
0.9769 + 0.1863i
0.5007 + 0.8098i
0.1092 + 0.6174i
0.5940 + 0.1059i
0.0982 + 0.9745i


x25
0.9452 + 0.2057i
0.5528 + 0.8347i
0.1074 + 0.6307i
0.7215 + 0.1100i
0.2842 + 0.9344i


x26
1.0100 + 0.2182i
0.4843 + 0.8486i
0.1109 + 0.6996i
0.5863 + 0.1138i
0.1142 + 1.1448i


x27
0.9795 + 0.2417i
0.5304 + 0.8759i
0.1076 + 0.7345i
0.6909 + 0.1166i
0.3385 + 1.0973i


x28
0.8241 + 0.4856i
0.1715 + 0.9147i
0.3291 + 0.6264i
0.5843 + 0.3604i
0.6062 + 0.7465i


x29
0.8232 + 0.4837i
0.1540 + 0.9510i
0.3126 + 0.6373i
0.6970 + 0.3592i
0.4607 + 0.8538i


x30
0.8799 + 0.5391i
0.1964 + 0.9438i
0.3392 + 0.6999i
0.5808 + 0.3250i
0.7263 + 0.8764i


x31
0.8796 + 0.5356i
0.1788 + 0.9832i
0.3202 + 0.7282i
0.6678 + 0.3290i
0.5450 + 1.0067i


x32
0.1376 + 0.3342i
0.3752 + 0.1667i
0.9652 + 0.1066i
0.1406 + 1.6182i
0.2655 + 0.0746i


x33
0.1383 + 0.3292i
0.3734 + 0.1667i
0.9075 + 0.1666i
0.1272 + 1.2984i
0.2664 + 0.0759i


x34
0.1363 + 0.3322i
0.3758 + 0.1661i
0.9724 + 0.1171i
0.1211 + 0.9644i
0.4571 + 0.0852i


x35
0.1370 + 0.3273i
0.3746 + 0.1649i
0.9186 + 0.1752i
0.1220 + 1.0393i
0.4516 + 0.1062i


x36
0.1655 + 0.3265i
0.4013 + 0.1230i
0.6342 + 0.1372i
0.1124 + 0.6101i
0.2559 + 0.1790i


x37
0.1656 + 0.3227i
0.4001 + 0.1230i
0.6550 + 0.1495i
0.1177 + 0.6041i
0.2586 + 0.1772i


x38
0.1634 + 0.3246i
0.4037 + 0.1230i
0.6290 + 0.1393i
0.1136 + 0.7455i
0.3592 + 0.2811i


x39
0.1636 + 0.3208i
0.4019 + 0.1218i
0.6494 + 0.1504i
0.1185 + 0.7160i
0.3728 + 0.2654i


x40
0.1779 + 0.6841i
0.6025 + 0.3934i
1.3127 + 0.1240i
0.4324 + 1.5679i
0.7706 + 0.0922i


x41
0.1828 + 0.6845i
0.5946 + 0.3928i
0.9572 + 0.4344i
0.3984 + 1.2825i
0.7407 + 0.2260i


x42
0.1745 + 0.6828i
0.6116 + 0.3879i
1.2403 + 0.2631i
0.3766 + 0.9534i
0.6180 + 0.0927i


x43
0.1793 + 0.6829i
0.6019 + 0.3837i
1.0254 + 0.4130i
0.3668 + 1.0301i
0.6019 + 0.1658i


x44
0.3547 + 0.6009i
0.7377 + 0.1618i
0.6096 + 0.4214i
0.3667 + 0.5995i
0.6007 + 0.4980i


x45
0.3593 + 0.6011i
0.7298 + 0.1582i
0.6773 + 0.4284i
0.3328 + 0.5960i
0.6673 + 0.3928i


x46
0.3576 + 0.5990i
0.7274 + 0.1782i
0.5995 + 0.4102i
0.3687 + 0.7194i
0.4786 + 0.3935i


x47
0.3624 + 0.5994i
0.7165 + 0.1746i
0.6531 + 0.4101i
0.3373 + 0.6964i
0.5176 + 0.3391i


x48
0.2697 + 0.1443i
0.1509 + 0.2425i
0.1250 + 0.1153i
0.1065 + 0.1146i
0.0757 + 0.1003i


x49
0.2704 + 0.1433i
0.1503 + 0.2400i
0.1252 + 0.1158i
0.1145 + 0.1108i
0.0753 + 0.1004i


x50
0.2644 + 0.1442i
0.1515 + 0.2437i
0.1245 + 0.1152i
0.1053 + 0.1274i
0.0777 + 0.4788i


x51
0.2650 + 0.1432i
0.1503 + 0.2425i
0.1247 + 0.1156i
0.1134 + 0.1236i
0.0867 + 0.4754i


x52
0.2763 + 0.1638i
0.1285 + 0.2388i
0.3768 + 0.1244i
0.1111 + 0.3821i
0.1023 + 0.2243i


x53
0.2768 + 0.1626i
0.1279 + 0.2419i
0.3707 + 0.1237i
0.1186 + 0.3867i
0.1010 + 0.2242i


x54
0.2715 + 0.1630i
0.1279 + 0.2431i
0.3779 + 0.1260i
0.1080 + 0.3431i
0.1950 + 0.3919i


x55
0.2719 + 0.1618i
0.1279 + 0.2406i
0.3717 + 0.1252i
0.1177 + 0.3459i
0.1881 + 0.3969i


x56
0.6488 + 0.1696i
0.3394 + 0.5764i
0.1161 + 0.3693i
0.3644 + 0.1080i
0.0930 + 0.8122i


x57
0.6462 + 0.1706i
0.3364 + 0.5722i
0.1157 + 0.3645i
0.3262 + 0.1104i
0.2215 + 0.7840i


x58
0.6456 + 0.1745i
0.3328 + 0.5758i
0.1176 + 0.3469i
0.3681 + 0.1173i
0.0937 + 0.6514i


x59
0.6431 + 0.1753i
0.3303 + 0.5698i
0.1171 + 0.3424i
0.3289 + 0.1196i
0.1540 + 0.6366i


x60
0.5854 + 0.3186i
0.1491 + 0.6316i
0.3530 + 0.3899i
0.3665 + 0.3758i
0.4810 + 0.6306i


x61
0.5862 + 0.3167i
0.1461 + 0.6280i
0.3422 + 0.3808i
0.3310 + 0.3795i
0.3856 + 0.7037i


x62
0.5864 + 0.3275i
0.1509 + 0.6280i
0.3614 + 0.3755i
0.3672 + 0.3353i
0.3527 + 0.5230i


x63
0.5873 + 0.3254i
0.1473 + 0.6225i
0.3509 + 0.3656i
0.3336 + 0.3402i
0.3100 + 0.5559i















x/Shape
R11/15
R12/15
R13/15






x0
0.3105 + 0.3382i
1.1014 + 1.1670i
0.3556 + 0.3497i



x1
0.4342 + 0.3360i
0.8557 + 1.2421i
0.3579 + 0.4945i



x2
0.3149 + 0.4829i
1.2957 + 0.8039i
0.5049 + 0.3571i



x3
0.4400 + 0.4807i
1.0881 + 0.8956i
0.5056 + 0.5063i



x4
0.1811 + 0.3375i
0.5795 + 1.2110i
0.2123 + 0.3497i



x5
0.0633 + 0.3404i
0.6637 + 1.4215i
0.2116 + 0.4900i



x6
0.1818 + 0.4851i
0.6930 + 1.0082i
0.0713 + 0.3489i



x7
0.0633 + 0.4815i
0.8849 + 0.9647i
0.0690 + 0.4960i



x8
0.3084 + 0.1971i
1.2063 + 0.5115i
0.3527 + 0.2086i



x9
0.4356 + 0.1993i
1.0059 + 0.4952i
0.3497 + 0.0713i



x10
0.3098 + 0.0676i
1.4171 + 0.5901i
0.4960 + 0.2123i



x11
0.4342 + 0.0691i
1.0466 + 0.6935i
0.4974 + 0.0698i



x12
0.1775 + 0.1985i
0.6639 + 0.6286i
0.2086 + 0.2079i



x13
0.0640 + 0.1978i
0.8353 + 0.5851i
0.2094 + 0.0690i



x14
0.1775 + 0.0676i
0.6879 + 0.8022i
0.0676 + 0.2079i



x15
0.0647 + 0.0669i
0.8634 + 0.7622i
0.0698 + 0.0683i



x16
0.7455 + 0.3411i
0.1213 + 1.4366i
0.3586 + 0.7959i



x17
0.5811 + 0.3396i
0.1077 + 1.2098i
0.3571 + 0.6392i



x18
0.7556 + 0.4669i
0.0651 + 0.9801i
0.5034 + 0.8271i



x19
0.5862 + 0.4756i
0.2009 + 1.0115i
0.5063 + 0.6600i



x20
0.9556 + 0.3280i
0.3764 + 1.4264i
0.2146 + 0.7862i



x21
1.1767 + 0.3091i
0.3237 + 1.2130i
0.2109 + 0.6340i



x22
0.9673 + 0.4720i
0.5205 + 0.9814i
0.0713 + 0.8093i



x23
1.2051 + 0.5135i
0.3615 + 1.0163i
0.0698 + 0.6467i



x24
0.7367 + 0.2015i
0.0715 + 0.6596i
0.2799 + 1.0862i



x25
0.5811 + 0.2015i
0.2116 + 0.6597i
0.2806 + 1.2755i



x26
0.7316 + 0.0669i
0.0729 + 0.8131i
0.4328 + 0.9904i



x27
0.5782 + 0.0669i
0.2158 + 0.8246i
0.4551 + 1.1812i



x28
0.9062 + 0.1971i
0.5036 + 0.6467i
0.2309 + 0.9414i



x29
1.2829 + 0.1185i
0.3526 + 0.6572i
0.1077 + 1.3891i



x30
0.9156 + 0.0735i
0.5185 + 0.8086i
0.0772 + 0.9852i



x31
1.1011 + 0.0735i
0.3593 + 0.8245i
0.0802 + 1.1753i



x32
0.3244 + 0.8044i
1.2545 + 0.1010i
0.8301 + 0.3727i



x33
0.4589 + 0.8218i
1.0676 + 0.0956i
0.8256 + 0.5256i



x34
0.3207 + 0.6415i
1.4782 + 0.1167i
0.6593 + 0.3668i



x35
0.4509 + 0.6371i
0.8981 + 0.0882i
0.6623 + 0.5182i



x36
0.1920 + 0.8196i
0.5518 + 0.0690i
1.0186 + 0.3645i



x37
0.0633 + 0.8167i
0.6903 + 0.0552i
1.0001 + 0.5242i



x38
0.1811 + 0.6371i
0.5742 + 0.1987i
1.1857 + 0.2725i



x39
0.0640 + 0.6415i
0.7374 + 0.1564i
1.3928 + 0.3408i



x40
0.3331 + 1.0669i
1.2378 + 0.3049i
0.8011 + 0.2227i



x41
0.4655 + 1.0087i
1.0518 + 0.3032i
0.7981 + 0.0735i



x42
0.3433 + 1.2865i
1.4584 + 0.3511i
0.6459 + 0.2198i



x43
0.5004 + 1.5062i
0.9107 + 0.2603i
0.6430 + 0.0713i



x44
0.1971 + 1.0051i
0.6321 + 0.4729i
0.9681 + 0.2205i



x45
0.0735 + 1.0298i
0.7880 + 0.4392i
0.9615 + 0.0735i



x46
0.1498 + 1.5018i
0.6045 + 0.3274i
1.3327 + 0.1039i



x47
0.0865 + 1.2553i
0.7629 + 0.2965i
1.1359 + 0.0809i



x48
0.7811 + 0.8080i
0.0596 + 0.0739i
0.8382 + 0.8709i



x49
0.6167 + 0.8153i
0.1767 + 0.0731i
0.8145 + 0.6934i



x50
0.7636 + 0.6255i
0.0612 + 0.2198i
0.6645 + 0.8486i



x51
0.6000 + 0.6327i
0.1815 + 0.2192i
0.6600 + 0.6786i



x52
0.9898 + 0.7680i
0.4218 + 0.0715i
1.1612 + 0.6949i



x53
1.5855 + 0.1498i
0.2978 + 0.0725i
0.9785 + 0.6942i



x54
0.9476 + 0.6175i
0.4337 + 0.2115i
1.3698 + 0.6259i



x55
1.4625 + 0.4015i
0.3057 + 0.2167i
1.2183 + 0.4841i



x56
0.8276 + 1.0225i
0.0667 + 0.5124i
0.7989 + 1.0498i



x57
0.6313 + 1.0364i
0.2008 + 0.5095i
0.4395 + 1.4203i



x58
0.8815 + 1.2865i
0.0625 + 0.3658i
0.6118 + 1.0246i



x59
0.6342 + 1.2705i
0.1899 + 0.3642i
0.6303 + 1.2421i



x60
1.0422 + 0.9593i
0.4818 + 0.4946i
1.0550 + 0.8924i



x61
1.2749 + 0.8538i
0.3380 + 0.5050i
0.8612 + 1.2800i



x62
1.1556 + 1.1847i
0.4571 + 0.3499i
1.2696 + 0.8969i



x63
1.4771 + 0.6742i
0.3216 + 0.3599i
1.0342 + 1.1181i




















TABLE 64





x/






Shape
7/15
9/15
11/15
13/15







x0
0.1256 + 0.2068i
0.0899 + 0.1337i
0.0582 + 0.3157i
1.4685 + 






0.3416i


x1
0.1255 + 0.2077i
0.0910 + 0.1377i
0.1170 + 0.3398i
0.9545 + 






1.2710i


x2
0.1265 + 0.2240i
0.0873 + 0.3862i
0.0609 + 0.4922i
1.2326 + 






0.9727i


x3
0.1264 + 0.2247i
0.0883 + 0.3873i
0.1505 + 0.4655i
0.9928 + 






1.0555i


x4
0.1557 + 0.2325i
0.1115 + 0.1442i
0.0809 + 0.2181i
0.5366 + 






1.2142i


x5
0.1555 + 0.2328i
0.1135 + 0.1472i
0.1912 + 0.2702i
0.7176 + 






1.2813i


x6
0.1594 + 0.2446i
0.2067 + 0.3591i
0.3444 + 0.3892i
0.5970 + 






1.0330i


x7
0.1590 + 0.2447i
0.1975 + 0.3621i
0.2678 + 0.3980i
0.7840 + 






1.0563i


x8
0.1383 + 0.6402i
0.1048 + 0.7533i
0.0827 + 0.7889i
1.3683 + 






0.5610i


x9
0.1417 + 0.6402i
0.1770 + 0.7412i
0.2390 + 0.7609i
0.9738 + 






0.6818i


x10
0.1344 + 0.6027i
0.1022 + 0.5904i
0.0692 + 0.6437i
1.2078 + 






0.7474i


x11
0.1366 + 0.6023i
0.1191 + 0.5890i
0.1982 + 0.6202i
1.0017 + 






0.8588i


x12
0.3330 + 0.5769i
0.4264 + 0.6230i
0.5248 + 0.6168i
0.6323 + 






0.6766i


x13
0.3228 + 0.5787i
0.3650 + 0.6689i
0.3946 + 0.7012i
0.7966 + 






0.6886i


x14
0.3099 + 0.5483i
0.3254 + 0.5153i
0.4297 + 0.5004i
0.6339 + 






0.8522i


x15
0.3017 + 0.5493i
0.2959 + 0.5302i
0.3236 + 0.5668i
0.8022 + 






0.8656i


x16
0.4168 + 0.1173i
0.3256 + 0.0768i
0.1081 + 0.0518i
0.0931 + 






1.1992i


x17
0.4168 + 0.1185i
0.3266 + 0.0870 i
0.2626 + 0.0685i
0.1140 + 






1.4121i


x18
0.4205 + 0.1200i
0.4721 + 0.0994i
0.4934 + 0.0753i
0.0730 + 






1.0061i


x19
0.4202 + 0.1215i
0.4721 + 0.1206i
0.4151 + 0.0899i
0.2219 + 






0.9648i


x20
0.3852 + 0.1797i
0.2927 + 0.1267i
0.0850 + 0.1018i
0.4566 + 






1.4370i


x21
0.3860 + 0.1803i
0.2947 + 0.1296i
0.2425 + 0.1555i
0.3039 + 






1.2999i


x22
0.3867 + 0.1760i
0.3823 + 0.2592i
0.4222 + 0.2788i
0.4348 + 






0.9883i


x23
0.3879 + 0.1765i
0.3944 + 0.2521i
0.3833 + 0.2175i
0.3152 + 






1.1094i


x24
0.7366 + 0.1528i
0.7755 + 0.1118i
0.8081 + 0.0814i
0.0615 + 






0.6426i


x25
0.7255 + 0.1884i
0.7513 + 0.2154i
0.7874 + 0.2263i
0.1977 + 






0.6403i


x26
0.7219 + 0.1535i
0.6591 + 0.1033i
0.6444 + 0.0779i
0.0627 + 






0.8195i


x27
0.7117 + 0.1857i
0.6446 + 0.1737i
0.6425 + 0.1804i
0.1944 + 






0.7950i


x28
0.5825 + 0.4149i
0.5906 + 0.4930i
0.6442 + 0.4989i
0.4831 + 






0.6579i


x29
0.6012 + 0.4001i
0.6538 + 0.4155i
0.7296 + 0.3793i
0.3357 + 






0.6420i


x30
0.5715 + 0.3988i
0.4981 + 0.3921i
0.5312 + 0.3831i
0.4793 + 






0.8186i


x31
0.5889 + 0.3864i
0.5373 + 0.3586i
0.5938 + 0.3038i
0.3334 + 






0.7968i


x32
0.1671 + 1.7095i
0.1630 + 1.6621i
0.1530 + 1.5695i
1.1496 + 






0.0911i


x33
0.4420 + 1.6137i
0.4720 + 1.5898i
0.4501 + 1.5218i
0.9665 + 






0.0743i


x34
0.1477 + 1.3374i
0.1268 + 1.3488i
0.1272 + 1.3199i
1.3416 + 






0.1182i


x35
0.3135 + 1.3073i
0.3752 + 1.2361i
0.3852 + 1.2808i
0.9586 + 






0.2173i


x36
1.0214 + 1.3784i
1.0398 + 1.2991i
1.0070 + 1.2418i
0.6270 + 






0.0735i


x37
0.7434 + 1.5169i
0.7733 + 1.4772i
0.7407 + 1.4062i
0.7860 + 






0.0714i


x38
0.7869 + 1.1040i
0.8380 + 1.0552i
0.8513 + 1.0461i
0.6272 + 






0.2168i


x39
0.6252 + 1.2041i
0.6242 + 1.2081i
0.6296 + 1.1796i
0.7881 + 






0.2183i


x40
0.1526 + 0.9140i
0.1103 + 0.9337i
0.0974 + 0.9545i
1.1689 + 






0.5018i


x41
0.1914 + 0.9087i
0.2415 + 0.9155i
0.2832 + 0.9168i
0.9701 + 






0.5155i


x42
0.1539 + 1.0179i
0.1118 + 1.1163i
0.1134 + 1.1215i
1.1870 + 






0.3104i


x43
0.2095 + 1.0084i
0.3079 + 1.0866i
0.3306 + 1.0823i
0.9885 + 






0.3629i


x44
0.5021 + 0.7976i
0.5647 + 0.7638i
0.6237 + 0.7456i
0.6391 + 






0.5151i


x45
0.4537 + 0.8238i
0.4385 + 0.8433i
0.4639 + 0.8471i
0.7973 + 






0.5176i


x46
0.5663 + 0.8802i
0.6846 + 0.8841i
0.7263 + 0.8835i
0.6279 + 






0.3616i


x47
0.5021 + 0.9146i
0.5165 + 1.0034i
0.5354 + 1.0001i
0.7944 + 






0.3650i


x48
1.7030 + 0.1913i
1.6489 + 0.1630i
1.5940 + 0.1623i
0.0681 + 






0.0655i


x49
1.6116 + 0.5458i
1.5848 + 0.4983i
1.5379 + 0.4722i
0.2048 + 






0.0670i


x50
1.3788 + 0.1641i
1.3437 + 0.1389i
1.3481 + 0.1336i
0.0671 + 






0.1968i


x51
1.3121 + 0.4320i
1.2850 + 0.4025i
1.2955 + 0.3937i
0.2016 + 






0.2008i


x52
1.2722 + 1.1406i
1.2728 + 1.0661i
1.2320 + 1.0203i
0.4809 + 






0.0710i


x53
1.4536 + 0.8623i
1.4509 + 0.7925i
1.4100 + 0.7594i
0.3406 + 






0.0684i


x54
1.0204 + 0.9076i
1.0249 + 0.8794i
1.0365 + 0.8610i
0.4779 + 






0.2121i


x55
1.1634 + 0.7212i
1.1758 + 0.6545i
1.1885 + 0.6421i
0.3377 + 






0.2072i


x56
0.9979 + 0.1554i
0.9629 + 0.1113i
0.9785 + 0.0944i
0.0625 + 






0.4797i


x57
0.9681 + 0.2706i
0.9226 + 0.2849i
0.9430 + 0.2764i
0.2019 + 






0.4853i


x58
1.0796 + 0.1672i
1.1062 + 0.1118i
1.1491 + 0.1124i
0.0650 + 






0.3344i


x59
1.0514 + 0.3095i
1.0674 + 0.3393i
1.1051 + 0.3293i
0.1977 + 






0.3399i


x60
0.7488 + 0.6179i
0.7234 + 0.6223i
0.7639 + 0.6120i
0.4860 + 






0.5021i


x61
0.8017 + 0.5596i
0.8211 + 0.4860i
0.8708 + 0.4567i
0.3403 + 






0.4928i


x62
0.8168 + 0.6803i
0.8457 + 0.7260i
0.8915 + 0.7259i
0.4763 + 






0.3547i


x63
0.8882 + 0.5928i
0.9640 + 0.5518i
1.0199 + 0.5389i
0.3343 + 






0.3477i









Table 59 indicates non-uniform 16-QAM, Tables 60, 61 and 62 indicate non-uniform 64-QAM, and Tables 63 and 64 indicate non-uniform 256-QAM, and different mapping methods may be applied according to a code rate.
On the other hand, when the non-uniform constellation is designed to have the x-axis and the y-axis symmetric to each other, constellation points may be expressed similarly to those of uniform QAM and an example is illustrated as in Tables 65 to 68 presented below:























TABLE 65







y0, q
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1


y2, q
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1


y4, q
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0


y6, q
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0


y8, q
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0


Re(zq)
−x15
−x14
−x13
−x12
−x11
−x10
−x9
−x8
−x7
−x6
−x5
−x4
−x3
−x2
−x1
−1 


y0, q
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0


y2, q
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0


y4, q
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0


y6, q
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0


y8, q
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0


Re(zq)
1
 x1
 x2
 x3
 x4
 x5
 x6
 x7
 x8
 x9
 x10
 x11
 x12
 x13
 x14
x15
































TABLE 66







y1, q
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1


y3, q
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1


y5, q
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0


y7, q
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0


y9, q
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0


Im(zq)
−x15
−x14
−x13
−x12
−x11
−x10
−x9
−x8
−x7
−x6
−x5
−x4
−x3
−x2
−x1
−1 


y1, q
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0


y3, q
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0


y5, q
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0


y7, q
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0


y9, q
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0


Im(zq)
1
 x1
 x2
 x3
 x4
 x5
 x6
 x7
 x8
 x9
 x10
 x11
 x12
 x13
 x14
x15
























TABLE 67





x/Shape
R6/15
R7/15
R8/15
R9/15
R10/15
R11/15
R12/15
R13/15























x1
1.0003
1
1.0005
1
1.0772
1.16666667
2.5983
2.85714286


x2
1.0149
1.04
2.0897
2.78571429
2.8011
3.08333333
4.5193
4.85714286


x3
1.0158
1.04
2.0888
2.78571429
2.9634
3.33333333
6.1649
6.85714286


x4
2.6848
3
3.995
4.85714286
4.8127
5.16666667
8.2107
8.85714286


x5
2.6903
3.04
3.9931
4.85714286
5.1864
5.75
9.9594
11


x6
2.882
3.28
5.3843
6.85714286
6.7838
7.41666667
12.0321
13.2857143


x7
2.8747
3.32
5.3894
6.85714286
7.5029
8.5
13.9574
15.7142857


x8
4.7815
5.24
7.5206
9.14285714
9.238
10.0833333
16.2598
18.1428571


x9
4.7619
5.32
7.6013
9.28571429
10.32
11.5833333
18.4269
20.7142857


x10
5.5779
6.04
9.3371
11.5714286
12.0115
13.3333333
20.9273
23.4285714


x11
5.6434
6.28
9.8429
12.2142857
13.5356
15.25
23.4863
26.2857143


x12
7.3854
8.24
11.9255
14.6428571
15.6099
17.3333333
26.4823
29.2857143


x13
7.8797
8.84
13.3962
16.4285714
17.7524
19.75
29.7085
32.4285714


x14
9.635
11.04
15.8981
19.4285714
20.5256
22.4166667
33.6247
35.7142857


x15
11.7874
13.68
19.1591
23.2857143
24.1254
25.5833333
38.5854
39.4285714



















TABLE 68





x/Shape
9/15
11/15
13/15


















x0
1
1.275373
2.968204


x1
2.753666
3.224572
4.986168


x2
2.754654 
3.680802
6.996148


x3
4.810415 
5.509975
9.073992


x4
4.814368 
6.346779
11.17465


x5
6.797569
8.066609
13.35998


x6
6.812391 
9.353538 
15.60908


x7
9.044328 
11.04938
17.97794


x8
9.195366
12.69977
20.46238


x9
11.42332
14.55533
23.10439


x10
12.08725
16.56972
25.93383


x11
14.46334
18.82536
28.98772


x12
16.26146
21.364
32.30898


x13
19.19229 
24.26295 
36.0013


x14
22.97401
27.70588
40.26307









Tables 65 and 66 are tables for determining the real number component Re(zq) and the imaginary number component Im(zq) when modulation is performed in the non-uniform 1024-QAM method. That is, Table 65 indicates the real number part of the 1024-QAM, and Table 66 indicates the imaginary number part of the 1024-QAM. In addition, Tables 67 and 68 illustrate an example of a case in which modulation is performed in the non-uniform 1024-QAM method, and show xi values of Tables 65 and 66.
Since the non-uniform constellation method asymmetrically map the modulation symbol onto the constellation point as shown in the above tables, modulation symbols mapped onto constellation points may have different decoding performance. That is, bits constituting a modulation symbol may have different performance.
For example, referring to FIG. 15 illustrating an example of a case in which modulation is performed in the non-uniform 64-QAM method, a modulation symbol 10 may be configured as (y0, y1, y2, y3, y4, y5)=(0, 0, 1, 0, 1, 0), and performance (e.g., capacity) of bits constituting the modulation symbol 10 may have a relationship of C(y0)>C(y1)>C(y2)>C(y3)>C(y4)>C(y5).
In addition, it is obvious that the constellation in the uniform constellation method and the non-uniform constellation method may be rotated and/or scaled (herein, the same or different scaling factor may be applied to a real number axis and an imaginary number axis), and other variations can be applied. In addition, the illustrated constellation indicates relevant locations of the constellation points and another constellation can be derived by rotation, scaling and/or other appropriate conversion.
As described above, the modulator 130 may map modulation symbols onto constellation points by using uniform constellation methods and non-uniform constellation methods. In this case, bits constituting a modulation symbol may have different performance as described above.
LDPC codeword bits may have different codeword characteristics according to a 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 columns of the parity check matrix, that is, a column degree.
Accordingly, the interleaver 120 may interleave to map the LDPC codeword bits onto modulation symbols by considering both the codeword characteristic of the LDPC codeword bits and the reliability of the bits constituting a modulation symbol.
In particular, since bits constituting a modulation symbol have different performance when a non-uniform QAM is used, the block interleaver 124 configures the number of columns to be identical to the number of bits constituting a modulation symbol such that one of a plurality of groups of an LDPC codeword can be mapped onto bits each of which exists on a same location of each modulation symbol.
That is, when LDPC codeword bits of high decoding performance are mapped onto high reliability bits from among bits of each modulation symbol, a receiver side may show high decoding performance, but there is a problem that the LDPC codeword bits of the high decoding performance are not 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 reception 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 a receiver side. Accordingly, the receiver side can achieve both the high reception performance and the high decoding performance.
In this case, since the LDPC codeword is divided into groups each formed of M (=360) number of bits having the same codeword characteristic and the bits are mapped respectively onto a bit of a specific location of each modulation symbol in group units, bits having a specific codeword characteristic can be mapped onto the specific location of each modulation symbol more effectively. In addition, the number of bits constituting the group may be an aliquot part of M as described above. However, the number of codeword bits constituting the group is limited to M for convenience of explanation.
That is, the modulator 130 can map at least one bit included in a predetermined group from among the plurality of groups constituting the LDPC codeword onto a predetermined bit of each modulation symbol. Herein, each of the plurality of groups may be formed of M (=360) number of bits.
For example, in the case of 16-QAM, at least one bit included in a predetermined group from among the plurality of groups may be mapped onto a first bit of each modulation symbol, or may be mapped onto a first bit and a second bit.
The modulator 130 can map at least one bit included in a predetermined group from among the plurality of groups onto a predetermined bit of each modulation symbol for the following reasons.
As described above, the block interleaver 124 interleaves a plurality of groups of an LDPC codeword in group units, the demultiplexer (not shown) demultiplexes bits output from the block interleaver 124, and the modulator 130 maps demultiplexed bits (that is, cells) onto modulation symbols serially.
Accordingly, the group interleaver 122, which is placed before the block interleaver 124, interleaves the LDPC codeword in group units such that groups including bits to be mapped onto bits of specific locations of a modulation symbol can be written in the same column of the block interleaver 124, considering a demultiplexing operation of the demultiplexer (not shown).
Specifically, the group interleaver 122 may rearrange the order of a plurality of groups of an LDPC codeword in group units such that at least one group including bits to be mapped onto the same location of different modulation symbols are serially arranged adjacent to one another, thereby allowing the block interleaver 122 to write a predetermined group on a predetermined column.
That is, the group interleaver 122 interleaves the plurality of groups of the LDPC codeword in group units based on the above-described Tables 28 to 43, so that at least one group including bits to be mapped onto the same location of each modulation symbol are arranged to be adjacent to one another, and the block interleaver 124 interleaves by writing the adjacent at least one group on the same column.
Accordingly, the modulator 130 may generate a modulation symbol by mapping a bit output from a predetermined column of the block interleaver 124 onto a predetermined bit of the modulation symbol. In this case, bits included in one group may be mapped onto one bit of each modulation symbol or may be mapped onto two bits of each modulation symbol.
To explain detail, a case in which an LDPC codeword having a length of 16200 is modulated in the non-uniform 64-QAM method will be explained.
The group interleaver 122 divides the LDPC codeword into 16200/360(=45) groups, and interleaves the plurality of groups in group units.
In this case, the group interleaver 122 determines the number of groups to be written in each column of the block interleaver 124 based on the number of columns of the block interleaver 124, and interleaves the plurality of groups in group units based on the determined number of groups.
Herein, groups written in a same column of the block interleaver 124 may be mapped onto a single specific bit or two specific bits from among bits constituting each modulation symbol according to the number of columns of the block interleaver 124. Thus, the group interleaver 122 interleaves the plurality of groups in group units such that groups including bits required to be mapped onto a predetermined bit of each modulation symbol are adjacent to one another and serially arranged, considering bit characteristic of the modulation symbol. In this case, the group interleaver 122 may use the above-described Table 29.
Accordingly, the groups which are adjacent to one another in the LDPC codeword interleaved in group units may be written in the same column of the block interleaver 124, and the bits written in the same column may be mapped onto a single specific bit or two specific bits of each modulation symbol by the modulator 130.
For example, it is assumed that the block interleaver 124 includes as many columns as the number of bits constituting a modulation symbol, that is, six (6) columns. In this case, each column of the block interleaver 124 may be divided into a first part including 2520 rows and a second part including 180 rows, as shown in Table 44 or 48.
Accordingly, the group interleaver 122 performs group interleaving such that 2520/360(=7) groups to be written in the first part of each column of the block interleaver 124 from among the plurality of groups are serially arranged to be adjacent to one another. Accordingly, the block interleaver 124 writes the seven (7) groups on the first part of each column and divides the bits included in the other three (3) groups and writes these bits on the second part of each column.
Thereafter, the block interleaver 124 reads the bits written in each row of the first part of the plurality of columns in the row direction, and reads the bits written in each row of the second part of the plurality of columns in the row direction.
That is, the block interleaver 124 may output the bits written in each row of the plurality of columns, from the bit written in the first row of the first column to the bit written in the first row of the sixth column, sequentially like (q0, q1, q2, q3, q4, q5, q6, q7, q8, q9, q10, q11, . . . ).
In this case, when the demultiplexer (not shown) is not used or the demultiplexer (not shown) outputs serially bits input to the demultiplexer (not shown) without changing the order of the bits, the LDPC codeword bits output from the block interleaver 124, (q0, q1, q2, q3, q4, q5), (q6, q7, q8, q9, q10, q11), . . . , etc. are modulated by the modulator 130. That is, the LDPC codeword bits output from the block interleaver 124, (q0, q1, q2, q3, q4, q5), (q6, q7, q8, q9, q10, q11), . . . , etc. configure cells (y0,0, y1,0, . . . , y5,0), (y0,1, y1,1, . . . , y5,1), . . . , etc. and the modulator 130 generates a modulation symbol by mapping the cells onto constellation points.
Accordingly, the modulator 130 may map bits output from a same column of the block interleaver 124 onto a single specific bit of bits constituting each modulation symbol. For example, the modulator 130 may map bits included in a group written in the first column of the block interleaver 124, that is, (q0, q6, . . . ), onto the first bit of each modulation symbol, and also, bits written in the first column may be bits which are determined to be mapped onto the first bit of each modulation symbol according to a codeword characteristic of the LDPC codeword bits and the reliability of the bits constituting the modulation symbol.
As described above, the group interleaver 122 may interleave a plurality of groups of an LDPC codeword in group units such that the groups including bits to be mapped onto a single bit of a specific location of each modulation symbol are written in a specific column of the block interleaver 124.
On the other hand, it is assumed that the block interleaver 124 includes as many columns as half of the number of bits constituting a modulation symbol, that is, three (3) columns. In this case, each column of the block interleaver 124 is not divided into parts as shown in Table 47 and 5400 bits are written in each column.
Accordingly, the group interleaver 122 performs group interleaving such that 5400/360(=15) groups to be written in each column of the block interleaver 124 from among the plurality of groups are serially arranged to be adjacent to one another. Accordingly, the block interleaver 124 writes the 15 groups on each column.
Thereafter, the block interleaver 124 may read bits written in each row of the plurality of columns in the row direction.
That is, the block interleaver 124 may output the bits written in each row of the plurality of columns, from the bit written in the first row of the first column to the bit written in the first row of the third column, sequentially like (q0, q1, q2, q3, q4, q5, q6, q7, q8, q9, q10, q11, . . . ).
In this case, the demultiplexer (not shown) demultiplexes the LDPC codeword bits output from the block interleaver 124 based on Table 50, and output cells likes (y0,0, y1,0, . . . , y5,0)=(q0, q2, q4, q1, q3, q5), (y0,1, y1,1, . . . , y5,1)=(q6, q8, q10, q7, q9, q11) . . . , etc. and the modulator 130 generates a modulation symbol by mapping the cells onto constellation points.
Accordingly, the modulator 130 may map bits output from the same column of the block interleaver 124 onto two specific bits of each modulation symbol. For example, the modulator 130 may map (q0, q6, . . . ) from among the bits (q0, q3, q6, q9, . . . ) included in the group written in the first column in the block interleaver 124 onto the first bit of each modulation symbol, and may map (q3, q9, . . . ) on the fifth bit of each modulation symbol. The bits written in the first column are bits which are determined to be mapped onto the first bit and the fifth bit of each modulation symbol according to the codeword characteristic of the LDPC codeword bits and the reliability of the bits constituting the modulation symbol. Herein, the first bit of the modulation symbol is a bit for determining a sign of the real number component Re(zq) of a constellation point onto which the modulation symbol is mapped, and the fifth bit of the modulation symbol is a bit for determining a relatively small size of the constellation point onto which the modulation symbol is mapped.
As described above, the group interleaver 122 may interleave the plurality of groups of the LDPC codeword in group units such that groups including bits to be mapped onto two bits of specific locations of a modulation symbol are written in a specific column of the block interleaver 124.
Hereinafter, it is assumed that the encoder 110 performs LDPC encoding at a code rate of 10/15, 11/15, 12/15, and 13/15 and generates an LDPC codeword (Nldpc=16200) formed of 16200 bits, and the modulator 130 uses the non-uniform 16-QAM modulation method corresponding to the code rate based on Table 59.
Hereinafter, exemplary embodiments will be explained in detail.
According to an exemplary embodiment, it is assumed that the encoder 110 performs LDPC encoding at a code rate of 10/15, 11/15, 12/15 and 13/15 and generates an LDPC codeword formed of 16200 bits (Nldpc=16200), and the modulator 130 uses the non-uniform 16-QAM modulation method corresponding to the code rate based on Table 59.
In this case, the group interleaver 122 may perform group interleaving by using Equation 11 and Table 28. The block interleaver 124 in which the number of columns is four (4), the number of rows of the first part is 3960(=360×11), and the number of rows of the second part is 180 according to Table 44 or 48 may be used. Accordingly, 11 groups (X35, X31, X39, X19, X29, X20, X36, X0, X9, X13, X5) constituting an LDPC codeword are input to the first part of the first column of the block interleaver 124, 11 groups (X37, X17, X43, X21, X41, X25, X1, X33, X24, X12, X30) are input to the first part of the second column of the block interleaver 124, 11 groups (X16, X32, X10, X28, X4, X26, X8, X40, X42, X3, X6) are input to the first part of the third column of the block interleaver 124, and 11 groups (X2, X38, X14, X34, X22, X18, X27, X23, X7, X11, X15) are input to the first part of the fourth column of the block interleaver 124.
In addition, a group X44 is input to the second part of the block interleaver 124. Specifically, bits constituting the group X44 are input to the rows of the first column of the second part serially, input to the rows of the second column serially, input to the rows of the third column serially, and finally input to the rows of the fourth column serially. In this case, the group X44 is formed of 360 bits and 90 bits are input to the second part of each column.
In addition, the block interleaver 124 may output the bits input to the first row to the last row of each column serially, and the bits output from the block interleaver 124 may be input to the modulator 130 serially. In this case, the demultiplexer (not shown) may be omitted or the demultiplexer (not shown) may output the input bits serially without changing the order of the bits.
Accordingly, one bit included in each of groups X35, X37, X16 and X2 constitute a single modulation symbol.
According to an exemplary embodiment, one bit included in each of the groups X35, X37, X16 and X2 constitute a single modulation symbol based on group interleaving and block interleaving. In addition to the above-described method, other methods for constituting a single modulation symbol with one bit included in each of the groups X35, X37, X16 and X2 may be included in the inventive concept.
Meanwhile, in case of Equation 11, Π(j) value defined by Tables 28 to 35 is applied as an index of an input group, and in case of Equation 12, Π(j) value defined by Tables 36 to 43 is applied as an index of an output group. Thus, they have an inverser relation.
The transmitting apparatus 100 may modulate a signal mapped onto a constellation and may transmit the signal to a receiving apparatus (for example, a receiving apparatus 2700 of FIG. 24). For example, the transmitting apparatus 100 may map a signal mapped onto a constellation onto an Orthogonal Frequency Division Multiplexing (OFDM) frame by using the OFDM method, and may transmit the signal to the receiving apparatus 2700 via an allocated channel.
To achieve this, the transmitting apparatus 100 may further include a frame mapper (not shown) to map the signal mapped onto the constellation onto the OFDM frame, and a transmitter (not shown) to transmit the signal of the OFDM frame format to the receiving apparatus 2700.
Case in which a Block-Row Interleaver is Used
According to another exemplary embodiment, the interleaver 120 may interleave an LDPC codeword in other methods, different from the methods described in the previous exemplary embodiment in which a block interleaver is used, and may map bits included in a predetermined group from among a plurality of groups constituting the interleaved LDPC codeword onto a predetermined bit of a modulation symbol. This will be explained in detail with reference to FIG. 20.
Referring to FIG. 20, the interleaver 120 includes a parity interleaver 121, a group interleaver (or a group-wise interleaver 122), a group twist interleaver 123 and a block-row interleaver 125. Herein, the parity interleaver 121 and the group twist interleaver 123 perform the same functions as in the exemplary embodiment 1 described above, and thus, a detailed description of these elements is omitted.
The group interleaver 122 may divide a parity-interleaved LDPC codeword into a plurality of groups, and may rearrange the order of the plurality of groups.
In this case, the operation of dividing the parity-interleaved LDPC codeword into the plurality of groups is the same as in the exemplary embodiment 1, and thus, a detailed description thereof is omitted.
The group interleaver 122 interleaves an LDPC codeword in group units. That is, the group interleaver 122 may rearrange the order of the plurality of groups in the LDPC codeword in group units by changing locations of the plurality of groups constituting the LDPC codeword.
In this case, the group interleaver 122 may rearrange an order of a plurality of groups in group units so that groups including bits mapped onto the same modulation symbol from among a plurality of groups are placed sequentially.
In this case, the group interleaver 122 may rearrange the order of the plurality of groups in group units so that the groups including bits mapped onto the same modulation symbol from among a plurality of groups are placed sequentially, by considering the number of columns and rows constituting the block-row interleaver 124, the number of groups constituting the LDPC codeword, and the number of bits included in each group.
For doing this, the group interleaver 122 may interleave the LDPC codeword in group units by using Equation 13

Yj=Xπ(j)(0≦j<Ngroup)  (13),

where Xj is the jth group before group interleaving, and Yj is the jth 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 code rate and a modulation method.

Accordingly, Xħ(j) is a Π(j)th group before group interleaving, and Equation 13 means that the pre-interleaving Π(j)th group is interleaved into the jth group.
According to an exemplary embodiment, an example of Π(j) may be defined as in Tables 69 to 76 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 group units 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 10/15 to generate an LDPC codeword of a length of 16200, 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 10/15 in tables 69 to 76 presented below.
For example, when the length of the LDPC codeword is 16200, the code rate is 10/15, and the modulation method is 16-QAM, the group interleaver 122 may perform interleaving by using Π(j) defined as in table 69.
An example of Π(j) is as follows:
For example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 69 presented below:








TABLE 69






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



10/15, 11/15,
7
42
37
18
17
20
40
36
33
9
19
0
31
35
16
6
26
43
27
11
10
22
39


12/15, 13/15
34
32
12
25
2
41
38
4
29
28
3
21
15
8
5
1
30
24
14
23
13
44









In the case of Table 69, Equation 13 may be expressed as Y0=XΠ(0)=X7, Y1=XΠ(1)=X42, Y2=XΠ(2)=X37, . . . , Y43=XΠ(43)=X13, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 7th group to the 0th group, the 42th group to the 1st group, the 37th group to the 2nd group, . . . , the 13th group to the 43rd group, and the 44th group to the 44th group.

In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 64-QAM, Π(j) may be defined as in Table 70 presented below:








TABLE 70






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



6/15, 7/15,
32
39
8
38
25
3
4
5
6
30
28
20
23
22
37
26
12
19
27
33
34
14
1


8/15, 9/15
36
35
40
0
17
29
11
24
18
15
10
9
7
16
13
21
31
41
2
42
43
44









In the case of Table 70, Equation 13 may be expressed as Y0=XΠ(0)=X32, Y=XΠ(1)=X39, Y2=XΠ(2)=X8, . . . , Y43=XΠ(43)=X43, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 32nd group to the 0th group, the 39th group to the 1st group, the 8th group to the 12th group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.

In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 71 presented below:








TABLE 71






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



10/15, 11/15,
28
22
7
33
34
17
30
14
6
37
12
13
9
29
2
20
15
35
27
11
39
25
4


12/15, 13/15
3
8
21
1
10
38
5
19
31
0
26
32
18
24
16
23
36
40
41
42
43
44









In the case of Table 71, Equation 13 may be expressed as Y0=XΠ(0)=X28, Y1=XΠ(1)=X22, Y2=XΠ(2)=X7, . . . , Y43=XΠ(43)=X43, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 28th group to the 0th group, the 22nd group to the 1st group, the 7th group to the 2nd group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.

In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 72 presented below:








TABLE 72






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



6/15, 7/15,
16
24
32
22
27
12
5
2
19
31
13
33
30
18
11
15
6
26
3
34
1
4
0


8/15, 9/15
8
37
10
36
14
21
28
25
29
17
9
35
20
38
7
23
39
40
41
42
43
44









In the case of Table 72, Equation 1 may be expressed as Y0=XΠ(0)=X16, Y1=XΠ(1)=X2, Y2=XΠ(2)=X32, . . . , Y43=XΠ(43)=X43, and Y44=XΠ(44)=X44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 16th group to the 0th group, the 24th group to the 1st group, the 32nd group to the 2nd group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.

In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 73 presented below:








TABLE 73






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
48
152
156
93
113
74
172
31
4
138
155
66
39
10
81
34
15
35
110


7/15,
122
69
68
158
166
97
62
80
123
140
102
171
128
108
43
58
79
14
67


8/15,
32
117
76
86
42
141
131
30
119
125
1
153
40
51
82
151
37
2
174


9/15
73
103
21
134
18
24
13
61
173
159
17
55
148
130
85
0
165
83
45



33
129
136
88
59
23
127
126
50
53
161
135
71
49
111
150
84
100
75



116
120
91
63
3
28
22
29
12
175
5
89
168
105
107
9
163
167
169



11
41
6
16
47
20
114
99
52
106
145
170
60
46
65
77
144
124
7



143
162
87
98
149
142
64
27
121
146
94
101
26
133
147
132
38
137
139



19
36
154
78
92
57
157
54
112
25
44
8
115
95
70
72
109
56
96



118
164
90
104
160
176
177
178
179









In the case of Table 73, Equation 13 may be expressed as Y0=XΠ(0)=X48, Y1=XΠ(1)=X152, Y2=XΠ(2)=X156, . . . , Y178=XΠ(178)=X178, and Y179=XΠ(179)=X179. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 48th group to the 0th group, the 152nd group to the 1st group, the 156th group to the 2nd group, . . . , the 178th group to the 178th group, and the 179th group to the 179th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 74 presented below:








TABLE 74






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
121
65
152
0
39
19
149
14
139
32
99
178
61
20
73
176
52
4
83


7/15,
173
45
110
89
136
44
23
177
161
66
76
72
16
55
11
77
168
50
158


8/15,
17
104
153
53
6
138
40
1
128
18
68
150
85
100
58
86
137
163
38


9/15
84
164
154
43
22
78
129
171
30
81
147
42
36
169
141
69
92
82
124



166
29
80
170
64
28
75
10
148
25
155
98
46
167
95
133
47
127
140



151
130
35
93
156
117
62
119
159
179
31
132
165
57
24
87
120
63
59



115
7
67
142
172
122
126
118
91
114
144
26
113
106
112
21
160
145
48



33
103
108
135
88
123
111
175
27
13
102
60
134
107
131
71
174
101
3



49
15
74
5
54
37
97
116
51
90
105
9
41
56
125
79
157
94
109



12
162
8
143
2
70
34
146
96









In the case of Table 74, Equation 13 may be expressed as Y0=XΠ(0)=X121, Y1=XΠ(1)=X65, Y2=XΠ(2)=X152, . . . , Y178=XΠ(178)=X146, and Y179=XΠ(179)=X96. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 121st group to the 0th group, the 65th group to the 1st group, the 152nd group to the 2nd group, . . . , the 146th group to the 178th group, and the 96th group to the 179th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 64-QAM, Π(j) may be defined as in Table 75 presented below:








TABLE 75






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
49
69
61
130
5
159
105
59
23
8
150
157
104
118
57
146
10
161
47


11/15,
70
93
138
168
156
116
45
66
39
174
127
112
24
97
96
155
178
84
91


12/15,
107
132
22
19
113
87
115
94
6
125
75
88
14
77
148
166
99
44
110


13/15
72
162
120
52
63
50
29
179
126
4
32
36
90
11
173
76
101
43
28



152
158
102
73
68
140
21
170
7
86
67
3
171
172
35
81
80
142
145



122
48
56
103
17
16
121
109
37
78
137
129
124
106
79
64
46
0
20



42
139
60
135
151
126
86
98
100
95
144
131
51
38
82
65
9
160
31



13
71
119
149
12
143
54
74
154
177
175
41
114
108
2
153
169
33
89



25
133
165
163
134
40
53
136
18
1
30
85
117
26
147
167
62
55
34



128
141
164
92
27
111
58
15
176









In the case of Table 75, Equation 13 may be expressed as Y0=XΠ(0)=X49, Y1=XΠ(1)=X69, Y2=XΠ(2)=X61, . . . , Y178=XΠ(178)=X15, and Y179=XΠ(179)=X176. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 49th group to the 0th group, the 69th group to the 1st group, the 61st group to the 2nd group, . . . , the 15th group to the 178th group, and the 176th group to the 179th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 76 presented below:








TABLE 76






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
45
46
71
84
87
152
116
28
160
163
37
122
3
86
97
8
106
105
119


11/15,
179
31
56
79
82
133
90
146
115
140
118
51
50
73
70
103
102
127
25


12/15,
174
161
47
23
63
78
143
10
120
111
168
173
53
42
147
6
95
104
15


13/15
117
170
17
43
44
77
68
99
139
114
107
18
159
55
52
65
149
13
98



150
151
20
175
59
40
26
74
89
100
112
126
162
138
49
58
81
72
11



12
153
131
178
156
24
30
67
62
128
132
29
113
155
169
39
36
83
4



91
14
123
109
22
171
0
32
61
66
93
136
134
135
176
167
2
38
145



64
7
94
27
148
16
144
33
11
69
76
9
88
130
124
172
177
41
34



75
154
137
142
108
129
164
21
57
48
5
60
101
96
110
158
157
19
35



54
85
80
141
92
125
121
166
165









In the case of Table 76, Equation 13 may be expressed as Y0=XΠ(0)=X45, Y1=XΠ(1)=X46, Y2=XΠ(2)=X71, . . . , Y178=XΠ(178)=X166, and Y179=XΠ(179)=X165. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 45th group to the 0th group, the 46th group to the 1st group, the 71st group to the 2nd group, . . . , the 166th group to the 178th group, and the 165th group to the 179th group.
As described above, the group interleaver 122 may rearrange the order of the plurality of groups in group units by using Equation 13 and Tables 69 to 76.
Meanwhile, the group interleaver 122 may interleave the LDPC codeword in group units by using Equation 14 presented below:

Yπ(j)=Xj (0≦j<Ngroup  (14)

where Xj is the jth group before group interleaving, and Yj is the jth 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 code rate and a modulation method.

Accordingly, YΠ(j) is a Π(j)th group before group interleaving, and Equation 14 means that the pre-interleaving jth group is interleaved into the Π(j)th group.
According to an exemplary embodiment, an example of Π(j) may be defined as in Tables 77 to 84 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 group units 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 10/15 to generate an LDPC codeword of a length of 16200, 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 10/15 in Tables 77 to 84 presented below.
For example, when the length of the LDPC codeword is 16200, the code rate is 10/15, and the modulation method is 16-Quadrature Amplitude Modulation (QAM), the group interleaver 122 may perform interleaving by using Π(j) defined as in table 77.
An example of Π(j) is as follows:
For example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 77 presented below:








TABLE 77






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



10/15, 11/15,
11
38
27
33
30
37
15
0
36
9
20
19
25
43
41
35
14
4
3
10
5
34
21


12/15, 13/15
42
40
26
16
18
32
31
39
12
24
8
23
13
7
2
29
22
6
28
1
17
44









In the case of Table 77, Equation 14 may be expressed as X0=YΠ(0)=Y11, X1=YΠ(1)=Y38, X2=YΠ(2)=Y27, . . . , X43=YΠ(43)=Y13, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 11th group, the 1st group to the 38th group, the 2nd group to the 27th group, . . . , the 43th group to the 17th group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 64-QAM, Π(j) may be defined as in Table 78 presented below:








TABLE 78






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



6/15, 7/15,
26
22
41
5
6
7
8
35
2
34
33
29
16
37
21
32
36
27
31
17
11
38
13


8/15, 9/15
12
30
4
15
18
10
28
9
39
0
19
20
24
23
14
3
1
25
40
42
43
44









In the case of Table 78, Equation 14 may be expressed as X0=YΠ(0)=Y26, X1=YΠ(1)=Y22, X2=YΠ(2)=Y41, . . . , X43=YΠ(43)=Y43, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 26th group, the 1st group to the 22nd group, the 2nd group to the 41th group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 79 presented below:








TABLE 79






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



10/15, 11/15,
32
26
14
23
22
29
8
2
24
12
27
19
10
11
7
16
37
5
35
30
15
25
1


12/15, 13/15
38
36
21
33
18
0
13
6
31
34
3
4
17
39
9
28
20
40
41
42
43
44









In the case of Table 79, Equation 14 may be expressed as X0=YΠ(0)=Y32, X1=YΠ(1)=Y26, X2=YΠ(2)=Y14, . . . , X43=YΠ(43)=Y43, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 32nd group, the 1st group to the 26th group, the 2nd group to the 14th group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 16200, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 80 presented below:








TABLE 80






Order of bits group to be block interleaved



π(j) (0 ≦ j < 45)






































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22


Rate
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44



6/15, 7/15,
22
20
7
18
21
6
16
37
23
33
25
14
5
10
27
15
0
32
13
8
35
28
3


8/15, 9/15
38
1
30
17
4
29
31
12
9
2
11
19
34
26
24
36
39
40
41
42
43
44









In the case of Table 80, Equation 14 may be expressed as X0=YΠ(0)=Y22, X1=YΠ(1)=Y20, X2=YΠ(2)=Y7, . . . , X43=YΠ(43)=Y43, and X44=YΠ(44)=Y44. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 22nd group, the 1st group to the 20th group, the 2nd group to the 7th group, . . . , the 43rd group to the 43rd group, and the 44th group to the 44th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 256-QAM, Π(j) may be defined as in Table 81 presented below:








TABLE 81






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
72
48
55
99
8
105
116
132
163
110
13
114
103
63
36
16
117
67
61


7/15,
152
119
59
101
81
62
161
145
140
100
102
45
7
38
76
15
17
153
54


8/15,
149
12
50
115
42
33
162
75
127
118
0
89
84
51
122
85
159
68
169


9/15
157
34
80
126
64
25
98
139
128
11
37
21
20
166
88
167
57
5
94



40
129
155
35
26
14
52
74
92
71
41
135
79
106
173
97
156
3
143



165
170
24
136
121
93
144
29
58
174
108
123
109
32
168
18
90
160
4



120
164
95
39
171
46
96
141
19
27
131
47
83
82
31
77
70
44
148



146
60
87
78
150
9
151
28
43
138
133
130
124
142
147
69
137
91
53



1
49
154
10
2
158
22
66
175
86
134
111
172
73
23
112
107
113
125



30
6
65
56
104
176
177
178
179









In the case of Table 81, Equation 14 may be expressed as X0=YΠ(0)=Y72, X1=YΠ(1)=Y48, X2=YΠ(2)=Y55, . . . , X178=YΠ(178)=Y178, and X179=YΠ(179)=Y179. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 72nd group, the 1st group to the 48th group, the 2nd group to the 55th group, . . . , the 178th group to the 178th group, and the 179th group to the 179th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 6/15, 7/15, 8/15 and 9/15, and the modulation method is 16-QAM, Π(j) may be defined as in Table 82 presented below:








TABLE 82






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












6/15,
3
45
175
151
17
155
42
115
173
163
83
33
171
142
7
153
31
38
47


7/15,
5
13
129
61
25
109
85
125
141
81
77
65
105
9
133
177
97
69
157


8/15,
56
4
44
164
68
60
24
20
88
92
132
152
36
160
16
41
156
32
165


9/15
108
52
113
144
12
101
112
80
1
28
116
48
72
176
148
30
14
154
82



29
34
62
167
78
66
74
18
57
50
53
110
137
22
161
122
73
98
169



90
179
158
87
10
51
150
143
134
39
162
127
146
135
170
21
139
128
126



123
114
159
100
121
102
111
0
119
138
75
166
120
93
46
63
96
147
106



91
145
136
23
54
43
8
94
71
117
174
124
131
178
67
84
6
49
95



2
40
59
86
99
168
37
103
130
27
172
55
58
107
76
89
35
70
79



64
118
19
149
140
15
26
11
104









In the case of Table 82, Equation 14 may be expressed as X0=YΠ(0)=Y3, X1=YΠ(1)=Y45, X2=YΠ(2)=Y175, . . . , X178=YΠ(178)=Y11, and X179=YΠ(179)=Y104. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 3rd group, the 1st group to the 45th group, the 2nd group to the 175th group, . . . , the 179th group to the 11th group, and the 179th group to the 104th group.

In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 64-QAM, Π(j) may be defined as in Table 83 presented below:








TABLE 83






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
112
161
147
87
66
4
46
84
9
130
16
70
137
133
50
178
100
99
160


11/15,
41
113
82
40
8
31
152
165
175
75
63
162
132
67
150
170
90
68
103


12/15,
127
27
157
144
114
74
55
25
1
19
134
57
79
140
48
72
51
104
109


13/15
92
91
128
85
36
163
120
43
49
151
69
37
174
20
45
123
33
32
121



54
122
73
78
98
12
6
108
38
146
102
56
176
30
42
145
44
24
164



13
135
59
101
95
119
107
47
65
29
171
106
3
125
39
153
156
117
159



105
44
24
164
13
135
59
101
95
119
107
47
65
29
171
106
3
125
39



153
156
117
159
105
21
115
81
172
93
138
124
94
15
16
52
136
10
118



76
148
141
34
23
11
77
55
131
17
58
155
173
154
53
167
22
149
83



88
89
71
28
143
179
142
35
64









In the case of Table 83, Equation 14 may be expressed as X0=YΠ(0)=Y112, X1=YΠ(1)=Y161, X2=YΠ(2)=Y147, . . . , , X178=YΠ(178)=Y35, and X179=YΠ(179)=Y64. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 112nd group, the 1st group to the 161st group, the 2nd group to the 147th group, . . . , the 178th group to the 35th group, and the 179th group to the 64th group.
In another example, when the length Nldpc of the LDPC codeword is 64800, the code rate is 10/15, 11/15, 12/15 and 13/15, and the modulation method is 1024-QAM, Π(j) may be defined as in Table 84 presented below:








TABLE 84






Order of bits group to be block interleaved



π(j) (0 ≦ j < 180)


































Code
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Rate
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37



38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56



57
58
59
60
61
62
63
64
65
66
67
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












10/15,
120
141
130
12
113
162
53
134
15
144
45
94
95
74
115
56
138
59
68


11/15,
169
78
159
118
41
100
37
82
136
7
106
101
20
121
140
151
170
111
10


12/15,
131
110
81
150
51
61
61
0
1
40
161
90
31
30
71
50
171
70
21


13/15
160
91
80
163
122
103
42
133
72
123
102
63
142
33
2
93
32
83
152



143
62
43
22
173
92
23
112
3
172
13
4
145
84
25
114
175
124
135



54
165
14
75
64
85
164
35
34
55
17
16
67
156
117
166
47
86
107



66
27
6
57
29
18
46
177
11
116
147
176
87
36
104
157
146
97
105



24
126
127
125
154
89
65
28
174
155
44
139
132
26
52
137
73
76
77



5
96
153
108
99
168
167
69
8
39
88
9
158
179
178
129
48
109
58



119
148
49
38
79
128
149
98
19









In the case of Table 84, Equation 14 may be expressed as X0=YΠ(0)=Y120, X1=YΠ(1)=Y141, X2=YΠ(2)=Y130, . . . , X178=YΠ(178)=Y98, and X179=YΠ(179)=Y19. Accordingly, the group interleaver 122 may rearrange the order of the plurality of groups in group units by changing the 0th group to the 120th group, the 1st group to the 141st group, the 2nd group to the 130th group, . . . , the 178th group to the 98th group, and the 179th group to the 19th group.
As described above, the group interleaver 122 may rearrange the order of the plurality of groups in group units by using Equation 14 and Tables 77 to 84.
On the other hand, since the order of the groups constituting the LDPC codeword is rearranged in group units by the group interleaver 122, and then, the 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 69 to 84 in relation to Π(j).
When the group interleaving is performed based on tables 69 to 84 as described above, the order of the groups constituting the group-interleaved LDPC codeword is different from that of the groups constituting the LDPC code group-interleaved based on tables 28 to 43.
This is because the block-row interleaver 125 is used in the present exemplary embodiment instead of the block interleaver 124 in FIG. 4. That is, since the interleaving method used in the block interleaver 124 and the interleaving method used in the block-row interleaver 125 are different from each other, the group interleaver 122 of the present exemplary embodiment rearranges the order of the plurality of groups constituting the LDPC codeword based on tables 69 to 84.
Specifically, the group interleaver 122 may rearrange the order of the plurality of groups in such that that an arrangement unit, in which at least one group including bits to be mapped onto the same modulation symbol is serially arranged in group units, is repeated.
That is, the group interleaver 122 may serially arrange one of a plurality of first groups including bits to be mapped onto a first specific location of each modulation symbol, one of a plurality of second groups including bits to be mapped onto a second specific location of each modulation symbol, . . . , one of a plurality of nth groups including bits to be mapped onto an nth specific location of each modulation symbol, and may arrange the other groups repeatedly in the same method.
The block-row interleaver 125 interleaves the plurality of groups the order of which has been rearranged. In this case, the block-row interleaver 125 may interleave the plurality of groups the order of which has been rearranged by using at least one row including a plurality of columns. This will be explained in detail below with reference to FIGS. 21 to 23.
FIGS. 21 to 23 are views to illustrate a configuration of a block-row interleaver and an interleaving method according to an exemplary embodiment.
First, when Ngroup/m is an integer, the block-row interleaver 125 includes an interleaver 125-1 including m number of rows each including M number of columns as shown in FIG. 21, and the block-row interleaver 125 may interleave by using Ngroup/m number of interleavers 125-1 having the configuration of FIG. 21.
Herein, Ngroup is the total number of groups constituting an LDPC codeword. In addition, M is the number of bits included in a single group and may be 360, for example. m may be identical to the number of bits constituting a modulation symbol or may be 1/2 of the number of bits constituting a modulation symbol. For example, when a non-uniform QAM is used, performance of the bits constituting a modulation symbol is different, and thus, by setting m to be identical to the number of bits constituting a modulation symbol, a single group can be mapped onto a single bit of the modulation symbol.
Specifically, the block-row interleaver 125 may interleave by writing each of a plurality of groups constituting an LDPC codeword in each row in the row direction in group units, and reading each column of the plurality of rows in which the plurality of groups are written in group units in the column direction.
For example, as shown in FIG. 21, the block-row interleaver 125 writes m number of continuous groups from among the plurality of groups in each of the m number of rows of the interleaver 125-1 in the row direction, and reads each column of m number of rows in which bits are written in the column direction. In this case, as many interleavers 125-1 as the number of groups divided by the number of rows, that is, Ngroup/m, may be used.
As described above, when the number of groups constituting an LDPC codeword is an integer multiple of the number of rows, the block-row interleaver 125 may interleave by writing as many groups as the number of rows from among a plurality of groups constituting the LDPC codeword serially.
On the other hand, when the number of groups constituting an LDPC codeword is not an integer multiple of the number of rows, the block-row interleaver 125 interleaves by using N number of interleavers (N is an integer greater than or equal to 2) including different number of columns.
For example, as shown in FIGS. 22 and 23, the block-row interleaver 125 may interleave by using a first interleaver 125-2 including m number of rows each including M number of columns, and a second interleaver 125-3 including m number of rows each including a×M/m number of columns. Herein, a is Ngroup−└Ngroup/m┘×m, and └Ngroup/m┘ is the largest integer below Ngroup/m.
In this case, the first interleaver 125-2 may be used as many as └Ngroup/m┘ and one second interleaver 125-3 may be used.
Specifically, the block-row interleaver 125 may interleave a plurality of groups constituting an LDPC codeword by writing each of └Ngroup/m┘×m number of groups from among the plurality of groups constituting the LDPC codeword in each row in the row direction in group units, and reading each column of the plurality of rows in which └Ngroup/m┘×m number of groups are written in group units in the column direction.
For example, as shown in FIGS. 22 and 23, the block-row interleaver 125 may write the same m number of continuous groups as the number of rows from among └Ngroup/m┘×m number of groups in each row of the first interleaver 125-2 in the row direction, and may read each column of the plurality of rows of the first interleaver 125-2 in which m number of groups are written in the column direction. In this case, the first interleaver 125-2 having the configuration FIGS. 22 and 23 may be used as many as └Ngroup/m┘.
In addition, in the case of a system using a plurality of antennas, m may be a product of the number of bits constituting a modulation method and the number of antennas
Thereafter, the block-row interleaver 125 may divide bits included in the other groups except the groups written in the first interleaver 125-2, and may write these bits in each row of the second interleaver 125-3 in the row direction. In this case, the same number of bits are written in each row of the second interleaver 125-3. That is, a single bit group may be input to the plurality of rows of the second interleaver 125-3.
For example, as shown in FIG. 22, the block-row interleaver 125 may write a×M/m number of bits from among the bits included in the other groups except the groups written in the first interleaver 125-2 in each of m number of rows of the second interleaver 125-3 in the row direction, and may read each column of m number of rows of the second interleaver 125-3 in which the bits are written in the column direction. In this case, one second interleaver 125-3 having the configuration of FIG. 22 may be used.
However, according to another exemplary embodiment, as shown in FIG. 23, the block-row interleaver 125 may write the bits in the first interleaver 125-2 in the same method as explained in FIG. 22, but may write the bits in the second interleaver 125-3 in a method different from that of FIG. 22.
That is, the block-row interleaver 125 may write the bits in the second interleaver 125-3 in the column direction.
For example, as shown in FIG. 23, the block-row interleaver 125 may write the bits included in the other groups except the groups written in the first interleaver 125-2 in each column of m number of rows each including a×M/m number of columns of the second interleaver 125-3 in the column direction, and may read each column of m number of rows of the second interleaver 125-3 in which the bits are written in the column direction. In this case, one second interleaver 125-3 having the configuration of FIG. 23 may be used.
In the method shown in FIG. 23, the block-row interleaver 125 interleaves by reading in the column direction after writing the bits in the second interleaver in the column direction. Accordingly, the bits included in the groups interleaved by the second interleaver are read in the order they were written and output to the modulator 130. Accordingly, the bits included in the groups belonging to the second interleaver are not rearranged by the block-row interleaver 125 and may be mapped onto the modulation symbols serially. As such, the block-row interleaver 125 may interleave at least a part of a plurality of groups, and may not interleave the other groups. Specifically, the block-row interleaver 125 may perform interleaving by sequentially writing LDPC codewords constituting at least a part of a plurality of groups in a plurality of rows and reading the LDPC codewordsin a column direction, but may not perform interleaving with respect to the other groups. That is, since the block-row interleaver 125 writes and reads bits included in the other groups in the same direction, the bits included in the other groups may be outputted without changing order thereof.
In addition, in the aforementioned exemplary embodiment, the bits included in the other groups are written and read in a column direction, but this is merely an example. That is, the block-row interleaver 125 may write and read the bits included in the other groups in a row direction, and in this case, the bits included in the other groups may be outputted without changing order thereof
As described above, the block-row interleaver 125 may interleave the plurality of groups of the LDPC codeword by using the methods described above with reference to FIGS. 21 to 23.
According to the above-described method, the output of the block-row interleaver 125 may be the same as the output of the block interleaver 124. Specifically, when the block-row interleaver 125 interleaves as shown in FIG. 21, the block-row interleaver 125 may output the same value as that of the block interleaver 124 which interleaves as shown in FIG. 8. In addition, when the block-row interleaver 125 interleaves as shown in FIG. 22, the block-row interleaver 125 may output the same value as that of the block interleaver 124 which interleaves as shown in FIG. 9. In addition, when the block-row interleaver 125 interleaves as shown in FIG. 23, the block-row interleaver 125 may output the same value as that of the block interleaver 124 which interleaves as shown in FIG. 10.
Specifically, when the group interleaver 122 is used based on Equation 11 and the block interleaver 124 is used, and the output groups of the group interleaver 122 are Yi(0≦j<Ngroup) and when the group interleaver 122 is used based on Equation 13 and the block-row interleaver 125 is used, and the output groups of the group interleaver 122 are Zi(0≦j<Ngroup) a relationship between the output groups Zi and Yi after group interleaving may be expressed as in Equations 15 and 16, and as a result, the same value may be output from the block interleaver 124:

Zi+m×j=Yα×i+j(0≦i<m,0≦j<α)  (15)

Zi=Yi(α×m≦i<Ngroup)  (16),

where α is └Ngroup/m┘ and is the number of groups written in a single column of the first part when the block interleaver 124 is used, and └Ngroup/m┘ is the largest integer below Ngroup/m. Here, m is identical to the number of bits constituting a modulation symbol or half of the bits constituting a modulation symbol. In addition, m is the number of columns of the block interleaver 124 and m is the number of rows of the block-row interleaver 125.

Accordingly, the modulator 130 may map the bits output from the block-row interleaver 125 onto a modulation symbol in the same method as when the block interleaver 124 is used.
The bit interleaving method suggested in the exemplary embodiments is performed by the parity interleaver 121, the group interleaver 122, the group twist interleaver 123, and the block interleaver 124 as shown in FIG. 4 (the parity interleaver 121 or group twist interleaver 123 may be omitted according to circumstances). However, this is merely an example and the bit interleaving method is not limited to three modules or four modules described above.
For example, when the block interleaver is used and the group interleaving method expressed as in Equation 11 is used, regarding the bit groups Xj(0≦j<Ngroup) defined as in Equation 9 and Equation 10, bits belonging to m number of bit groups, for example, {XΠ(i), XΠ(α+i), . . . , XΠ((m−1)×α+i)} (0≦i<α), may constitute a single modulation symbol.
Herein, α is the number of bit groups constituting the first part of the block interleaver, and α=└Ngroup/m┘. In addition, m is the number of columns of the block interleaver and may be equal to the number of bits constituting the modulation symbol or half of the number of bits constituting the modulation symbol.
Therefore, for example, regarding parity-interleaved bits ui, {uΠ(i)+j, uΠ(α+i)+j, . . . , uΠ((m−1)×α+i)+j} (0<i≦m, 0<j≦M) may constitute a single modulation symbol. As described above, there are various methods for constituting a single modulation symbol.
In addition, the bit interleaving method suggested in the exemplary embodiments is performed by the parity interleaver 121, the group interleaver 122, the group twist interleaver 123, and the block-row interleaver 125 as shown in FIG. 20 (the group twist interleaver 123 may be omitted according to circumstances). However, this is merely an example and the bit interleaving method is not limited to three modules or four modules described above.
For example, when the block-row interleaver is used and the group interleaving method expressed as in Equation 13 is used, regarding the bit groups Xj(0≦Ngroup) defined as in Equation 9 and Equation 10, bits belonging to m number of bit groups, for example, {XΠ(m×i), XΠ(m×i+1), . . . , XΠ(m×i+(m−1))} (0≦i<α), may constitute a single modulation symbol.
Herein, α is the number of bit groups constituting the first part of the block interleaver, and α=└Ngroup/m┘. In addition, m is the number of columns of the block interleaver and may be equal to the number of bits constituting the modulation symbol or half of the number of bits constituting the modulation symbol.
Therefore, for example, regarding parity-interleaved bits ui, {uΠ(m×i)+j, uΠ(m×i+1)+j, . . . , uΠ(m×i+(m−1))+j} (0<i≦m, 0≦j≦M) may constitute a single modulation symbol. As described above, there are various methods for constituting a single modulation symbol.
The transmitting apparatus 100 may perform a different interleaving method according to a set comprising at least one of a code rate, a length of an LDPC codeword and a modulation method.
For example, the transmitting apparatus 100 performs interleaving using the block interleaver 124 in a first set comprising a first predetermined code rate, a first predetermined length of an LDPC codeword and a first predetermined modulation method and, performs interleaving using the block-row interleaver 125 in a second set comprising a second predetermined code rate, a second predetermined length of an LDPC codeword and a second predetermined modulation method different with the first set.
FIG. 24 is a block diagram to illustrate a configuration of a receiving apparatus according to an exemplary embodiment. Referring to FIG. 24, the receiving apparatus 2700 includes a demodulator 2710, a multiplexer 2720, a deinterleaver 2730 and a decoder 2740.
The demodulator 2710 receives and demodulates a signal transmitted from the transmitting apparatus 100. Specifically, the demodulator 2710 generates a value corresponding to an LDPC codeword by demodulating the received signal, and outputs the value to the multiplexer 2720. In this case, the demodulator 2710 may use a demodulation method corresponding to a modulation method used in the transmitting apparatus 100. For doing this, the transmitting apparatus 100 may transmit information on the modulation method to the receiving apparatus 2700. In addition, the transmitting apparatus 100 may perform modulation by using a modulation method predefined between the transmitting apparatus 100 and the receiving apparatus 2700.
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 2720 multiplexes the output value of the demodulator 2710 and outputs the value to the deinterleaver 2730.
Specifically, the multiplexer 2720 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). Accordingly, when the demultiplexer (not shown) is omitted from the transmitting apparatus 100, the multiplexer 2720 may be omitted from the receiving apparatus 2700.
That is, the multiplexer 2720 converts the output value of the demodulator 2710 into cell-to-bit and outputs an LLR value on a bit basis.
In this case, when the demultiplexer (not shown) does not change the order of the LDPC codeword bits as shown in FIG. 13, the multiplexer 2720 may output the LLR values serially on the bit basis without changing the order of the LLR values corresponding to the bits of the cell. Alternatively, the multiplexer 2720 may rearrange the order of the LLR values corresponding to the bits of the cell to perform an inverse operation to the demultiplexing operation of the demultiplexer (not shown) based on Table 50. Meanwhile, information on performance of the demultiplexing operation may be provided from the transmitting apparatus 100, or may be predefined between the transmitting apparatus 100 and the receiving apparatus 2700.
The deinterleaver 2730 deinterleaves the output value of the multiplexer 2720 and outputs the values to the decoder 2740.
Specifically, the deinterleaver 2730 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 2730 deinterleaves the LLR value by performing the interleaving operation of the interleaver 120 inversely.
For doing this, the deinterleaver 2730 may include elements as shown in FIGS. 25 and 27.
First, as shown in FIG. 25, the deinterleaver 2730 includes a block deinterleaver 2731, a group twist deinterleaver 2732, a group deinterleaver 2733, and a parity deinterleaver 2734, according to an exemplary embodiment.
The block deinterleaver 2731 deinterleaves the output of the multiplexer 2720 and outputs a value to the group twist deinterleaver 2732.
Specifically, the block deinterleaver 2731 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 2731 deinterleaves by using at least one row formed of a plurality of columns, that is, by writing the LLR value output from the multiplexer 2720 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.
In this case, when the block interleaver 124 interleaves by dividing a column into two parts, the block deinterleaver 2731 may deinterleave by dividing a row into two parts.
In addition, when the block interleaver 124 performs writing and reading with respect to a group which does not belong to the first part in the row direction, the block deinterleaver 2731 may deinterleave by writing and reading a value corresponding to the group which does not belong to the first part in the row direction.
Hereinafter, the block deinterleaver 2731 will be explained with reference to FIG. 26. However, this is merely an example and the block deinterleaver 2731 may be implemented in other methods.
An input LLR vi (0≦i<Nldpc) is written in a ri row and a ci column of the block deinterleaver 2431. Herein, ci=(i mod Nc) and




  
    
      r
      i
    
    =
    
      ⌊
      
        i
        
          N
          c
        
      
      ⌋
    
  
  ,




On the other hand, an output LLR qi(0≦i<Nc×Nr1) is read from a ci column and a ri row of the first part of the block deinterleaver 2431. Herein,




  
    
      c
      i
    
    =
    
      ⌊
      
        i
        
          N
          
            r
            ⁢
            
              
            
            ⁢
            1
          
        
      
      ⌋
    
  
  ,




ri=(i mod Nr1)

In addition, an output LLR qi(Nc×Nr1≦i<Nldpc) is read from a ci column and a ri row of the second part. Herein,




  
    
      c
      i
    
    =
    
      ⌊
      
        
          (
          
            i
            -
            
              
                N
                c
              
              ×
              
                N
                
                  r
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
            
          
          )
        
        
          N
          
            r
            ⁢
            
              
            
            ⁢
            2
          
        
      
      ⌋
    
  
  ,




ri=Nr1+{(i−Nc×Nr1) mode Nr2}.

The group twist deinterleaver 2732 deinterleaves the output value of the block deinterleaver 2731 and outputs the value to the group deinterleaver 2733.
Specifically, the group twist deinterleaver 2732 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 2732 may rearrange the LLR values of the same group by changing the order of the LLR values existing in the same group. When the group twist operation is not performed in the transmitting apparatus 100, the group twist deinterleaver 2732 may be omitted.
The group deinterleaver 2733 (or the group-wise deinterleaver) deinterleaves an output value of the group twist deinterleaver 2732 and outputs a value to the parity deinterleaver 2734.
Specifically, the group deinterleaver 2733 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 2733 may rearrange the order of the plurality of groups in group units. In this case, the group deinterleaver 2733 may rearrange the order of the plurality of groups in group units by applying the interleaving method of Tables 28 to 43 inversely according to a length of the LDPC codeword, a modulation method and a code rate.
As described above, in the parity check matrix having the format shown in FIGS. 2 and 3, the order of column groups is changeable and the column group corresponds to a bit group. Accordingly, when the order of column groups of the parity check matrix is changed, the order of bit groups is changed accordingly and the group deinterleaver 2733 may rearrange the order of the plurality of groups in group units with reference to this.
The parity deinterleaver 2734 performs parity deinterleaving with respect to an output value of the group deinterleaver 2733 and outputs a value to the decoder 2740.
Specifically, the parity deinterleaver 2734 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 2734 may deinterleave the LLR values corresponding to the parity bits from among the LLR values output from the group deinterleaver 2733. In this case, the parity deinterleaver 2734 may deinterleave the LLR values corresponding to the parity bits in an inverse method of the parity interleaving method of Equation 8. However, the parity deinterleaver 2734 may be omitted according to a decoding method and implementation of the decoder 2740.
Although the deinterleaver 2730 of FIG. 24 includes three (3) or four (4) elements as shown in FIG. 25, operations of the elements may be performed by a single element. For example, when bits each of which belongs to each of bit groups Xa, Xb, Xc, and Xd constitute a single modulation symbol, the deinterleaver 2730 may deinterleave these bits to locations corresponding to their bit groups based on the received single modulation symbol.
For example, when a code rate is 12/15 and a modulation method is 16-QAM, the group deinterleaver 2733 may perform deinterleaving based on table 28.
In this case, bits each of which belongs to each of bit groups X35, X37, X16, and X2 constitute a single modulation symbol. Since one bit in each of the bit groups X35, X37, X16, and X2 constitutes a single modulation symbol, the deinterleaver 2730 may map bits onto decoding initial values corresponding to the bit groups X35, X37, X16, and X2 based on the received single modulation symbol.
The deinterleaver 2730 may include a block-row deinterleaver 2735, a group twist deinterleaver 2732, a group deinterleaver 2733 and a parity deinterleaver 2734, as shown in FIG. 27. In this case, the group twist deinterleaver 2732 and the parity deinterleaver 2734 perform the same functions as in FIG. 25, and thus, a redundant explanation is omitted.
The block-row deinterleaver 2735 deinterleaves an output value of the multiplexer 2720 and outputs a value to the group twist deinterleaver 2732.
Specifically, the block-row deinterleaver 2735 is an element corresponding to the block-row interleaver 125 provided in the transmitting apparatus 100 and may perform the interleaving operation of the block-row interleaver 125 inversely.
That is, the block-row deinterleaver 2735 may deinterleave by using at least one column formed of a plurality of rows, that is, by writing the LLR values output from the multiplexer 2720 in each column in the column direction and reading each row of the plurality of columns in which the LLR value is written in the column direction.
However, when the block-row interleaver 125 performs writing and reading with respect to a group which does not belong to the first part in the column direction, the block-row deinterleaver 2735 may deinterleave by writing and reading a value corresponding to the group which does not belong to the first part in the column direction.
The group deinterleaver 2733 deinterleaves the output value of the group twist deinterleaver 2732 and outputs the value to the parity deinterleaver 2734.
Specifically, the group deinterleaver 2733 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 2733 may rearrange the order of the plurality of groups in group units. In this case, the group deinterleaver 2733 may rearrange the order of the plurality of groups in group units by applying the interleaving method of Tables 69 to 84 inversely according to a length of the LDPC codeword, a modulation method and a code rate.
Although the deinterleaver 2730 of FIG. 24 includes three (3) or four (4) elements as shown in FIG. 27, operations of the elements may be performed by a single element. For example, when bits each of which belongs to each of bit groups Xa, Xb, Xc, and Xd constitute a single modulation symbol, the deinterleaver 2730 may deinterleave these bits to locations corresponding to their bit groups based on the received single modulation symbol.
For doing this, the transmitting apparatus 100 may transmit various pieces of information used for performing interleaving by the interleaver 120 to the receiving apparatus 2700. In addition, the transmitting apparatus 100 may perform interleaving by using a method predefined between the transmitting apparatus 100 and the receiving apparatus 2700.
The decoder 2740 may perform LDPC decoding by using the output value of the deinterleaver 2730. To achieve this, the decoder 2740 may include a separate LDPC decoder (not shown) to perform the LDPC decoding.
Specifically, the decoder 2740 is an element corresponding to the encoder 110 of the transmitting apparatus 200 and may correct an error by performing the LDPC decoding by using the LLR value output from the deinterleaver 2730.
For example, the decoder 2740 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 2740 may use a parity check matrix when performing the LDPC decoding. In this case, an information word submatrix in the parity check matrix is defined as in Tables 4 to 27 according to a code rate and a length of the LDPC codeword, and a parity submatrix may have a dual diagonal configuration.
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 2700 or may be provided by the transmitting apparatus 100.
FIG. 28 is a flowchart to illustrate a signal processing method according to an exemplary embodiment.
First of all, an LDPC codeword is generated by performing LDPC encoding (S3010).
Subsequently, the LDPC codeword is interleaved (S3020), and a modulation symbol is generated by modulating the interleaved LDPC codeword according to a modulation method (S3030).
Herein, in S3020, the interleaving includes performing interleaving by grouping the LDPC codeword into a plurality of groups, rearranging an order of the plurality of groups in group units, and dividing the plurality of rearranged groups based on a modulation order according to the modulation method.
In this case, the interleaving may include performing interleaving by dividing each of the plurality of columns into a first part and a second part, sequentially writing a plurality of groups in bit group units in a plurality of columns constituting the first part, dividing bits constituting the other groups into bit groups each consisting of predetermined number of bits based on the number of the plurality of columns, and sequentially writing the divided bit groups in a plurality of columns constituting the second part.
The first part may be formed of rows as many as the number of bits included in at least a part of bit group which is writable in bit group units in each of the plurality of columns among a plurality of bit groups constituting the LDPC codeword according to the number of the plurality of columns, the number of the bit groups, and the number of bits constituting each bit group, in each of the plurality of columns. In addition, the second part may be formed of rows as many as rows excluding the number of bits included in at least a part of bit group which is writable in bit group units in each of the plurality of columns of the rows constituting each of the plurality of columns, in each of the plurality of columns.
In this case, the number of rows of the second part may have the same value as a quotient obtained by dividing the number of bits included in all bit groups excluding the bit group corresponding to the first part by the number of columns.
In S3020, the interleaving may include performing interleaving by sequentially writing the bits included in the at least a part of bit group which is writable in bit group units in each of the plurality of columns constituting the first part, dividing the bits included in the other bit group excluding at least a part of bit group from a plurality of bit groups based on the number of columns, and sequentially writing the divided bits in each of the plurality of columns constituting the second part.
In this case, in S3020, the interleaving may include performing interleaving by dividing bits included in the other bit groups by the number of the plurality of columns, writing each of the divided bits in each of the plurality of columns constituting the second part in a column direction, and reading the plurality of columns constituting the first part and the second part in a row direction.
Meanwhile, in response to the modulation method being QPSK, 16-QAM, 64-QAM, 256-QAMN, 1024-QAM, and 4096-QAM, the modulation order may be 2, 4, 6, 8, 10, and 12.
A non-transitory computer readable medium, which stores a program for performing the above signal processing 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.
Components, elements or units represented by a block as illustrated in FIGS. 1, 4, 12, 13, 23 and 27-29 may be embodied as the various numbers of hardware, software and/or firmware structures that execute respective functions described above, according to exemplary embodiments. For example, 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. 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 the above 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.
Although a bus is not illustrated in the block diagrams of the transmitting apparatus and the receiving apparatus, communication may be performed between each element of each apparatus via the bus. In addition, each apparatus may further include a processor such as a Central Processing Unit (CPU) or a microprocessor to perform the above-described various operations.
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.

## Claims

1. A transmitting apparatus comprising:
an interleaver configured to interleave a codeword; and
a constellation mapper configured to map bits of the interleaved codeword onto constellation points,
wherein the interleaver comprises:
a group-wise interleaver configured to split the codeword into a plurality of bit groups and interleave the plurality of bit groups, and
a block interleaver configured to interleave the interleaved plurality of bit groups using a plurality of columns, each of the plurality of columns comprising a first part and a second part,
wherein some bit groups of the interleaved plurality of bit groups are interleaved in the first part, and at least one remaining bit group is interleaved in the second part, and
wherein bits of the some bit groups and bits of the at least one remaining bit group are written in a column direction and are read out in a row direction.

2. The apparatus as claimed in claim 1, wherein each of the plurality of bit groups comprises 360 bits.

3. The apparatus as claimed in claim 1, wherein the interleaver further comprises a parity interleaves configured to interleave the parity bits, and
wherein the group-wise interleaves is configured to split an LDPC codeword comprising the interleaved parity bits into a plurality of bit groups.