Patent Publication Number: US-2023155724-A1

Title: Method and apparatus for channel encoding/decoding in a communication or broadcasting system

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
     This application is a continuation application of prior application Ser. No. 17/564,531 filed on Dec. 29, 2021, which will be issued as U.S. Pat. No. 11,575,465 on Feb. 7, 2023; which is a continuation application of prior application Ser. No. 17/109,476 filed on Dec. 2, 2020, which has issued as U.S. Pat. No. 11,233,604 on Jan. 25, 2022; which is a continuation application of prior application Ser. No. 16/730,412 filed on Dec. 30, 2019, which has issued as U.S. Pat. No. 10,903,936 on Jan. 26, 2021; which is a continuation application of prior application Ser. No. 15/361,283 filed on Nov. 25, 2016, which has issued as U.S. Pat. No. 10,574,389 on Feb. 25, 2020; and which was based on and claimed priority under 35 U.S.C. § 119(a) of a Korean patent application number 10-2015-0165114, filed on Nov. 24, 2015, in the Korean Intellectual Property Office, a Korean patent application number 10-2016-0002929, filed on Jan. 8, 2016, in the Korean Intellectual Property Office, a Korean patent application number 10-2016-0102635, filed on Aug. 11, 2016, in the Korean Intellectual Property Office, a Korean patent application number 10-2016-0105807, filed on Aug. 19, 2016, in the Korean Intellectual Property Office, and a Korean patent application number 10-2016-0149882, filed on Nov. 10, 2016, in the Korean Intellectual Property Office, the disclosures of each of which are incorporated by reference herein in its entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to a method and an apparatus for channel encoding/decoding in a communication or broadcasting system. More particularly, the present disclosure relates to a method and an apparatus for low density parity check (LDPC) encoding and decoding, which support various input lengths and various code rates. 
     BACKGROUND 
     To satisfy demands for wireless data traffic, which have been increasing since commercialization of a 4 th  generation (4G) communication system, efforts have been made to develop an improved 5 th  generation (5G) or pre-5G communication system. That is why the 5G or pre-5G communication system is called a beyond 4G network communication system or a post long term evolution (LTE) system. 
     To achieve high data rates, deployment of the 5G communication system in a millimeter wave (mmWave) band (for example, a 60-GHz band) is under consideration. In order to mitigate propagation path loss and increase a propagation distance in the mmWave band, beamforming, massive multiple input multiple output (MIMO), full dimensional MIMO (FD-MIMO), array antenna, analog beamforming, and large-scale antenna technology have been discussed for the 5G communication system. 
     Further, to improve a system network, techniques such as evolved small cell, advanced small cell, cloud radio access network (cloud RAN), ultra-dense network, device-to-device (D2D) communication, wireless backhaul, moving network, cooperative communication, coordinated multi-point (CoMP), and received interference cancelation have been developed for the 5G communication system. 
     Besides, advanced coding modulation (ACM) techniques, such as hybrid frequency shift keying (FSK) and quadrature amplitude modulation (QAM) modulation (FQAM) and sliding window superposition coding (SWSC), and advanced access techniques, such as filter bank multi carrier (FBMC) and non-orthogonal multiple access (NOMA), and sparse code multiple access (SCMA) have been developed for the 5G communication system. 
     In a communication or broadcasting system, link performance may be degraded greatly by noise, fading, and inter-symbol interference (ISI). Accordingly, a technique for overcoming noise, fading, and ISI is required to implement high-speed digital communication or broadcasting systems that require high data throughput and high reliability, such as future-generation mobile communication, digital broadcasting, and portable Internet. To overcome noise, error correction codes have recently been studied actively as a method for increasing communication reliability by efficiently recovering information distortion. 
     Low density parity check (LDPC) codes were originally developed by Gallager in 1960s and largely ignored for a long time because their computational complexity was too high for the hardware technology at the time. However, in 1993, turbo codes developed by Berrou, Glavieux, and Thitimajshima were the first codes to be shown to perform close to the Shannon limit or channel capacity. Along with many interpretations regarding the performance and characteristics of turbo codes, extensive research was made on iterative decoding and graph-based channel encoding. The success of turbo codes led to the rediscovery of LDPC codes in the late 1990s. It was revealed that iterative decoding using a sum-product algorithm on a Tanner graph representing an LDPC code performs close to the Shannon limit. 
     Although an LDPC code is generally defined by a party heck matrix, a bipartite graph known as a Tanner graph may be used to represent the LDPC code. 
       FIG.  1    is a view illustrating a structure of a systematic LDPC codeword according to the related art. 
     Referring to  FIG.  1   , the systematic LDPC codeword will be described below. 
     An LDPC codeword  100  including N ldpc  bits or symbols is generated by LDPC-encoding a received information word  102  including K ldpc  bits or symbols. For convenience of description, it is assumed that for the input of the information word  102  including K ldpc  bits or symbols, the codeword  100  including N ldpc  bits or symbols is generated. For example, LDPC encoding of the information word  102  including K ldpc  bits, I=[i 0 , i 1 , i 2 , . . . L K     ldpc     −1 ] results in the codeword  100 , c=[c 0 , c 1 , c 2 , . . . , C N     ldpc     −1 ]. For example, a codeword is a bit stream including a plurality of bits, and a codeword bit is bit of the codeword. Further, an information word is a bit stream including a plurality of bits, and an information word bit is a bit of the information word. In the case of a systematic code, the codeword  100  is given as c=[c 0 , c 1 , c 2 , . . . , C N     ldpc     −1 ]=[i 0 , i 1 , i 2 , . . .  1   K     ldpc     −1 , p 0 , p 1 , p 2 , . . . P N     ldpc     −K     ldpc     −1 ] where P=[p 0 , p 1 , p 2 , . . . p N     1dpc     −K     1dpc     −1 ] represents parity bits  104 . The number of parity bits  104 , N parity  may be calculated by N parity =N ldpc −K ldpc . 
     An LDPC code is a form of linear block code, and LDPC encoding involves determining a codeword satisfying the condition described by Equation 1. 
     
       
         
           
             
               
                 
                   
                     H 
                     · 
                     
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                       T 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             h 
                             0 
                           
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                             1 
                           
                           , 
                           
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                             2 
                           
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                             … 
                             ⁢ 
                                 
                             
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                   Equation 
                   ⁢ 
                       
                   1 
                 
               
             
           
         
       
       
         
           
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             , 
             
               c 
               = 
               
                 [ 
                 
                   
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                   , 
                   
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     In Equation 1, H is a parity check matrix, C is a codeword, c i  is an i th  bit of the codeword C, and N lpdc  is the length of the LDPC codeword. Herein, h i  is an i th  column of the parity check matrix H. 
     The parity check matrix H includes as many columns as the number of bits of the LDPC codeword, that is, N ldpc  columns. According to Equation 1, the sum of the products between the columns h i  and the codeword bits c i  is ‘0’, which means that each i th  column h i  is related to each i th  codeword bit c i . 
     With reference to  FIG.  2   , a graph representation of an LDPC code will be described. 
       FIG.  2    illustrates a parity check matrix H 1  with 4 rows by 8 columns, and a Tanner graph representing the parity check matrix H 1  according to the related art. 
     Referring to  FIG.  2   , since the parity check matrix H 1  includes 8 columns, a codeword of length 8 is generated. A code generated from the parity check matrix H 1  is an LDPC code, and the columns correspond to 8 coded bits. 
     Referring to  FIG.  2   , the Tanner graph representing the LDPC code for encoding and decoding based on the parity check matrix H 1  includes eight variable nodes x 1    202 , x 2    204 , x 3    206 , x 4    208 , x 5    210 , x 6    212 , x 7    214 , and x 8    216  and four check nodes  218 ,  220 ,  222  and  224 . An i th  column and a j th  row in the parity-check matrix H 1  represent a variable node x i  and a j th  check node, respectively. If an entry at the i th  column and the j th  row in the parity-check matrix H 1  is one, i.e., non-zero, this means that an edge is drawn between the variable node x i  and the j th  check node on the Tanner graph illustrated in  FIG.  2   . 
     The degree of a variable node or a check node on the Tanner graph of the LDPC code is the number of edges connected to the node. The degree of a node is equal to the number of non-zero entries in a column or row corresponding to the node in the parity-check matrix of the LDPC code. For example, the degrees of the variable nodes x 1    202 , x 2    204 , x 3    206 , x 4    208 , x 5    210 , x 6    212 , x 7    214 , and x 8    216  are 4, 3, 3, 3, 2, 2, 2 and 2, respectively, and the degrees of the check nodes  218 ,  220 ,  222  and  224  are 6, 5, 5 and 5, respectively. Similarly, the numbers of non-zeroes in the columns of the parity-check matrix H 1  of  FIG.  2   , corresponding to the variable nodes of  FIG.  2    are 4, 3, 3, 3, 2, 2, 2 and 2, respectively, and the numbers of non-zeroes in the rows of the parity-check matrix of  FIG.  2   , corresponding to the check nodes of  FIG.  2    are 6, 5, 5 and 5, respectively. 
     The LDPC code may be decoded using an iterative decoding algorithm based on a sum-product algorithm on the bipartite graph illustrated in  FIG.  2   . The sum-product algorithm is a form of message passing algorithm in which messages are exchanged through an edge on a bipartite graph, and an output message is calculated and updated from messages input to a variable node or a check node. 
     The value of an i th  coded bit may be determined based on a message of an i th  variable node. The value of the i th  coded bit may be determined by either of hard decision and soft decision. Accordingly, the performance of the i th  bit, c i  of the LDPC code corresponds to the performance of the i th  variable node of the Tanner graph. The performance may be determined according to the positions and number of ones in the i th  column of the parity check matrix. In other words, the performance of N ldpc  codeword bits of a codeword may depend on the positions and number of ones in the parity check matrix, which means that the performance of the LDPC code is affected significantly by the parity check matrix. Therefore, to design an LDPC code with excellent performance, there is a need for a method for designing a good parity check matrix. 
     For implementation simplicity, a communication or broadcasting system generally adopts a quasi-cyclic LDPC (QC-LDPC) code using a QC parity check matrix. 
     A QC-LDPC code characteristically has a parity check matrix including zero matrices or circulant permutation matrices, which are small square matrices. 
     A detailed description will be given of a QC-LDPC code. 
     First, an L×L circulant permutation matrix P=(P i,j ) is defined as Equation 2. P i,j  represents an entry in an i th  row and a j th  column of the matrix P (0≤i, j&lt;L). 
     
       
         
           
             
               
                 
                   
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                   ⁢ 
                       
                   2 
                 
               
             
           
         
       
     
     For the permutation matrix P as defined above, P i  (0≤i&lt;L) is a circulant permutation matrix obtained by cyclically shifting the elements of an L×L identity matrix to the right by i positions. 
     The simplest parity check matrix H of a QC-LDPC code may be represented as Equation 3. 
     
       
         
           
             
               
                 
                   H 
                   = 
                   
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                   Equation 
                   ⁢ 
                       
                   3 
                 
               
             
           
         
       
     
     Let P −1  be defined as an L×L zero matrix. The exponent a i,j  of each circulant permutation matrix or zero matrix in Equation 3 has one of the values of {−1, 0, 1, 2, . . . , L−1}. The parity check matrix H described in Equation 3 has m row blocks by n column blocks, and thus its size is mL×nL. 
     If the parity check matrix of Equation 3 is of full rank, the size of the information word bits of the QC-LDPC code corresponding to the parity check matrix is obviously (n−m)L. For convenience of description, (n−m) column blocks corresponding to the information word bits are referred to as information word column blocks, and m column blocks corresponding to the other parity bits are referred to as parity column blocks. 
     In general, an m×n binary matrix produced by replacing each circulant permutation matrix and each zero matrix by one and zero, respectively in the parity check matrix of Equation 3 is called a mother matrix M(H) of the parity check matrix H, and an m×n integer matrix produced by selecting the exponent of each circulant permutation matrix or zero matrix is called an exponent matrix E(H) of the parity check matrix H, as expressed as Equation 4. 
     
       
         
           
             
               
                 
                   
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                     ⁡ 
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                   = 
                   
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                   ⁢ 
                       
                   4 
                 
               
             
           
         
       
     
     Meanwhile, the performance of an LDPC code may be determined according to its parity check matrix. Therefore, it is necessary to design a proper parity check matrix for an LDPC code with excellent performance. Further, an LDPC encoding or decoding method supporting various input lengths and code rates is required. 
     Lifting is used to efficiently design a QC-LDPC code. The lifting is a technique of efficiently designing a very large parity check matrix by setting L determining the size of a circulant permutation matrix or zero matrix from a given small mother matrix in a specific rule. A lifting scheme of the related art and the characteristics of a QC-LDPC code designed in the lifting scheme of the related art are summarized as follows. 
     Given an LDPC code C 0 , let S QC-LDPC codes to be designed by lifting be denoted by C 1 , . . . , C S  and the size of row and column blocks of each of the QC-LDPC codes be denoted by Lk. The LDPC code C 0  is the smallest LDPC code having the mother matrices of the LDPC codes C 1 , . . . , C S  as a parity check matrix, and the size L 0  of row and column block of the LDPC code C 0  is 1. For convenience of description, the parity check matrix H k  of each code C k  includes an m×n exponent matrix E(H k )=(e i,j   (k) )) where each exponent e i,j   (k)  has a value selected from the values of {−1, 0, 1, 2, . . . , L k −1}. 
     Lifting is performed in the order of C 0 -&gt;C 1 -&gt; . . . -&gt;C S  and characterized by L (k+1) =q (k+1) L k (q (k&#39;1)  is a positive integer, k=0, 1, . . . , S−1). In view of the nature of lifting, once a parity check matrix Hs of a code C s  is stored, all of the QC-LDPC codes C 0 , C 1 , . . . , C S  may be represented according to the lifting scheme by Equation 5. 
     
       
         
           
             
               
                 
                   
                     E 
                     ⁡ 
                     ( 
                     
                       H 
                       k 
                     
                     ) 
                   
                   ≡ 
                   
                     ⌊ 
                     
                       
                         
                           L 
                           k 
                         
                         
                           L 
                           S 
                         
                       
                       ⁢ 
                       
                         E 
                         ⁡ 
                         ( 
                         
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                           S 
                         
                         ) 
                       
                     
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                   Equation 
                   ⁢ 
                       
                   5 
                 
               
             
           
         
       
       
         
           Or 
         
       
       
         
           
             
               
                 
                   
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                     ⁢ 
                        
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                       k 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   6 
                 
               
             
           
         
       
     
     In the lifting scheme described by Equation 5 or Equation 6, since L k  values being row block sizes or column block sizes of the parity check matrices of the QC-LDPC codes C k  are in a multiple relationship, the exponent matrices are also selected in a specific method. This lifting scheme of the related art facilitates designing of a QC-LDPC code with improved error floor characteristics, because the algebraic or graph characteristics of each parity check matrix designed by lifting are improved. 
     However, a shortcoming with the lifting scheme of the related art is that the length of each code is limited greatly because of the multiple relationship between the L k  values. For example, it is assumed that a minimum lifting scheme, such as L (k+1) =2*L k  is applied to each value of L k . In this case, the size of the parity check matrix of each QC-LDPC code may be 2 k m×2 k n. For example, if lifting is applied at 10 levels (S=10), 10 sizes may result. 
     For the above reason, the lifting scheme of the related art is not viable in designing a QC-LDPC code supporting various lengths. However, a typical communication system requires very high-level length compatibility in consideration of transmission of various types of data. As a result, it is difficult to apply an LDPC code to the communication system in the method of the related art. 
     Therefore, a need exists for a method and an apparatus for LDPC encoding and decoding, which support various input lengths and various code rates. 
     The above information is presented as background information only to assist with an understanding of the present disclosure. No determination has been made, and no assertion is made, as to whether any of the above might be applicable as prior art with regard to the present disclosure. 
     SUMMARY 
     An aspect of the present disclosure is to address at least the above-mentioned problems and/or disadvantages and to provide at least the advantages described below. Accordingly, an aspect of the present disclosure is to provide a method and an apparatus for low density parity check (LDPC) encoding and decoding, which support various input lengths and various code rates. 
     Another aspect of the present disclosure is to provide a method and an apparatus for LDPC encoding and decoding, which support various input lengths and various code rates, using a parity check matrix. 
     In accordance with an aspect of the present disclosure, a channel encoding method in a communication or broadcasting system is provided. The channel encoding method includes reading a first sequence corresponding to a parity check matrix, converting the first sequence to a second sequence by applying a predetermined rule to a block size corresponding to a parity check matrix and the first sequence, and encoding information bits based on the second sequence. The block size has at least two different integer values. 
     In accordance with another aspect of the present disclosure, a channel encoder in a communication or broadcasting system is provided. The channel encoder includes a transceiver configured to transmit and receive data, a memory configured to store the data, and at least one processor configured to read a first sequence corresponding to a parity check matrix, convert the first sequence to a second sequence by applying a predetermined rule to a block size corresponding to a parity check matrix and the first sequence, and encode information bits based on the second sequence. The block size has at least two different integer values. 
     In accordance with another aspect of the present disclosure, a channel decoding method in a communication or broadcasting system is provided. The channel decoding method includes receiving a codeword, the codeword being encoded based on a second sequence to which a first sequence corresponding to a parity check matrix is converted by applying a predetermined rule to a block size corresponding to a parity check matrix and the first sequence, and decoding the received codeword. The block size has at least two different integer values. 
     In accordance with another aspect of the present disclosure, a channel decoder in a communication or broadcasting system is provided. The channel decoder includes a transceiver configured to transmit and receive data, a memory configured to store the data, and at least one processor configured to receive a codeword, the codeword being encoded based on a second sequence to which a first sequence corresponding to a parity check matrix is converted by applying a predetermined rule to a block size corresponding to a parity check matrix and the first sequence, and decode the received codeword. The block size has at least two different integer values. 
     In accordance with another aspect of the present disclosure, a channel decoding method in a communication or broadcasting system is provided. The channel decoding method includes receiving a codeword, determining a block size corresponding to a parity check matrix; determining a set including the determined block size, determining a first sequence corresponding to the determined set, converting the first sequence to a second sequence by applying a certain rule to the block size and the first sequence, and decoding the received codeword based on the second sequence. The codeword being encoded based on the block size and the second sequence, and the block size has at least two different integer values. 
     In accordance with another aspect of the present disclosure, a channel encoding method in a communication or broadcasting system is provided. The channel encoding method includes determining a block size corresponding to a parity check matrix, determining a set including the determined block size, determining a first sequence corresponding to the determined set, converting the first sequence to a second sequence by applying a certain rule to the block size and the first sequence, and encoding information bits using the second sequence. The block size has at least two different integer values. 
     In accordance with another aspect of the present disclosure, a channel decoding method in a communication or broadcasting system is provided. The channel decoding method includes receiving a codeword, determining a block size corresponding to a parity check matrix, determining a set including the determined block size, determining a first sequence corresponding to the determined set, converting the first sequence to a second sequence by applying a certain rule to the block size and the first sequence, and decoding the received codeword based on the second sequence. The codeword being encoded based on the block size and the second sequence, and the block size has at least two different integer values. 
     In accordance with another aspect of the present disclosure, a channel encoding method in a communication or broadcasting system is provided. The channel encoding method includes determining a block size corresponding to a parity check matrix, determining a set including the determined block size, determining a first sequence corresponding to the determined set, converting the first sequence to a second sequence by applying a certain rule to the block size and the first sequence, and encoding information bits using the second sequence. The block size has at least two different integer values. 
     Other aspects, advantages, and salient features of the disclosure will become apparent to those skilled in the art from the following detailed description, which, taken in conjunction with the annexed drawings, discloses various embodiments of the present disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features, and advantages of certain embodiments of the present disclosure will be more apparent from the following description taken in conjunction with the accompanying drawings, in which: 
         FIG.  1    is a view illustrating a structure of a systematic low density parity check (LDPC) codeword according to the related art; 
         FIG.  2    is a view illustrating a parity check matrix H 1  of an LDPC code, with four rows and eight columns, and a Tanner graph representing the parity check matrix H 1  according to the related art; 
         FIG.  3    is a block diagram of a transmitter according to an embodiment of the present disclosure; 
         FIG.  4    is a block diagram of a receiver according to an embodiment of the present disclosure; 
         FIGS.  5 A and  5 B  are message structure diagrams illustrating message passing operations at a check node and a variable node for LDPC decoding according to various embodiments of the present disclosure; 
         FIG.  6    is a block diagram of an LDPC encoder according to an embodiment of the present disclosure; 
         FIGS.  7  and  8    are views illustrating structures of transport blocks according to various embodiments of the present disclosure; 
         FIGS.  9 A and  9 B  are block diagrams of interleavers according to various embodiments of the present disclosure; 
         FIG.  10    is a block diagram of an LDPC decoder according to an embodiment of the present disclosure; 
         FIG.  11    is a block diagram of an LDPC decoder according to an embodiment of the present disclosure; 
         FIG.  12    is a view illustrating a structure of a transport block according to another embodiment of the present disclosure 
         FIGS.  13 A and  13 B  illustrate a parity check matrix with ID=6 and R=1/3 according to various embodiments of the present disclosure; 
         FIGS.  14 A,  14 B,  14 C,  14 D, and  14 E  are views illustrating a parity check matrix (an exponent matrix) designed in consideration of lifting according to various embodiments of the present disclosure; 
         FIGS.  15 A,  15 B,  15 C,  15 D and  15 E  are views illustrating a parity check matrix (an exponent matrix) designed in consideration of lifting according to various embodiments of the present disclosure; 
         FIGS.  16 A,  16 B,  16 C, and  16 D  are views illustrating a parity check matrix (an exponent matrix) designed in consideration of lifting according to various embodiments of the present disclosure; 
         FIGS.  17 A and  17 B  are views illustrating a cycle property of a quasi-cyclic LDPC (QC-LDPC) code according to various embodiments of the present disclosure; 
         FIG.  18    is an view illustrating an extended Tanner graph according to an embodiment of the present disclosure; 
         FIG.  19    is a flowchart illustrating a sequence-based LDPC encoding method according to an embodiment of the present disclosure; and 
         FIG.  20    is a block diagram of a transmitter for performing sequence-based LDPC encoding according to an embodiment of the present disclosure. 
     
    
    
     Throughout the drawings, like reference numerals will be understood to refer to like parts, components, and structures. 
     DETAILED DESCRIPTION 
     The following description with reference to the accompanying drawings is provided to assist in a comprehensive understanding of various embodiments of the present disclosure as defined by the claims and their equivalents. It includes various specific details to assist in that understanding but these are to be regarded as merely exemplary. Accordingly, those of ordinary skill in the art will recognize that various changes and modifications of the various embodiments described herein can be made without departing from the scope and spirit of the present disclosure. In addition, descriptions of well-known functions and constructions may be omitted for clarity and conciseness. 
     The terms and words used in the following description and claims are not limited to the bibliographical meanings, but, are merely used by the inventor to enable a clear and consistent understanding of the present disclosure. Accordingly, it should be apparent to those skilled in the art that the following description of various embodiments of the present disclosure is provided for illustration purpose only and not for the purpose of limiting the present disclosure as defined by the appended claims and their equivalents. 
     It is to be understood that the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, reference to “a component surface” includes reference to one or more of such surfaces. 
     By the term “substantially” it is meant that the recited characteristic, parameter, or value need not be achieved exactly, but that deviations or variations, including for example, tolerances, measurement error, measurement accuracy limitations and other factors known to those of skill in the art, may occur in amounts that do not preclude the effect the characteristic was intended to provide. 
     The following exponent matrix is equivalent to a sequence corresponding to a parity-check matrix or the exponent matrix. 
     The following block size can have at least two different integer values. 
     Those skilled in the art will understand that the subject matter of the present disclosure can be implemented in other systems having a similar technical background with a slight modification without departing from the scope of the present disclosure. 
     The advantages and features of the present disclosure, and a method for achieving them will be apparent from the attached drawings and the following detailed description of embodiments. However, embodiments of the present disclosure may be implemented in various ways, not limited to the following embodiments. The various embodiments of the present disclosure are provided to assist in a comprehensive understanding of the scope and spirit of the present disclosure, and the present disclosure is defined only by the appended claims and their equivalents. Like reference numeral denotes the same components through the specification. 
     While the following description will be given of the present disclosure with the appreciation that there is only one circulant permutation matrix corresponding to one block for convenience of description, the same thing is applicable to the case where a plurality of circulant permutation matrices are included in one block. 
     According to embodiments of the present disclosure, a parity check matrix may be extracted using a memory, given preliminarily in a transmitter or receiver, or generated directly in the transmitter or receiver. The transmitter may store or generate a sequence or integer matrix corresponding to the parity check matrix, and apply the sequence or integer matrix to encoding. Similarly, the receiver may store or generate the sequence or square matrix corresponding to the parity check matrix, and apply the sequence or square matrix to decoding. 
       FIG.  3    is a block diagram of a transmitter according to an embodiment of the present disclosure. 
     Referring to  FIG.  3   , a transmitter  300  may include a segmenter  310 , a zero padder  320 , a low density parity check (LDPC) encoder  330 , a rate matcher  340 , and a modulator  350  in order to process input bits of a variable length. The rate matcher  340  may include an interleaver  341  and a puncturer/repeater/zero remover  342 . 
     The components illustrated in  FIG.  3    are components that encode and modulate input bits of a variable length. A component may be omitted from, modified in, or added to the transmitter  300 . 
       FIG.  4    is a block diagram of a receiver according to an embodiment of the present disclosure. 
     Referring to  FIG.  4   , a receiver  400  may include a demodulator  410 , a rate dematcher  420 , an LDPC decoder  430 , a zero remover  440 , and a desegmenter  450  in order to process information of a variable length. The rate dematcher  420  may include a log likelihood ratio (LLR) inserter  422 , an LLR combiner  423 , and a deinterleaver  424 . 
     The components illustrated in  FIG.  4    execute functions corresponding to their counterparts illustrated in  FIG.  3   . A component may be omitted from, modified in, or added to the receiver  400 . 
     Let S LDPC codes to be designed by lifting be denoted by C 1 , . . . , C S , and let the size of a row block or a column block in a parity check matrix H z  of each LDPC code C Z  be denoted by Z (Z=1, . . . , S). The parity check matrix H z  of each code C Z  has an m×n exponent matrix E(H Z )=(e i,j   (Z) ) where each exponent e i,j   (Z)  is a value selected from among the values of {−1, 0, 1, 2, . . . , Z−1}. Although an exponent indicating a zero matrix is represented as −1 in the present disclosure, the exponent may be changed to a different value for the convenience of a system. 
     Therefore, the exponent matrix of an LDPC code Cs having a largest parity check matrix is given as E(H S )=(e i,j   (S) ). 
     A general lifting scheme for acquiring E(H S )=(e i,j   (S) ) may be expressed as Equation 7. 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       ⁡ 
                       ( 
                       
                         H 
                         Z 
                       
                       ) 
                     
                     = 
                     
                       ( 
                       
                         e 
                         ij 
                         
                           ( 
                           Z 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
                       e 
                       ij 
                       
                         ( 
                         Z 
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   e 
                                   ij 
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     e 
                                     ij 
                                     
                                       ( 
                                       s 
                                       ) 
                                     
                                   
                                   ≤ 
                                   0 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   f 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       e 
                                       ij 
                                       
                                         ( 
                                         s 
                                         ) 
                                       
                                     
                                     , 
                                     Z 
                                   
                                   ) 
                                 
                                 , 
                                 
                                   
                                     e 
                                     ij 
                                     
                                       ( 
                                       s 
                                       ) 
                                     
                                   
                                   &gt; 
                                   0 
                                 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   7 
                 
               
             
           
         
       
       
         
           Or 
         
       
       
         
           
             
               
                 E 
                 ⁡ 
                 ( 
                 
                   H 
                   Z 
                 
                 ) 
               
               = 
               
                 ( 
                 
                   e 
                   ij 
                   
                     ( 
                     Z 
                     ) 
                   
                 
                 ) 
               
             
             , 
             
               
                 e 
                 ij 
                 
                   ( 
                   Z 
                   ) 
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         
                           
                             e 
                             ij 
                             
                               ( 
                               s 
                               ) 
                             
                           
                           , 
                           
                             
                               e 
                               ij 
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                     
                     
                       
                         
                           
                             f 
                             ⁡ 
                             ( 
                             
                               
                                 e 
                                 ij 
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                               , 
                               Z 
                             
                             ) 
                           
                           , 
                           
                             
                               e 
                               ij 
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                             ≥ 
                             0 
                           
                         
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
     In Equation 7, a lifting function f(x, Z) is an integer function defined by integers x and Z. For example, the lifting function f(x,Z) is a function determined by the exponents of the parity check matrix of a given quasi-cyclic LDPC (QC-LDPC) code and the size of a circulant matrix included in the parity check matrix of the QC-LDPC code. In this context, a lifting method of the present disclosure will be described briefly. In the lifting method, the exponents of an exponent matrix given to define an LDPC code are converted using integers corresponding to the exponents and Z determined from the size Z×Z of a circulant matrix, and LDPC encoding or decoding is performed using the converted exponents. 
     An embodiment of the present disclosure provides a method for appropriately selecting the function f(x,Z) as an exponent matrix conversion rule and designing a parity check matrix according to the selected function f(x,Z). When the function f(x,Z) has a different value for every Z value, implementation of the parity check matrix in a system increases complexity. Therefore, the present disclosure deals with a method for minimizing performance degradation with reduced implementation complexity by using the same f(x,Z) value for different Z values. In other words, the function f(x,Z) of the present disclosure is characterized by conversion to the same exponent matrix at least for different Z values. However, it is not necessary to always impose this constraint on f(x,Z). 
     Exponents representing a circulant permutation matrix and a zero matrix included in the parity check matrix of each LDPC code may be determined by Equation 8 or Equation 9. 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   2 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &gt; 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             Z 
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   8 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   2 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≥ 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             Z 
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   9 
                 
               
             
           
         
       
     
     In Equation 8 and Equation 9, mod(e i,j   (S) , 2 k ) represents the remainder of dividing e i,j   (S)  by 2 k  where k is 0, 1, . . . , └log 2 S┘. └x┘ represents a largest integer smaller than X. 
     First, a block size Z is determined. The block size Z may be determined based on exponent matrix information or the size of an information word. 
     Once the block size Z is determined, a range of numbers to which the block size Z belongs is determined. More specifically, referring to Equation 8 or Equation 9, if all of the exponents of circulant permutation matrices included in the parity check matrix of the largest QC-LDPC code are set, the range of numbers to which the block size Z belongs is first determined. Subsequently, a representative value of the determined range (a specific value or predetermined value in the determined range) is determined, and if the representative value is not a value representing a zero matrix, the exponents of circulant permutation matrices of a final desired QC-LDPC code may be determined by performing a modulo operation on the representative value. While in the embodiment of the present disclosure, the first value in a range is set as a representative value of the range, various other values may be available as the representative value. 
     For reference, the range of numbers to which the block size Z belongs in Equation 8 or Equation 9 may be determined in various methods. For example, the determination may be made easily by defining k according to Z as k=Ålog 2  Z┘ as illustrated in Equation 10 or Equation 11. For example, the operation for determining a range to which the block size Z belongs and the operation for determining a representative value for the range may be performed simply by applying a system-set calculation method to the block size Z. 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   2 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               &gt; 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ⁢ 
                                 Z 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   10 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   2 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               ≥ 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ⁢ 
                                 Z 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   11 
                 
               
             
           
         
       
     
     As described above, an embodiment of the present disclosure may configure a parity check matrix of every possible block size Z using the circulant permutation matrices included in the parity check matrix of the largest QC-LDPC code. 
     While a modulo operation is taken as an example in the present disclosure, many other operations are also applicable. 
     For example, a flooring operation described in Equation 12 or Equation 13 may be used. 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &gt; 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             Z 
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   12 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               s 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≥ 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             Z 
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   13 
                 
               
             
           
         
       
     
     In Equation 12 or Equation 13, k s  is a constant preset by the system. Although it is typical that k S =└log 2  S┘, k s  may be changed according to a system requirement. 
     
       
         
           
             ⌊ 
             
               
                 e 
                 
                   i 
                   , 
                   j 
                 
                 
                   ( 
                   S 
                   ) 
                 
               
               
                 2 
                 
                   
                     k 
                     s 
                   
                   - 
                   k 
                 
               
             
             ⌋ 
           
         
       
     
     represents the quotient of dividing e i,j   (S)  by 2 k     s     −k  (k may be 0, 1, . . . , └log 2  S┘. 
     For reference, the operation for determining the range of numbers to which the block size Z belongs by Equation 12 or Equation 13 may be performed easily by defining k according to Z as k=└log 2  Z┘ as illustrated in Equation 14 or Equation 15. 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               &gt; 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ⁢ 
                                 Z 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   14 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               ≥ 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ⁢ 
                                 Z 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   15 
                 
               
             
           
         
       
     
     The above process of the present disclosure is summarized as follows. 
     If information about a parity check matrix (that is, information about an exponent matrix) is given in a given communication or broadcasting system, the block size Z of the parity check matrix is determined, and an integer k is determined based on the block size Z by k=└log 2  Z┘ according to a system-set method. A sequence corresponding to the blocks of the parity check matrix is converted by applying a predefined computation method based on the integer k=└log 2  Z┘, and encoding and decoding are performed using the converted sequence. 
     For reference, the reason for using 2 k     s     −k  as the denominator in Equation 12 to Equation 15 will be described briefly as follows. 
     If the floor lifting of the related art as described in Equation 5 is applied, each entry of a given exponent matrix is multiplied by Z/S. A general integer division and multiplication increases implementation complexity. For complexity reduction, approximation of a value to a form with base 2, such as 2 X  or 2 −X , integer division and multiplication may be implemented easily. 
     If S=2 k     s    from k s =└log 2  Z┘ where 2 k ≤Z&lt;2 k+1 , it is obvious that 2 k     s     −k−1 &lt;S/Z≤2 k     s     −k . Thus, ┌S/Z┐=2 −(k     s     −k) , and Z/S may be approximated to Z/S≈2 k     s     −k . Floor lifting using Z/S≈2 −(k     s     −k)  simplifies implementation. It is obvious that the approximation is possible using flooring according to S or a Z range. 
     Various embodiments of implementing Equation 10, Equation 11, Equation 14, and Equation 15 in hardware will be described below. 
     In Equation 10 and Equation 11 based on a modulo operation, calculation of the remainder of a given exponent e ij   (S)  by 2 k  is equivalent to selection and output of only bits at k th  and lower digits, when the exponent ee ij   (S)  is expressed as a binary number. For example, if a given exponent is 118, its binary number is 1110110. Herein, the remainder of dividing the exponent by 2 6  (=64) is obtained by selecting only bits at 5 th  and lower digits, that is, 110110(=2 5 +2 4 +2 2 +2 1 =54). 
     Calculation of the quotient of dividing a given exponent e ij   (S)  by in Equation 14 and Equation 15 based on flooring is equivalent to selection and output of only bits at digits higher than a (k s −k) th  digit from the start, when the exponent e ij   (S)  is expressed as a binary number. For example, if S=256, k S =└log 2 256┘=8, and the given exponent is 157, the binary number is 10011101. If for Z=96, a flooring operation is performed on the exponent 10011101, calculating the quotient of dividing the exponent 10011101 by 2 2  (=4) is equivalent to selecting only bits at digits higher than a second digit in the exponent, 100111(=2 5 +2 2 +2 1 +1=39), considering that k=└log 2 96┘=6 and k s −k=2. Flooring-based lifting may be regarded as selecting k bits from the start, when an exponent is expressed as a binary number of k s  bits. For example, if S=256, k S =└log 2 256┘=8, a given exponent is 00100101, and a flooring operation is performed for Z=96, calculation of the quotient of dividing the exponent by 2 2  (=4) is equivalent to selection of the first 6 bits of the exponent 00100101, 001001(=9), considering that k=└log 2 96┘=6 and k s −k=2. 
     Further, it is obvious that although ranges are defined on a 2 k  basis, the ranges may also be defined on a 3 k  basis or on an any other unit basis. The ranges may not need to be set always in the same rule. According to a lifting process, ranges may be set differently, such as 2 k ≤Z&lt;2 k+1 , 2 k+1 ≤Z&lt;3·2 k+1 , and 3·2 k+1 ≤Z&lt;2 k+2 . 
     While it has been described that when ranges of the block size Z to which lifting is applied are defined as l i ≤Z&lt;l i +1 (i=1, 2 . . . ), the representative value of each i th  range is set as l i , the representative value may be changed according to a system requirement. 
     If S LDPC codes designed by lifting are C 1 , . . . , C S , and Z values being row block sizes or column block sizes increment sequentially by D at each time, such as Z={D, 2*D, 3*D, 4*D, . . . , S*D}, rather than the Z values sequentially increases, such as 1, 2, 3, . . . , lifting may be performed in the manner expressed as Equation 16 to Equation 23. 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     2 
                                     k 
                                   
                                   ⁢ 
                                   D 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≥ 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             
                               Z 
                               D 
                             
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   16 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     2 
                                     k 
                                   
                                   ⁢ 
                                   D 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &gt; 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             
                               Z 
                               D 
                             
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   17 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     2 
                                     k 
                                   
                                   ⁢ 
                                   D 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               ≥ 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ( 
                                 
                                   Z 
                                   D 
                                 
                                 ) 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   18 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                               
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     2 
                                     k 
                                   
                                   ⁢ 
                                   D 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               &gt; 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ( 
                                 
                                   Z 
                                   D 
                                 
                                 ) 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   19 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≥ 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             
                               Z 
                               D 
                             
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   20 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &gt; 
                             
                               0 
                               ⁢ 
                                   
                               and 
                               ⁢ 
                                   
                               
                                 2 
                                 k 
                               
                             
                             ≤ 
                             
                               Z 
                               D 
                             
                             &lt; 
                             
                               2 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   21 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               ≥ 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ( 
                                 
                                   Z 
                                   D 
                                 
                                 ) 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   22 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               S 
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   S 
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     S 
                                     ) 
                                   
                                 
                               
                               &gt; 
                               
                                 0 
                                 ⁢ 
                                     
                                 and 
                                 ⁢ 
                                     
                                 k 
                               
                             
                             = 
                             
                               ⌊ 
                               
                                 
                                   log 
                                   2 
                                 
                                 ( 
                                 
                                   Z 
                                   D 
                                 
                                 ) 
                               
                               ⌋ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   23 
                 
               
             
           
         
       
     
     The lifting method has been described above on the assumption that there is one parity check matrix. However, if a plurality of parity check matrices are used, lifting may support more excellent coding performance. 
     Let S LDPC codes designed by lifting be denoted by C 1 , . . . , C S . If the size of row blocks and column blocks, Z increases in the order of 1, 2, 3, . . . , a method for supporting lifting using a plurality of parity check matrices, instead of a single parity check matrix, will be described. For convenience of description, application of lifting based on two parity check matrices will be described. An LDPC code corresponds to at least two parity check matrices of different sizes, and the parity check matrices may be defined using different row block (or column block) sizes and the same sequence (or integer matrix). The lifting method described by Equation 8 to Equation 23 will be summarized briefly. If 2 k ≤Z&lt;2 k+1  or 2 k ≤Z/D&lt;2 k+1 , an exponent matrix corresponding to Z may be identical to an exponent matrix with Z=2 k  or Z=2 k D. In other words, up to 2 k  parity check matrices may be acquired from the same exponent matrix according to the range of Z. 
     However, the algebraic characteristics of a parity check matrix are determined according to an exponent matrix and the size Z of a permutation matrix included in the parity check matrix. If more parity check matrices have the same exponent matrix, the probability of performance degradation may be increased. 
     Therefore, the following method may be used in order to reduce occurrences of the same exponent matrix according to each Z value. It is first assumed that two exponent matrices E(H S1 )=(e i,j   (S1) ), E(H S2 )=(e i,j   (S2) ) are given to apply sequence conversion. Notably, it is assumed that the mother matrices of the exponent matrices are the same. As in Equation 24 or Equation 25, conversion of different exponent matrices may be applied according to Z values. 
     
       
         
           
             
               
                 
                   Equation 
                   ⁢ 
                       
                   24 
                 
               
               
                  
               
             
             
               
                 
                   
                     
                       2 
                       k 
                     
                     ≤ 
                     Z 
                     &lt; 
                     
                       
                         3 
                         · 
                         
                           2 
                           
                             k 
                             - 
                             1 
                           
                         
                       
                       ⁢ 
                           
                       
                         e 
                         
                           i 
                           , 
                           j 
                         
                         
                           ( 
                           Z 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               
                                 S 
                                 ⁢ 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                                
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     
                                       S 
                                       ⁢ 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 , 
                                 
                                   2 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             ≥ 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   i 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       3 
                       · 
                       
                         2 
                         
                           k 
                           - 
                           1 
                         
                       
                     
                     ≤ 
                     Z 
                     &lt; 
                     
                       
                         2 
                         
                           k 
                           + 
                           1 
                         
                       
                       ⁢ 
                           
                       
                         e 
                         
                           i 
                           , 
                           j 
                         
                         
                           ( 
                           Z 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               
                                 S 
                                 ⁢ 
                                 2 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                 
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             mod 
                             ⁢ 
                                
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     
                                       S 
                                       ⁢ 
                                       2 
                                     
                                     ) 
                                   
                                 
                                 , 
                                 
                                   3 
                                   · 
                                   
                                     2 
                                     k 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             ≥ 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ii 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Equation 
                   ⁢ 
                       
                   25 
                 
               
               
                  
               
             
             
               
                 
                   
                     
                       2 
                       k 
                     
                     ≤ 
                     Z 
                     &lt; 
                     
                       
                         3 
                         · 
                         
                           2 
                           
                             k 
                             - 
                             1 
                           
                         
                       
                       ⁢ 
                           
                       
                         e 
                         
                           i 
                           , 
                           j 
                         
                         
                           ( 
                           Z 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               
                                 S 
                                 ⁢ 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     1 
                                   
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             ≥ 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   i 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       3 
                       · 
                       
                         2 
                         
                           k 
                           - 
                           1 
                         
                       
                     
                     ≤ 
                     Z 
                     &lt; 
                     
                       
                         2 
                         
                           k 
                           + 
                           1 
                         
                       
                       ⁢ 
                           
                       
                         e 
                         
                           i 
                           , 
                           j 
                         
                         
                           ( 
                           Z 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               
                                 S 
                                 ⁢ 
                                 2 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 3 
                                 · 
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     
                                       S 
                                       ⁢ 
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                   + 
                                   1 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     ⁢ 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             ≥ 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ii 
                   ) 
                 
               
             
           
         
       
     
     Equation 24 and Equation 25 will be described below. 
     First, ranges of Z values are determined, and an integer representing each range is determined. In Equation 24 and Equation 25, the first value of each range is determined to be a representative value of the range. Subsequently, one of a plurality of exponent matrices is selected according to a Z-value range or a representative value, and exponent matrix conversion is performed using the selected exponent matrix. 
     As two exponent matrices are used as described in Equation 24 and Equation 25, if 2 k ≤Z&lt;2 k+1 , 2 k−1  parity check matrices have the same exponent matrix. Since the number of occurrences of the same exponent matrix is reduced in this manner, design of a QC-LDPC code may be facilitated and performance degradation may further be reduced. On the other hand, since there should be a plurality of exponent matrices and Z-value ranges should be defined more elaborately, complexity is slightly increased. Accordingly, lifting should be applied in proper consideration of performance and complexity. 
     For reference, ii) of Equation 25 may be changed to another similar equation, such as Equation 26 in order to reduce implementation complexity. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ii 
                         ) 
                       
                       ⁢ 
                          
                       
                         3 
                         · 
                         
                           2 
                           
                             k 
                             - 
                             1 
                           
                         
                       
                     
                     ≤ 
                     Z 
                     &lt; 
                     
                       2 
                       
                         k 
                         + 
                         1 
                       
                     
                   
                   ⁢ 
                     
                   
                     
                       e 
                       
                         i 
                         , 
                         j 
                       
                       
                         ( 
                         Z 
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               e 
                               
                                 i 
                                 , 
                                 j 
                               
                               
                                 ( 
                                 
                                   S 
                                   ⁢ 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                    
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     
                                       S 
                                       ⁢ 
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                               &lt; 
                               0 
                             
                           
                         
                         
                           
                             
                               
                                 ⌊ 
                                 
                                   
                                     e 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                     
                                       ( 
                                       
                                         S 
                                         ⁢ 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   
                                     2 
                                     
                                       
                                         k 
                                         s 
                                       
                                       - 
                                       k 
                                     
                                   
                                 
                                 ⌋ 
                               
                               + 
                               
                                 ⌊ 
                                 
                                   
                                     e 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                     
                                       ( 
                                       
                                         S 
                                         ⁢ 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   
                                     2 
                                     
                                       
                                         k 
                                         s 
                                       
                                       - 
                                       k 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ⌋ 
                               
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                    
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     
                                       S 
                                       ⁢ 
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                               ≥ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   26 
                 
               
             
           
         
       
     
     Another embodiment of supporting lifting using a plurality of exponent matrices will be described. 
     It is assumed that values available as a row block size or a column block size are given as Equation 27. 
         A,A+ 1, A+ 2, A+ 3, . . . ,2 A− 2,2 A− 1 
       2 A, 2( A+ 1),2( A+ 2), . . . ,2(2 A− 2),2(2 A− 1) 
       4 A, 4( A+ 1),4( A+ 2), . . . ,4(2 A− 2),4(2 A− 1) 
       . . . 
       2 S   A, 2 S ( A+ 1),2 S ( A+ 2), . . . ,2 S (2 A− 2),2 S (2 A− 1)  Equation27
 
     In Equation 27, A and S are any positive integers. The block sizes are classified into A sets, as expressed as Equation 28. 
         X   i ={( A+i ),2( A+i ),2 2 ( A+i ) . . . ,2 S ( A+i )}, i= 0,1,2, . . . , A− 1   Equation 28
 
     In a set X i , integers are in a factor or multiple relationship. Therefore, it is noted that one exponent matrix may be generated by applying the lifting scheme of the related art for the block sizes of each set X i . In other words, all exponent matrices supporting the block sizes included in the set X i  may be generated out of a single exponent matrix. Therefore, once a total of A exponent matrices are obtained, exponent matrices supporting the block sizes included in the A sets, X i (i=0, . . . , A−1) may be generated. In general, A exponent matrices may be converted to exponent matrices for a total of A*S block sizes. 
     While it has been described that both a supported minimum block size and the number of elements in each of the sets into which block sizes are classified are equally A in the above embodiment of the present disclosure, this should not be construed as limiting the present disclosure. 
     Accordingly, once a transmitter and a receiver determine a block size according to an information word size, they determine a block size set to which the block size belongs (an exponent matrix to be used), and apply lifting using the exponent matrix defined for the block size set, thereby achieving an exponent matrix suitable for the block size. 
     For example, if block sizes are classified as described in Equation 28 and a block size Z is determined according to a given information word size in the transmitter and the receiver, non-negative integers b and i satisfying Z=2 b (A+i) for a given minimum block size A are obtained and b th  lifting is applied using an i th  exponent matrix, thus achieving an exponent matrix or a parity check matrix corresponding to the block size Z. For reference, the non-negative integers b and i satisfying Z=2 b (A+i) may be obtained in various manners. For example, b may be easily obtained by setting b=x−1 for a first x satisfying Z/2 X &lt;A, while the determined Z value is sequentially divided by 2. After b is obtained, i may be easily obtained by Z/2 b −A=i. 
     As described before, the foregoing method needs a plurality of exponent matrices, thus increasing complexity slightly. However, the method advantageously improves performances because lifting almost optimum for an information word length belonging to each set X i  may be applied. 
     Another embodiment of supporting lifting using a plurality of exponent matrices will be described. 
     To get a plurality of exponent matrices according to a block size, the block size Z may be classified according to an integer type. For example, the block size Z may be expressed as Z=qa+b where q, a, and b are all non-negative integers. For q=4, block sizes may be classified as enumerated in Equation 29. 
         1 , 2 , 3 , 4 , 
         5 , 6 , 7 , 8 , 
       . . . 
         4   a+ 1,4 a+ 2,4 a+ 4,4( a+ 1) 
       . . .  Equation 29
 
     The block sizes may be classified into a plurality of sets described in Equation 30. For example, the block sizes Z are grouped into one or more sets each including 4 block sizes, and each set is mapped to a base matrix (for example, an exponent matrix). 
         X   b   ={x|x=q ( a− 1)+ b,a= 1,2 . . . }, b= 1,2, . . . , q   Equation 30
 
     It is assumed that the block sizes Z are classified by Equation 30, each set X b  has a specific exponent matrix, and b exponent matrices are given as E(H S     b   )=(e i,j   (S     b     ) ), for sequence conversion. Notably, it is assumed that the same mother matrix corresponds to the exponent matrices. Conversion of different exponent matrices may be applied according to Z values, as expressed as Equation 31 or Equation 32. 
     
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               
                                 S 
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     b 
                                   
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               mod 
                               ⁡ 
                               ( 
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     b 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             , 
                             
                               
                                 q 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       k 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               + 
                               b 
                             
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     
                                       S 
                                       b 
                                     
                                     ) 
                                   
                                 
                               
                               ≥ 
                               0 
                             
                             , 
                             
                               Z 
                               ∈ 
                               
                                 X 
                                 b 
                               
                             
                             , 
                             
                               
                                 2 
                                 k 
                               
                               ≤ 
                               a 
                               &lt; 
                               
                                 2 
                                 
                                   k 
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   31 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     e 
                     
                       i 
                       , 
                       j 
                     
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             e 
                             
                               i 
                               , 
                               j 
                             
                             
                               ( 
                               
                                 S 
                                 b 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                                   
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     b 
                                   
                                   ) 
                                 
                               
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             ⌊ 
                             
                               
                                 e 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 
                                   ( 
                                   
                                     S 
                                     b 
                                   
                                   ) 
                                 
                               
                               
                                 2 
                                 
                                   
                                     k 
                                     s 
                                   
                                   - 
                                   k 
                                 
                               
                             
                             ⌋ 
                           
                         
                         
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   e 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                   
                                     ( 
                                     
                                       S 
                                       b 
                                     
                                     ) 
                                   
                                 
                               
                               ≥ 
                               0 
                             
                             , 
                             
                               Z 
                               ∈ 
                               
                                 X 
                                 b 
                               
                             
                             , 
                             
                               
                                 2 
                                 k 
                               
                               ≤ 
                               a 
                               &lt; 
                               
                                 2 
                                 
                                   k 
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   32 
                 
               
             
           
         
       
     
     While the foregoing lifting method has been described on the assumption that lifting is applied to an entire exponent matrix corresponding to a parity check matrix, for convenience of description, lifting may be applied to a part of the exponent matrix. For example, a partial matrix corresponding to parity bits of a parity check matrix generally has a special structure, for efficient encoding. In this case, lifting may cause a change in an encoding method or complexity. Therefore, to maintain the same encoding method or the same complexity, lifting may not be applied or lifting different from lifting applied to a part of an exponent matrix corresponding to information word bits may be applied to a part of the exponent matrix corresponding to a parity of a parity check matrix. In other words, lifting applied to a sequence corresponding to information word bits, and lifting applied to a sequence corresponding to parity bits may be set differently for an exponent matrix. Under circumstances, lifting may not be applied to the whole or part of the sequence corresponding to the parity bits, and thus the sequence may be used fixedly without sequence conversion. 
     Information of a parity check matrix to be used for encoding and decoding may be generated by performing the foregoing lifting method in the same manner in a transmitter and a receiver. For example, if both the transmitter and the receiver are aware of the same exponent matrix and the same lifting method, once the receiver acquires information about Z used in the transmitter, the receiver may acquire information about an exponent matrix used by the transmitter by converting the stored exponent matrix. Although the transmitter may directly transmit information about the Z value, the receiver may determine the Z value in a different manner. 
     If the number of information word column blocks is Kb, a supported information word size is K b Z in the lifting method of the present disclosure. For example, the granularity of supported information words is K b  bits. Thus, to support a smaller information word granularity than K b  bits, a method, such as shortening may be used. For example, if an information word length to be supported is K, a Z value satisfying K b Z≥K is first determined. When shortening is needed, an information word is shortened by K b Z−K bits. Thus, a K-bit information word may be applied easily. Accordingly, the maximum length of shortened bits may be K b −1. 
     This operation may be summarized briefly as follows. 
     Step 1) Z is determined by Equation 33. 
         Z=┌K/K   b ┐  Equation 33
 
     Step 2) An exponent matrix of a parity check matrix supporting a length K b Z is generated by applying lifting with respect to Z. 
     Step 3) In LDPC encoding/decoding based on the exponent matrix, a shortened information word of a size K b Z−K is considered. 
     Meanwhile, an LDPC code may be decoded by an iterative decoding algorithm based on a sum-product algorithm on the bipartite graph illustrated in  FIG.  2   , and the sum-product algorithm is a form of message passing algorithm. 
     With reference to  FIGS.  5 A and  5 B , a general message passing operation used for LDPC decoding will be described below. 
       FIGS.  5 A and  5 B  illustrate message passing operations at any check node and variable node, for LDPC decoding according to various embodiments of the present disclosure. 
     Referring to  FIG.  5 A , a check node m  500 , and a plurality of variable nodes  510 ,  520 ,  530 , and  540  connected to the check node m  500  are shown. T n′,m  is a message passed from the variable node n′  510  to the check node m  500 , and E n,m  is a message passed from the check node m  500  to the variable node n  530 . A set of all variable nodes connected to the check node m  500  is defined as N(m), and a set obtained by excluding the variable node n  530  from the set N(m) is defined as N(m)\n. 
     In this case, a message update rule based on the sum-product algorithm may be expressed as Equation 34. 
       | E   n,m |=Φ[Σ n′ϵN(m)\n Φ(| T   n′,m |]
 
       Sign( E   n,m )=Π n′ϵN(m)\n sign( T   n′,m)   Equation 34
 
     In Equation 34, Sign(E n,m ) represents the sign of the message E n.m , and |E n,m | represents the magnitude of the message E n,m . Meanwhile, a function Φ(x) may be given by Equation 35. 
     
       
         
           
             
               
                 
                   
                     Φ 
                     ⁡ 
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     - 
                     
                       log 
                       ⁡ 
                       ( 
                       
                         tanh 
                         ⁡ 
                         ( 
                         
                           X 
                           2 
                         
                         ) 
                       
                       ) 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   35 
                 
               
             
           
         
       
     
     Meanwhile, referring to  FIG.  5 B , a variable node x  550  and a plurality of check nodes  560 ,  570 ,  580 , and  590  connected to the variable node x  550 . E y′,x  represents a message passed from the check node y′  560  to the variable node x  550 , and T y,x  represents a message passed from the variable node x  550  to the check node y  580 . A set of all variable nodes connected to the variable node x  550  is defined as M(x), and a set obtained by excluding the check node y  530  from the set M(x) is defined as M(x)\y. 
     In this case, a message update rule based on the sum-product algorithm may be expressed as Equation 36. 
         T   y,x   =E   x +Σ y′ϵM(x)\y   E   y′,x   Equation 36
 
     In Equation 36, Ex represents an initial message value of the variable node x. 
     A bit value of the node x may be decided by Equation 37. 
     
       
         
           
             
               
                 
                   
                     P 
                     x 
                   
                   = 
                   
                     
                       E 
                       x 
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             y 
                             ′ 
                           
                           ∈ 
                           
                             M 
                             ⁡ 
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                         E 
                         
                           
                             y 
                             ′ 
                           
                           , 
                           x 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   37 
                 
               
             
           
         
       
     
     In this case, a coded bit corresponding to the node x may be decided according to P x . 
     The method described above with reference to  FIGS.  5 A and  5 B  is a general decoding method and thus will not be described herein. However, aside from the method illustrated in  FIGS.  5 A and  5 B , other methods may be used in determining a message value passed between a variable node and a check node, as disclosed in Frank R. Kschischang, Brendan J. Frey, and Hans-Andrea Loeliger, “Factor Graphs and the Sum-Product Algorithm,” IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 2, February 2001, pp 498-519). 
       FIG.  6    is a block diagram of an LDPC encoder according to an embodiment of the present disclosure. 
     Referring to  FIG.  6   , K ldpc  bits may form Kwa LDPC information bits I=(i 0 ,i 1 , . . . , i Kldpc−1 ) for an LDPC encoder  610 . The LDPC encoder  610  may generate an LDPC codeword including N ldpc  bits, Λ=(c 0 ,c 1 , . . . , c Nkldpc−1 )=(i 0 ,i 1 , . . . , i Kldpc−1 ,p 0 ,p 1 , . . . ,p Nldpc−Kldpc−1 ) by systematically LDPC-encoding the Kwa LDPC information word bits. 
     As described in Equation 1, LDPC encoding involves an operation for determining a codeword in such a manner that the product between the LDPC codeword and a parity check matrix may be a zero vector. The parity check matrix of the present disclosure is in the form as defined by Equation 3 and Equation 4. Hereinbelow, a description will be given of a method for designing a parity check matrix and a method for using the same in order to address the length compatibility issue of the lifting method of the related art. 
     It is assumed that there are a mother matrix H 1  of a parity check matrix and an exponent matrix E(H 1 )=(e ij   (1) ) of the parity check matrix. Since the mother matrix H 1  obviously includes only 0s and 1s as its entries, the exponent matrix E(H 1 ) includes only −1s representing zero matrices or 0s representing identity matrices. The following is a modified modulo-based lifting method according to the present disclosure. 
     For convenience of description, ranges of numbers for lifting are defined as 2 k ≤Z&lt;2 k+1 , (k=0, 1, 2 . . . ). A maximum Z value is Z max . 
     Step 1). If e i,j   (1) =−1, e i,j   (Z) =−1 Z=2, 3, . . . , Z max  for E(H z )=(e i,j   (Z) ). 
     Step 2) k=1. E(H 2     k   )=e i,j   (2     k     ) , E(H 2     k     +1 )=e i,j   (2     k     +1 ), (H 2     k     +2 )=e i,j   (2     k     +2) , E(H 2     k+1     −11 )=e i,j   (2     k+1     −1)  are set so that the following conditions may be satisfied. 
     Condition 1: If e i,j   (2     k     −1) ≠1, e i,j   (2     k     )  is determined to be one of e i,j   (2     k     −1)  and e i,j   (2     k     −1) +2 k−1 . 
     Condition 2: For every i and j, each exponent e i,j   (2     k     ) , e i,j   (2     k     +1) , . . . , e i,j   (2     k+1     −1)  satisfies e ij   (2     k     ) =e ij   (2     k     +1) =e ij   (2     k+1     −1) . 
     Condition 3: If k&gt;A, a Tanner graph for each parity check matrix H 2     k   , H 2     k+1   , . . . , H 2     k+1     −1  does not include a short cycle between variable nodes (bit nodes) with orders of 2 and 3 (a short cycle is a predetermined value. Although the short cycle typically refers to a cycle of length 4 or 6, it may have a longer length according to the size of a given mother matrix to apply lifting. A is a constant determined according to the size of the given mother matrix to apply lifting). 
     Condition 4: If the same cycle is generated for the exponents e ij   (2     k−1)    and e ij   (2     k−1)   +2 k−1  of Condition 1, a case with a larger sum of the orders of variable nodes forming the cycle is selected. 
     Step 3) k=k+1 is applied, and Step 2) is repeated until k=└log 2  Z max ┘. 
     The method is a simple design method for a case where a modulo-based lifting method is applied. If a flooring lifting method is applied, Condition 1 and Condition 4 of Step 2) are represented as follows. 
     Condition 1′: If e i,j   (2     k−1)   ≠1, the value of e i,j   (2     k     )  is determined to be one of 2e i,j   (2     k−1)    and 2e i,j   (2     k−1)   +1. 
     Condition 4′: If the same cycle is generated for the exponents 2e i,j   (2     k−1)    and 2e i,j   (2     k−1)   +1 of Condition 1, a case with a larger sum of the orders of variable nodes included in the cycle is selected. 
       FIG.  3    is a block diagram illustrating the detailed structure of a transmitter according to an embodiment of the present disclosure. 
     Referring to  FIG.  3   , the transmitter  300  may include the segmenter  310 , the zero padder  320 , the LDPC encoder  330 , the rate matcher  340 , and the modulator  350  in order to process input bits of a variable length. 
     The components illustrated in  FIG.  3    encode and modulate input bits of a variable length. When needed, a component may be omitted from, modified in, or added to the components illustrated in  FIG.  3   . 
     The LDPC encoder  330  illustrated in  FIG.  3    may perform an operation of the LDPC encoder  500  illustrated in  FIG.  5   . 
     Meanwhile, the transmitter  300  may determine necessary parameters (for example, an input bit length, a modulation and code rate (ModCod), a parameter for zero padding, a code rate/codeword length of an LDPC code, a parameter for interleaving, a parameter for repetition, a parameter for puncturing, and a modulation scheme), encode input bits based on the determined parameters, and transmit the coded bits to the receiver  400 . 
     If the variable number of input bits is larger than a predetermined value, the input bits may be segmented so that each segment may have a length equal to or less than the predetermined value. Each segmented block may correspond to one LDPC code block. However, if the number of input bits is equal to or less than the predetermined value, the input bits are not segmented. The input bits may correspond to one LDPC code block. 
     Now, a detailed description will be given of a segmentation method. 
     The segmenter  310  segments input bits. In the method for segmenting input bits in the segmenter  311 , B input bits b 0 , b 1 , b 2 , b 3 , . . . , b B−1  (B&gt;0) are input to the segmenter  310 . If B is larger than a predetermined value being a maximum number of input bits for encoding, K max , the input bits are segmented. The maximum number of input bits for encoding, K max  is determined according to a code rate, as listed in Table 1. 
     
       
         
           
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Code Rate 
                 K max   
                 K min   
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 ¼ 
                 2048 
                 8 
               
               
                 ½ 
                 4096 
                 16 
               
               
                 ¾ 
                 6144 
                 24 
               
               
                 ⅞ 
                 7168 
                 28 
               
               
                   
               
            
           
         
       
     
     Table 1 may be changed according to a system, and Table 2 may also be made. 
     
       
         
           
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Code Rate 
                 K max   
                 K min   
               
               
                   
               
             
            
               
                  8/9 
                 3072 
                 384 
               
               
                  6/9 
                 2304 
                 288 
               
               
                  4/9 
                 1536 
                 192 
               
               
                 ⅓ 
                 3072 
                 384 
               
               
                   
               
            
           
         
       
     
     If the number of segment blocks is C, the number of bits to be segmented is determined as follows. 
     If input bits are segmented into at least two segments, the two segments of input bits are separately LDPC-encoded, producing at least two forward error correction (FEC) frames. Accordingly, at least two FEC frames are required to transmit the input bits. 
     Therefore, the segmenter  310  may calculate the number C of FEC frames by Equation 38. 
         C=┌B /( K   max   −L )┐  Equation 38
 
     In Equation 38, ┌x┐ represents a smallest integer equal to or larger than x. 
     The following representation is possible. 
     if B≤K max  
         L=0   Number of code blocks: C=1       

     B′=B 
     else
         L=24   Number of code blocks: C=┌B/(K max −L)┘.   B′=B+C·L       

     end if 
     L represents the number of parity bits of a CRC code. The segment blocks are CRC-encoded separately. Therefore, the number of input bits, B is changed to B′ in consideration of the number of CRC bits. 
     To make the segment blocks have the same number of bits, &lt;Null&gt; bits may be inserted. The number of &lt;Null&gt; bits and the number of bits in each block may be calculated in the following manner. 
     Let an r th  block of output bits of the segmenter  310  be denoted by C r0 , C r1 , C r2 , C r3 , . . . , c r(K     r     −1)  where K r  is the number of bits in the r th  block. 
     The number of bits in each block is determined as follows. To make the lengths of all blocks equal, &lt;Null&gt; bits are inserted in the last block. For example, the segmenter  310  may fill F &lt;Null&gt; bits (that is, bits being zeroes). Accordingly, F&lt;Null&gt; bits may be filled as illustrated in  FIG.  7   . 
       FIGS.  7  and  8    illustrate structures of transport blocks according to various embodiments of the present disclosure. 
     Referring to  FIGS.  7  and  8   , since the length of a padding field is calculated and as many &lt;Null&gt; bits as the calculated length are padded in a padding part, input bits may be segmented into a plurality of blocks each having an equal number of bits, that is, K r  bits. 
     It is possible to pad &lt;Null&gt; bits at the start or end of Segmentation C in  FIG.  7   . 
     It is also possible to pad &lt;Null&gt; bits at the start or end of Segmentation C in  FIG.  8   . 
     if C=1,
         Kr=B′       

     else
         The number of filler bits F   F=ceiling(B′/C)×C−B′   B″=B′+F       

     Kr=B″/C 
     The filler bits &lt;NULL&gt; shall be inserted at the end of the last block (or at the beginning of the first block). 
     The position of &lt;NULL&gt; bits may be changed. For example, the &lt;Null&gt; bits may be inserted at the end or start of the last segment block. In the above, ceiling(x) represents a smallest integer equal to or larger than x. For example, ceiling(1.5)=2. 
     For k=Kr −F −1−L to Kr−1−L, 
     c (C−1)k =&lt;NULL&gt; 
     end for 
     end if 
     If the number of segment blocks is 2 or large, each segment is CRC-encoded. A CRC code may be omitted according to a transmission system. 
     for r=0 to C−1 
     k=0
         while k&lt;Kr −L
           c rk =b s //segmented bits are mapped.   k=k+1   
            s=s+1
           end while   
           if C &gt;1       

     CRC bits P r0 , P r1 , p r2 , . . . , p r(L−1)  are added to the bits of the r th  segment block c r0 ,c r1 ,c r2 ,c r3 , . . . ,c r(K     r     −L−1)  and mapped to c rk  as follows. 
     For CRC calculation, it is assumed that filler bits, if present, have the value 0. 
     while k&lt;K r  
         c rk =p r(k+L−K     r     )      k=k+1   end while       

     end if
         k=0       

     end for 
     Specifically, if C is larger than 1 as illustrated in  FIG.  7   , the segmenter  310  may group every K r  input bits into one block, thus segmenting total input bits into C blocks. The blocks of input bits are individually CRC-encoded. As a result of encoding, the number of input bits for the zero padder  320  of the transmitter  300  may be K=(K r +L) where L is the parity length of a CRC code, 24. 
     However, if L 1  detail signaling is not segmented, K=B. The segmented blocks may be encoded in the following procedure. 
     The zero padder  320  pads zero bits. Specifically, in the case of an LDPC code, a predetermined number of LDPC information word bits according to a code rate and a code length are required. This, if the number of bits in a segment block is smaller than the number of LDPC information word bits, the zero padder  320  may generate the predetermined number of LDPC information word bits by padding zero bits for LDPC encoding and output the LDPC information word bits to the LDPC encoder  330 . On the other hand, if the number of bits in one block received from the segmenter  310  is equal to the number of LDPC information word bits, the zero padder  320  does not perform zero padding. 
     Because the zero padder  320  pads zero bits for LDPC encoding, zero bits padded for shortening are not transmitted to the receiver  400 . 
     Specifically, Z is determined based on K ldpc_b  defined according to a code rate. Z is the size of a sub-matrix in a parity check matrix of an LDPC code, and K ldpc_b  is the number of column groups in an information word part of the parity check matrix. Therefore, the maximum of values obtained by dividing the length K of input bits by K ldpc_b is determined to be a sub-matrix size, thereby minimizing the number of &lt;Null&gt; bits. The submatrix size Z may be any integer between a minimum value and a maximum value. Hereinbelow, ZP represents the number of &lt;Null&gt; bits.    
         Z=┌K/K   ldpc_b ┐
 
     
       
      
       ZP=Z×K 
       ldpc_b 
       −K  
      
     
         K   ldpc   =Z×K   ldpc_b   Equation 39
 
     In Equation 39, the values of K ldbp_b  are listed in [Table 3] according to the code rates of 1/4, 1/2, 3/4, and 7/8. 
     
       
         
           
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                   
                 Code Rate 
                 K ldpc _b 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 ¼ 
                 8 
               
               
                   
                 ½ 
                 16 
               
               
                   
                 ¾ 
                 24 
               
               
                   
                 ⅞ 
                 28 
               
               
                   
               
            
           
         
       
     
     The parameters described in Table 3 may be changed according to a system, and Table 4 is also available. 
     
       
         
           
               
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                 Code Rate (R) 
                 K ldpc _b 
               
               
                   
               
             
            
               
                   
                  8/9 
                 32 
               
               
                   
                 ⅔ 
                 24 
               
               
                   
                  4/9 
                 16 
               
               
                   
                 ⅓ 
                 32 
               
               
                   
               
            
           
         
       
     
     In the foregoing embodiment of the present disclosure, all integers from the minimum value to the maximum value are available as the submatrix size, Z. If Z is a multiple of D, Z may be determined for the number of input bits, K as follows. D may be 12. 
         Z=┌K/K   ldpc_b   ×D┐×D    Equation 40
 
     //// zero padding for shortening 
     For j=0 to ZP−1 
     i k =&lt;NULL&gt; 
     end for 
     For j=ZP to K lpdc −1 
     i k =c {k−ZP}   
     end for 
     The &lt;NULL&gt; bits may be padded at a specific position in information word bits. For example, the &lt;NULL&gt; bits may be positioned at the end of the information word. 
     In another example, as &lt;NULL&gt; bits are padded and interleaved, the padded bits may be distributed uniformly across bit blocks corresponding to the column blocks of the parity check matrix. 
     //// zero padding for shortening 
     For j=0 to ZP−1 
     x k =&lt;NULL&gt; 
     end for 
     For j=ZP to K kdpc −1 
     x k =c {k−ZP}   
     end for 
     /// interleaving the information bits 
     For j=0 to K kdpc_b −1 
     For k=0 to Z−1
         i {j·Z+k} =x {k·Kldpc_b+j}         

     end for k 
     end for j 
     Z is a submatrix size calculated by Equation 39 or Equation 40. K ldpc_b  is the number of column blocks in an information word part of a parity check matrix, given as Table 3 or Table 4. 
     More specifically, as illustrated in  FIG.  8   , the segmenter  310  inserts &lt;Null&gt; bits in a segment block (including a CRC), to thereby making the length of the segment block equal to the information word length of an LDPC code. The smallest of integers equal to or larger than the number K of input bits among multiples of the number of column blocks in the parity check matrix of the LDPC code is selected as the information word length of the LDPC code. For example, K ldpc =┌K/K 1dpc_b ┐×K kdpc_b . 
     Now, a detailed description is given of the LDPC encoder  330  of the transmitter  300  or the LDPC encoder  500  of  FIG.  5   . 
     The LDPC encoder  330  LDPC-encoders outputs bits of the zero padder  320 . 
     Specifically, the LDPC encoder  330  may generate LDPC parity bits by LDPC-encoding LDPC information word bits received from the zero padder  320 , and output an LDPC codeword including the LDPC information word bits and the LDPC parity bits to the rate matcher  340 . 
     For example, K ldpc  bits output from the zero padder  320  may form K ldpc  LDPC information word bits I=(i 0 ,i 1 , . . . , i L     ldpc     −1 ) for the LDPC encoder  330 . 
     The LDPC encoder  330  may generate an LDPC codeword with N ldpc  bits, Λ=(c 0 ,c 1 , . . . , c N     ldpc     −1 )=(i 0 ,i 1 , . . . , i Kldpc−1 ,p 0 ,p 1 , . . . ,p Nldpc−Kldpc−1 ) by systematically LDPC-encoding the Kwa LDPC information word bits. 
     According to the present disclosure, parameters for a parity check matrix are listed in Table 5. Code Rate means the code rate of an LDPC code, N ldpc_b  represents the number of column blocks of the parity check matrix, equal to n in Equation 4, K ldpc_b  represents the number of column blocks in an information word part of the parity check matrix, equal to (n−m), and N parity_b  represents the number of column blocks or row blocks in a parity part of the parity check matrix. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                   
                 Code Rate 
                 N ldpc _b 
                 K ldpc _b 
                 N parity _b 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 ¼ 
                 32 
                 8 
                 28 
               
               
                   
                 ½ 
                 32 
                 16 
                 16 
               
               
                   
                 ¾ 
                 32 
                 24 
                 8 
               
               
                   
                 ⅞ 
                 32 
                 28 
                 4 
               
               
                   
               
            
           
         
       
     
     For the size of a circulant permutation matrix being a submatrix, Z (L×L=Z*Z) in Equation 3 is 256, Table 6 to Table 9 list the exponent of each circulant permutation matrix, a i,j (0≤i&lt;N ldpx −K ldpc , 0≤j&lt;N ldpc ). Table 6, Table 7, Table 8, and Table 9 describe parity check matrices of LDPC codes with code rates of 7/8, 3/4, 1/2, and 1/4, respectively. If the size of a circulant permutation matrix, Z is equal to or less than 255, the exponents of a parity check matrix, a i,j (Z) is determined by Equation 41. 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       
                         i 
                         , 
                         j 
                       
                     
                     ( 
                     z 
                     ) 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 a 
                                 
                                   i 
                                   , 
                                   j 
                                 
                               
                               ⁢ 
                                  
                               if 
                               ⁢ 
                                  
                               
                                 a 
                                 
                                   i 
                                   , 
                                   j 
                                 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               
                                 mod 
                                 ⁢ 
                                     
                                 
                                   ( 
                                   
                                     
                                       a 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                     
                                     , 
                                     
                                       2 
                                       k 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                     
                                 for 
                                 ⁢ 
                                     
                                 
                                   2 
                                   k 
                                 
                               
                               ≤ 
                               z 
                               &lt; 
                               
                                 2 
                                 
                                   k 
                                   + 
                                   1 
                                 
                               
                             
                             , 
                             
                               0 
                               ≤ 
                               k 
                               ≤ 
                               7 
                             
                             , 
                                
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   a 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                 
                               
                               &gt; 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   41 
                 
               
             
           
         
       
     
     In Equation 41, a i,j  represents an entry in an i th  row and a j th  column in [Table 6] to [Table 9], which is the exponent of a circulant permutation matrix in an i th  row and a j th  column, for a circulant permutation matrix size of 256. a i,j  (Z) represents the exponent of a circulant permutation matrix in an i th  row and a j th  column, for a circulant permutation matrix size of 255 or less (0≤Z&lt;256). 
     Specifically, for 2 0 ≤Z&lt;2 1 , if a i,j  is −1 or 0, a i,j  (Z) is a i,j  and if a i,j  is larger than 0, a i,j (Z) is mod(a i,j , 1). This means that a circulant matrix other than a zero matrix is 1 representing a 1×1 circulant matrix. 
     Specifically, for 2 1 ≤Z&lt;2 2 , if a i,j  is −1 or 0, a i,j (Z) is a i,j , and if a i,j  is larger than 0, a i,j (Z) is mod(a i,j , 2 1 ). 
     Specifically, for 2 2 ≤Z&lt;2 3 , if a i,j  is −1 or 0, a i,j (Z) is a i,j , and if a i,j  is larger than 0, a i,j (Z) is mod(a i,j , 2 2 ). 
     Specifically, for 2 7 ≤Z&lt;2 8 , if a i,j  is −1 or 0, a i,j (Z) is a i,j , and if a i,j  is larger than 0, a i,j (Z) is mod(a i,j , 2 7 ). 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 6 
               
               
                   
               
             
            
               
                 234 
                 32 
                 12 
                 24 
                 195 
                 162 
                 81 
                 237 
                 44 
                 251 
                 205 
                 172 
                 141 
                 10 
                 194 
                 −1 
               
               
                 241 
                 251 
                 105 
                 139 
                 112 
                 28 
                 22 
                 255 
                 227 
                 68 
                 31 
                 0 
                 182 
                 177 
                 131 
                 162 
               
               
                 252 
                 134 
                 245 
                 228 
                 250 
                 25 
                 252 
                 204 
                 184 
                 21 
                 94 
                 249 
                 10 
                 45 
                 63 
                 105 
               
               
                 155 
                 5 
                 75 
                 14 
                 70 
                 231 
                 227 
                 26 
                 101 
                 246 
                 161 
                 3S 
                 88 
                 123 
                 232 
                 32 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 71 
                 27 
                 −1 
                 210 
                 105 
                 0 
                 −1 
                 147 
                 78 
                 153 
                 178 
                 84 
                 1 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 169 
                 61 
                 −1 
                 211 
                 100 
                 92 
                 132 
                 −1 
                 174 
                 181 
                 −1 
                 0 
                 0 
                 0 
                 −1 
               
               
                   
                 99 
                 −1 
                 91 
                 44 
                 88 
                 −1 
                 101 
                 72 
                 47 
                 −1 
                 48 
                 79 
                 −1 
                 1 
                 0 
                 0 
               
               
                   
                 1 
                 184 
                 16 
                 192 
                 −1 
                 161 
                 80 
                 −1 
                 1 
                 168 
                 −1 
                 128 
                 1 
                 1 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 7 
               
               
                   
               
             
            
               
                 −1 
                 180 
                 175 
                 225 
                 −1 
                 22 
                 −1 
                 −1 
                 −1 
                 153 
                 −1 
                 −1 
                 86 
                 188 
                 −1 
                 147 
               
               
                 246 
                 −1 
                 14 
                 127 
                 61 
                 −1 
                 −1 
                 −1 
                 −1 
                 50 
                 13 
                 −1 
                 66 
                 159 
                 −1 
                 160 
               
               
                 245 
                 −1 
                 −1 
                 −1 
                 4 
                 −1 
                 152 
                 247 
                 108 
                 −1 
                 70 
                 174 
                 −1 
                 −1 
                 83 
                 −1 
               
               
                 −1 
                 231 
                 −1 
                 −1 
                 −1 
                 99 
                 249 
                 54 
                 144 
                 −1 
                 −1 
                 56 
                 −1 
                 −1 
                 193 
                 −1 
               
               
                 234 
                 32 
                 12 
                 24 
                 195 
                 162 
                 81 
                 237 
                 −1 
                 −1 
                 205 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 241 
                 251 
                 105 
                 139 
                 112 
                 28 
                 22 
                 255 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 131 
                 −1 
               
               
                 252 
                 134 
                 245 
                 228 
                 250 
                 205 
                 252 
                 204 
                 184 
                 21 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 105 
               
               
                 155 
                 5 
                 75 
                 14 
                 70 
                 231 
                 227 
                 26 
                 −1 
                 −1 
                 −1 
                 35 
                 88 
                 123 
                 −1 
                 −1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 158 
                 −1 
                 203 
                 −1 
                 218 
                 −1 
                 68 
                 −1 
                 1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 141 
                 4 
                 −1 
                 75 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 77 
                 −1 
                 55 
                 −1 
                 207 
                 −1 
                 103 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 120 
                 26 
                 92 
                 −1 
                 −1 
                 168 
                 −1 
                 106 
                 −1 
                 −1 
                 −1 
                 0 
                 1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 27 
                 −1 
                 210 
                 −1 
                 −1 
                 −1 
                 147 
                 0 
                 −1 
                 −1 
                 −1 
                 1 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 61 
                 −1 
                 211 
                 100 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 101 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
               
               
                   
                 1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 8 
               
               
                   
               
             
            
               
                 −1 
                 17 
                 243 
                 134 
                 152 
                 155 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 144 
                 −1 
                 −1 
                 −1 
                 254 
                 −1 
                 85 
                 −1 
                 4 
                 −1 
                 −1 
                 249 
                 −1 
                 70 
                 −1 
               
               
                 185 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 161 
                 −1 
                 171 
                 4 
                 167 
                 — 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                 243 
                 −1 
                 58 
                 181 
                 215 
                 −1 
                 −1 
                 −1 
                 −2 
                 — 
                 −1 
                 4 
                 −1 
                 6 
                 −1 
                 −1 
               
               
                 234 
                 −1 
                 −1 
                 −1 
                 195 
                 162 
                 81 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 54 
                 −1 
                 −1 
                 −1 
               
               
                 241 
                 251 
                 −1 
                 139 
                 112 
                 4 
                 −1 
                 −1 
                 −1 
                 — 
                 −1 
                 −1 
                 −1 
                 249 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 228 
                 −1 
                 −1 
                 252 
                 −1 
                 184 
                 21 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 105 
               
               
                 −1 
                 5 
                 75 
                 14 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 246 
                 −1 
                 −1 
                 88 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 180 
                 175 
                 225 
                 −1 
                 22 
                 −I 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −I 
                 −1 
                 −1 
               
               
                 246 
                 −1 
                 14 
                 127 
                 61 
                 −1 
                 −1 
                 −1 
                 −1 
                   
                 −1 
                 −1 
                 4 
                 −1 
                 4 
                 160 
               
               
                 245 
                 −1 
                 −1 
                 −1 
                 4 
                 −1 
                 −1 
                 247 
                 −1 
                 −1 
                 70 
                 174 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 231 
                 −1 
                 −1 
                 −1 
                 99 
                 −1 
                 −1 
                 144 
                 −1 
                 −1 
                 56 
                 −1 
                 −1 
                 193 
                 −1 
               
               
                 −2 
                 32 
                 12 
                 24 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 251 
                 205 
                 −1 
                 −1 
                 −1 
                 4 
                 −1 
               
               
                 −1 
                 −1 
                 105 
                 −1 
                 −1 
                 28 
                 −1 
                 255 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 131 
                 −1 
               
               
                 252 
                 134 
                 245 
                 −1 
                 250 
                 205 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 155 
                 −1 
                 −1 
                 −1 
                 70 
                 231 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 35 
                 −1 
                 123 
                 −1 
                 −1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −I 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −I 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −I 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −I 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 0 
                 −1 
                 −1 
                 −I 
                 −1 
                 −1 
                 −1 
                 −1 
                 1 
                 0 
                 −1 
                 −I 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 1 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
               
               
                   
                 1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −I 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 9 
               
               
                   
               
             
            
               
                 4 
                 221 
                 32 
                 251 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 187 
                 255 
                 31 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 239 
                 −1 
                 −1 
                 −1 
                 3 
                 243 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 134 
                 −1 
                 61 
                 −1 
                 −1 
                 −1 
                 247 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 −1 
                 121 
                 243 
                 197 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
               
               
                 4 
                 134 
                 123 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
               
               
                 −1 
                 121 
                 121 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
               
               
                 255 
                 −1 
                 223 
                 −1 
                 253 
                 130 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                 −1 
                 17 
                 243 
                 134 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 144 
                 −1 
                 −1 
                 −1 
                 254 
                 −1 
                 85 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 185 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 161 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 248 
                 −1 
                 58 
                 181 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 234 
                 −1 
                 −1 
                 −1 
                 195 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 241 
                 251 
                 −1 
                 139 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 228 
                 −1 
                 −1 
                 252 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 5 
                 75 
                 14 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 180 
                 175 
                 225 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 246 
                 −1 
                 14 
                 127 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 245 
                 −1 
                 −1 
                 −1 
                 4 
                 −1 
                 −1 
                 247 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 231 
                 −1 
                 −1 
                 −1 
                 99 
                 249 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 32 
                 12 
                 24 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 105 
                 −1 
                 −1 
                 28 
                 −1 
                 255 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 252 
                 134 
                 245 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 155 
                 −1 
                 −1 
                 −1 
                 70 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 1 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     In Table 6, Table 7, Table 8, and Table 9, all of the column permutations of the parity check matrices may be regarded as the same parity check matrix. 
     More specifically, the exponents of the 28 th  column in Table 6 may be changed from [1 0 −1 1] T  to [0 Y −1 0] T , as illustrated in Table 10. Y may be any integer, (Z−1). 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 10 
               
               
                   
               
             
            
               
                 234 
                 32 
                 12 
                 24 
                 195 
                 162 
                 81 
                 237 
                 44 
                 251 
                 205 
                 172 
                 141 
                 10 
                 194 
                 −1 
               
               
                 241 
                 251 
                 105 
                 139 
                 112 
                 28 
                 22 
                 255 
                 227 
                 68 
                 31 
                 0 
                 182 
                 177 
                 131 
                 162 
               
               
                 252 
                 134 
                 245 
                 228 
                 250 
                 205 
                 252 
                 204 
                 184 
                 21 
                 94 
                 249 
                 10 
                 45 
                 68 
                 105 
               
               
                 155 
                 5 
                 75 
                 14 
                 70 
                 231 
                 227 
                 26 
                 101 
                 246 
                 161 
                 35 
                 88 
                 123 
                 232 
                 32 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 71 
                 27 
                 −1 
                 210 
                 105 
                 0 
                 −1 
                 147 
                 78 
                 153 
                 178 
                 84 
                 0 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 169 
                 61 
                 −1 
                 211 
                 100 
                 92 
                 132 
                 −1 
                 174 
                 181 
                 −1 
                 Y 
                 0 
                 0 
                 −1 
               
               
                   
                 99 
                 −1 
                 91 
                 44 
                 88 
                 −1 
                 101 
                 72 
                 47 
                 −1 
                 48 
                 79 
                 −1 
                 −1 
                 0 
                 0 
               
               
                   
                 0 
                 184 
                 16 
                 192 
                 −1 
                 161 
                 80 
                 −1 
                 0 
                 168 
                 −1 
                 128 
                 0 
                 −1 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     More specifically, the exponents of the 24 th  column in Table 7 may be changed from [1 −1 −1 −1 −1 0 −1 −1 1]T to [0 −1 −1 −1 −1 Y −1 −1 0] T , and the exponents of the 28 th  column in Table 7 may be changed from [−1 −1 −1 1 1 −1 −1 −1] to [−1 −1 −1 0 0 −1 −1 −1] T  as illustrated in Table 11. Y may be any integer, (Z−1). 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 11 
               
               
                   
               
             
            
               
                 −1 
                 180 
                 175 
                 225 
                 −1 
                 22 
                 −1 
                 −1 
                 −1 
                 153 
                 −1 
                 −1 
                 86 
                 188 
                 −1 
                 147 
               
               
                 246 
                 −1 
                 14 
                 127 
                 61 
                 −1 
                 −1 
                 −1 
                 −1 
                 50 
                 13 
                 −1 
                 65 
                 159 
                 −1 
                 160 
               
               
                 245 
                 −1 
                 −1 
                 −1 
                 4 
                 −1 
                 152 
                 247 
                 108 
                 −1 
                 70 
                 174 
                 −1 
                 −1 
                 83 
                 −1 
               
               
                 −1 
                 231 
                 −1 
                 −1 
                 −1 
                 99 
                 249 
                 54 
                 144 
                 −1 
                 −1 
                 56 
                 −1 
                 −1 
                 193 
                 −1 
               
               
                 234 
                 32 
                 12 
                 24 
                 195 
                 162 
                 81 
                 237 
                 −1 
                 −1 
                 205 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 241 
                 251 
                 103 
                 139 
                 112 
                 28 
                 22 
                 255 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 131 
                 −1 
               
               
                 252 
                 134 
                 245 
                 228 
                 250 
                 205 
                 252 
                 204 
                 184 
                 21 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 105 
               
               
                 155 
                 5 
                 75 
                 14 
                 70 
                 231 
                 227 
                 26 
                 −1 
                 −1 
                 −1 
                 35 
                 88 
                 123 
                 −1 
                 −1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 158 
                 −1 
                 203 
                 −1 
                 218 
                 −1 
                 68 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 141 
                 4 
                 −1 
                 75 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 77 
                 −1 
                 55 
                 −1 
                 207 
                 −1 
                 103 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 120 
                 26 
                 92 
                 −1 
                 −1 
                 168 
                 −1 
                 106 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 27 
                 −1 
                 210 
                 −1 
                 −1 
                 −1 
                 147 
                 Y 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 61 
                 −1 
                 211 
                 100 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 101 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
               
               
                   
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     More specifically, the exponents of the 16 th  column and 24 th  column in Table 8 may be changed, as illustrated in Table 12. Y may be any integer, (Z−1). 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 12 
               
               
                   
               
             
            
               
                 −17 
                 17 
                 243 
                 134 
                 152 
                 155 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 144 
                 −1 
                 −1 
                 −1 
                 254 
                 −1 
                 85 
                 −1 
                 −1 
                 −1 
                 −1 
                 249 
                 −1 
                 70 
                 −1 
               
               
                 185 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 161 
                 −1 
                 171 
                 −1 
                 167 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                 248 
                 −1 
                 58 
                 181 
                 215 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 6 
                 −1 
                 −1 
               
               
                 234 
                 −1 
                 −1 
                 −1 
                 195 
                 162 
                 81 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 54 
                 −1 
                 −1 
                 −1 
               
               
                 241 
                 251 
                 −1 
                 139 
                 112 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 249 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 228 
                 −1 
                 −1 
                 252 
                 −1 
                 184 
                 21 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 105 
               
               
                 −1 
                 5 
                 75 
                 14 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 246 
                 −1 
                 −1 
                 88 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 180 
                 175 
                 225 
                 −1 
                 22 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 246 
                 −1 
                 14 
                 127 
                 61 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 160 
               
               
                 245 
                 −1 
                 −1 
                 −1 
                 4 
                 −1 
                 −1 
                 247 
                 −1 
                 −1 
                 70 
                 174 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 231 
                 −1 
                 −1 
                 −1 
                 99 
                 −1 
                 −1 
                 144 
                 −1 
                 −1 
                 56 
                 −1 
                 −1 
                 193 
                 −1 
               
               
                 −1 
                 32 
                 12 
                 24 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 251 
                 205 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 105 
                 −1 
                 −1 
                 28 
                 −1 
                 255 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 131 
                 −1 
               
               
                 252 
                 134 
                 245 
                 −1 
                 250 
                 205 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 155 
                 −1 
                 −1 
                 −1 
                 70 
                 231 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 35 
                 −1 
                 123 
                 −1 
                 −1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 Y 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
               
               
                   
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     More specifically, the exponents of the 16 th  column and 24 th  column in Table 9 may be changed, as illustrated in Table 13. Y may be any integer, (Z−1). 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 13 
               
               
                   
               
             
            
               
                 4 
                 221 
                 32 
                 251 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 187 
                 255 
                 35 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 239 
                 −1 
                 −1 
                 −1 
                 2 
                 243 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 134 
                 −1 
                 61 
                 −1 
                 −1 
                 −1 
                 247 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 −1 
                 121 
                 243 
                 197 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
               
               
                 4 
                 134 
                 123 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
                 −1 
               
               
                 −1 
                 121 
                 121 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 −1 
               
               
                 255 
                 −1 
                 223 
                 −1 
                 253 
                 130 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                 −1 
                 17 
                 243 
                 134 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 144 
                 −1 
                 −1 
                 −1 
                 254 
                 −1 
                 85 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 185 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 161 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 248 
                 −1 
                 58 
                 181 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 234 
                 −1 
                 −1 
                 −1 
                 195 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 241 
                 251 
                 −1 
                 139 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 −1 
                 228 
                 −1 
                 −1 
                 252 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 5 
                 75 
                 14 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 180 
                 175 
                 225 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 246 
                 −1 
                 14 
                 127 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 245 
                 −1 
                 −1 
                 −1 
                 4 
                 −1 
                 −1 
                 247 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 231 
                 −1 
                 −1 
                 −1 
                 99 
                 249 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 32 
                 12 
                 24 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 −1 
                 −1 
                 105 
                 −1 
                 −1 
                 28 
                 −1 
                 255 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 252 
                 134 
                 245 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                 155 
                 −1 
                 −1 
                 −1 
                 70 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 1 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 1 
                 −1 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 1 
                 0 
                 −1 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
                 −1 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
                 0 
               
               
                   
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 −1 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     Table 6, Table 7, Table 8, and Table 9 represent parity check matrices for LDPC codes with code rates of 7/8, 3/4, 1/2, and 1/4, respectively. The size of a circulant permutation matrix of each parity check matrix, Z is an integer ranging from 1 to Z max . A set of (Z 0 , Z 1 , . . . , Z l ) may be defined by selecting l values from among the integers from 1 to Z max . For example, (Z 0 , Z 1 , . . . , Z l )={2 0 , 2 1 , . . . , 2 l }. In the set, if i&lt;j, Z i &lt;Z j , Z l ≥Z max . 
     For Z 1 &gt;Z max , if a circulant matrix size of a parity check matrix is Z (1≤Z≤Z max ), the exponent of a circulant matrix in an i th  row block and a j th  column block is determined by Equation 42. 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       
                         i 
                         , 
                         j 
                       
                     
                     ( 
                     Z 
                     ) 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   a 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                 
                                 ( 
                                 
                                   Z 
                                   
                                     l 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                                   
                               if 
                               ⁢ 
                                   
                               
                                 
                                   a 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                 
                                 ( 
                                 
                                   Z 
                                   
                                     l 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               
                                 mod 
                                 ⁢ 
                                    
                                 
                                   ( 
                                   
                                     
                                       
                                         a 
                                         
                                           i 
                                           , 
                                           j 
                                         
                                       
                                       ( 
                                       
                                         Z 
                                         
                                           l 
                                           - 
                                           1 
                                         
                                       
                                       ) 
                                     
                                     , 
                                     
                                       Z 
                                       k 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                    
                                 for 
                                 ⁢ 
                                     
                                 
                                   Z 
                                   k 
                                 
                               
                               ≤ 
                               Z 
                               &lt; 
                               
                                 Z 
                                 
                                   k 
                                   + 
                                   1 
                                 
                               
                             
                             , 
                             
                               0 
                               ≤ 
                               k 
                               &lt; 
                               l 
                             
                             , 
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   
                                     a 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                   
                                   ( 
                                   
                                     Z 
                                     
                                       l 
                                       - 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               &gt; 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   42 
                 
               
             
           
         
       
     
     In Equation 42, mod(x,y)=x mod y, representing the remainder of dividing x by y. 
     Specifically, for Z 0 ≤Z≤Z 1 , if a i,j (Z l-1 ) is −1 or 0, a i,j (Z) is a i,j (Z l-1 ), and if a i,j  (Z i-1 ) is larger than 0, a i,j (Z) is mod(a i,j (Z l-1 ), Z 0 ). 
     For example, Z max =192, l=8, and {Z 0 , Z 1 , . . . , Z 8 }={2 0 , 2 1 , . . . , 2 8 }. 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       
                         i 
                         , 
                         j 
                       
                     
                     ( 
                     Z 
                     ) 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   a 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                 
                                 ( 
                                 
                                   2 
                                   7 
                                 
                                 ) 
                               
                               ⁢ 
                                   
                               if 
                               ⁢ 
                                   
                               
                                 
                                   a 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                 
                                 ( 
                                 
                                   2 
                                   7 
                                 
                                 ) 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               
                                 mod 
                                 ⁢ 
                                    
                                 
                                   ( 
                                   
                                     
                                       
                                         a 
                                         
                                           i 
                                           , 
                                           j 
                                         
                                       
                                       ( 
                                       
                                         Z 
                                         
                                           l 
                                           - 
                                           1 
                                         
                                       
                                       ) 
                                     
                                     , 
                                     
                                       2 
                                       k 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                    
                                 for 
                                 ⁢ 
                                     
                                 
                                   2 
                                   k 
                                 
                               
                               ≤ 
                               Z 
                               &lt; 
                               
                                 2 
                                 
                                   k 
                                   + 
                                   1 
                                 
                               
                             
                             , 
                             
                               0 
                               ≤ 
                               k 
                               &lt; 
                               l 
                             
                             , 
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   
                                     a 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                   
                                   ( 
                                   
                                     2 
                                     7 
                                   
                                   ) 
                                 
                               
                               &gt; 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   43 
                 
               
             
           
         
       
       
         
           
             
               
                 a 
                 
                   i 
                   , 
                   j 
                 
               
               ( 
               z 
               ) 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           a 
                           
                             i 
                             , 
                             j 
                           
                         
                         ⁢ 
                            
                         if 
                         ⁢ 
                            
                         
                           a 
                           
                             i 
                             , 
                             j 
                           
                         
                       
                       ≤ 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           mod 
                           ⁢ 
                               
                           
                             ( 
                             
                               
                                 a 
                                 
                                   i 
                                   , 
                                   j 
                                 
                               
                               , 
                               
                                 2 
                                 k 
                               
                             
                             ) 
                           
                           ⁢ 
                               
                           for 
                           ⁢ 
                               
                           
                             2 
                             k 
                           
                         
                         ≤ 
                         z 
                         &lt; 
                         
                           2 
                           
                             k 
                             + 
                             1 
                           
                         
                       
                       , 
                       
                         0 
                         ≤ 
                         k 
                         &lt; 
                         7 
                       
                       , 
                          
                       
                         
                           if 
                           ⁢ 
                               
                           
                             a 
                             
                               i 
                               , 
                               j 
                             
                           
                         
                         &gt; 
                         0 
                       
                     
                   
                 
               
             
           
         
       
     
     If Z l =Z max , for example, Z max =256, l=8, and {Z 0 , Z 1 , . . . , Z 8 }{2 0 , 2 1 , . . . , 2 8 }. If the size of a circulant permutation matrix in a parity check matrix is Z (1&lt;Z&lt;Z max ), the exponent a i,j  (Z) of a circulant matrix in an i th  row block and a j th  column block is determined by Equation 44. 
     
       
         
           
             
               
                 
                   
                     
                       a 
                       
                         i 
                         , 
                         j 
                       
                     
                     ⁢ 
                     
                       ( 
                       Z 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   a 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                 
                                 ( 
                                 
                                   Z 
                                   l 
                                 
                                 ) 
                               
                               ⁢ 
                                   
                               if 
                               ⁢ 
                                   
                               
                                 
                                   a 
                                   
                                     i 
                                     , 
                                     j 
                                   
                                 
                                 ( 
                                 
                                   Z 
                                   l 
                                 
                                 ) 
                               
                             
                             ≤ 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               
                                 mod 
                                 ⁢ 
                                    
                                 
                                   ( 
                                   
                                     
                                       
                                         a 
                                         
                                           i 
                                           , 
                                           j 
                                         
                                       
                                       ( 
                                       
                                         Z 
                                         l 
                                       
                                       ) 
                                     
                                     , 
                                     
                                       Z 
                                       k 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                    
                                 for 
                                 ⁢ 
                                     
                                 
                                   Z 
                                   k 
                                 
                               
                               ≤ 
                               Z 
                               &lt; 
                               
                                 Z 
                                 
                                   k 
                                   + 
                                   1 
                                 
                               
                             
                             , 
                             
                               0 
                               ≤ 
                               k 
                               &lt; 
                               l 
                             
                             , 
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   
                                     a 
                                     
                                       i 
                                       , 
                                       j 
                                     
                                   
                                   ( 
                                   
                                     Z 
                                     l 
                                   
                                   ) 
                                 
                               
                               &gt; 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   44 
                 
               
             
           
         
       
     
     In Equation 44, mod(x, y)=x mod y, representing the remainder of dividing x by y. 
     Specifically, for Z 0 ≤Z&lt;Z 1 , if a i,j (Z l−i ) is −1 or 0, a i,j (Z) is a i,j (Z l-1 ), and if a i,j (Z l−1 ) is larger than 0, a i,j (Z) is mod(a i,j (Z l−1 ), Z 0 ). 
     Specifically, for Z 1 ≤Z&lt;Z 2 , if a i,j (Z l−1 ) is −1 or 0, a i,j (Z) is a i,j (Z l−1 ), and if a i,j (Z l−1 ) is larger than 0, a i,j (Z) is mod(a i,j , 2 1 ). 
     Specifically, for 2 2 ≤Z&lt;2 3 , if a i,j  is −1 or 0, a i,j  (Z) is a i,j , and if a i,j  is larger than 0, a i,j  (Z) is mod(a i,j , 2 2 ). 
     Specifically, for 2 7 ≤Z&lt;2 8 , if a i,j  is −1 or 0, a i,j (Z) is a i,j , and if a i,j  is larger than 0, a i,j (Z) is mod(a i,j , 2 7 ). 
     Various parameters may be available for the parity check matrix, for example, as listed in Table 14 or Table 15. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 14 
               
               
                   
               
               
                   
                 ID 
                 Code Rate 
                 N ldpc _b 
                 K ldpc _b 
                 N parity _b 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 0 
                  8/9 
                 37 
                 32 
                 5 
               
               
                   
                 1 
                 ⅔ 
                 37 
                 24 
                 13 
               
               
                   
                 2 
                  4/9 
                 37 
                 16 
                 21 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 15 
               
               
                   
               
               
                   
                 ID 
                 Code Rate 
                 N ldpc _b 
                 K ldpc _b 
                 N parity _b 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 3 
                  8/9 
                 36 
                 32 
                 4 
               
               
                   
                 4 
                 ⅔ 
                 36 
                 24 
                 12 
               
               
                   
                 5 
                  4/9 
                 36 
                 16 
                 20 
               
               
                   
                 6 
                 ⅓ 
                 96 
                 32 
                 64 
               
               
                   
               
            
           
         
       
     
     Various embodiments of parity check matrices corresponding to the parameters listed in Table 14 and able  15  are illustrated in Table 1] to Table 3. Table 1 to Table 3 describe the exponent matrix of each parity check matrix (a small empty block represents a Z×Z zero matrix). For convenience of design, the numbers of columns in mother matrices are equally 36. Code rates of 8/9, 2/3, and 4/9 are set respectively for Table 1 to Table 3. For lifting, Z is set to 12, 24, 36, 48, 60, 72, 84, and 96, which means support of a total of 8 lengths. 
     For Z=96, Z being the size of a circulant permutation matrix which is a submatrix in Equation 3 (L×L=Z*Z), Table 16 to Table 18 list the exponents of circulant permutation matrices, a i,j (0≤i&lt;N ldpc −K lkdpc , 0≤j&lt;N kdpc ). If the circulant permutation matrix size, Z is equal to or less than 96, the exponents of a parity check matrix, a i,j (Z k ) is determined by Equation 45. 
         a   i,j ( Z   k )∝ a   i,j  mod  Z   k  
 
         Z   k =12· k ,( k= 1,2, . . . ,8)  Equation 45
 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 16 
               
               
                   
               
             
            
               
                 50 
                 47 
                 35 
                 49 
                 24 
                 13 
                 85 
                 30 
                 58 
                 84 
                 93 
                 44 
                 86 
                 65 
                 89 
                 57 
                 60 
                 15 
                 21 
                 8 
               
               
                 33 
                 48 
                 26 
                 3 
                 59 
                 11 
                 33 
                 19 
                 67 
                 0 
                 27 
                 61 
                 26 
                 23 
                 55 
                 13 
                 40 
                 20 
               
               
                 27 
                 76 
                 41 
                 24 
                 85 
                 54 
                 29 
                 28 
                 73 
                 16 
                 30 
                 92 
                 81 
                 61 
                 5 
                 95 
                 21 
                 45 
                 71 
                 15 
               
               
                 20 
                 73 
                 23 
                 87 
                 73 
                 33 
                 16 
                 26 
                 75 
                 42 
                 61 
                 63 
                 25 
                 86 
                 71 
                 8 
                 25 
                 20 
                 67 
                 95 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                   
                 55 
                 67 
                 79 
                 34 
                   
                 86 
                 3 
                   
                 28 
                 44 
                 29 
                 1 
                 0 
                   
                   
               
               
                   
                 83 
                 78 
                 77 
                 76 
                 5 
                 91 
                 65 
                 35 
                 33 
                 41 
                 12 
                   
                   
                 0 
                 0 
               
               
                   
                 71 
                   
                 85 
                 89 
                   
                 84 
                   
                 11 
                 8 
                   
                 71 
                 50 
                 0 
                   
                 0 
                 0 
               
               
                   
                 52 
                 35 
                   
                   
                 42 
                 70 
                 93 
                   
                 63 
                 61 
                   
                 63 
                 1 
                   
                   
                 0 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 17 
               
               
                   
               
             
            
               
                 29 
                 86 
                 48 
                 36 
                 34 
                 14 
                   
                   
                   
                   
                   
                   
                 52 
                   
                   
                   
                   
                   
                 64 
                 1 
               
               
                 54 
                 34 
                 78 
                 3 
                 10 
                 24 
                 9 
                   
                   
                 13 
                 29 
                   
                   
                   
                   
                 34 
                 60 
               
               
                 9 
                 94 
                 75 
                 58 
                 83 
                 62 
                   
                 21 
                   
                   
                   
                   
                   
                 68 
                   
                 14 
               
               
                 42 
                 48 
                 67 
                 30 
                 65 
                 66 
                   
                   
                 94 
                   
                   
                 17 
                   
                   
                 77 
                   
                 45 
                 88 
               
               
                 10 
                 10 
                 3 
                 57 
                 45 
                 8 
                   
                 49 
                 31 
                   
                   
                   
                   
                   
                   
                   
                   
                 38 
               
               
                 36 
                 44 
                 45 
                 58 
                 6 
                 3 
                 25 
                   
                   
                   
                   
                 76 
                   
                   
                 8 
                 35 
               
               
                 57 
                 64 
                 44 
                 53 
                 94 
                 77 
                   
                   
                   
                 94 
                   
                 55 
                 86 
                   
                 84 
               
               
                 39 
                 2 
                 73 
                 41 
                 54 
                 71 
                 63 
                   
                   
                   
                   
                   
                   
                 83 
                   
                   
                 37 
                   
                 91 
               
               
                 27 
                 85 
                 39 
                 42 
                 58 
                 40 
                   
                   
                 9 
                   
                 3 
                   
                 89 
                   
                   
                   
                   
                   
                 69 
               
               
                 68 
                 80 
                 22 
                 36 
                 54 
                 49 
                   
                   
                   
                   
                 43 
                   
                   
                   
                   
                   
                   
                 13 
                   
                 52 
               
               
                 62 
                 41 
                 83 
                 43 
                 72 
                 61 
                   
                   
                   
                 22 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 50 
               
               
                 20 
                 1 
                 52 
                 81 
                 76 
                 60 
                   
                 27 
                   
                   
                   
                   
                   
                 89 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                   
                 37 
                 28 
                   
                 1 
                 0 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                 17 
                   
                   
                 53 
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                 82 
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                 31 
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                 0 
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                 89 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                 95 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                 22 
                   
                 16 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                 93 
                   
                 40 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                 1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 18 
               
               
                   
               
             
            
               
                 50 
                 39 
                   
                 22 
                 49 
                   
                   
                   
                   
                 43 
                   
                   
                   
                   
                   
                   
                 1 
                 0 
                   
                   
               
               
                   
                   
                 23 
                 86 
                   
                   
                   
                   
                 39 
                 82 
                   
                   
                   
                   
                   
                 85 
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                 28 
                   
                   
                   
                 85 
                   
                 32 
                 45 
                 29 
                   
                   
                 0 
                 0 
               
               
                 63 
                   
                   
                 29 
                   
                 56 
                   
                   
                   
                   
                 0 
                   
                 93 
                   
                   
                   
                   
                   
                   
                 0 
               
               
                 13 
                 80 
                   
                 68 
                   
                   
                   
                   
                   
                   
                 68 
                   
                 88 
               
               
                   
                   
                 88 
                   
                 44 
                 89 
                 33 
                   
                   
                   
                   
                   
                   
                 91 
               
               
                 53 
                 86 
                 42 
                 40 
                   
                   
                   
                   
                   
                   
                 89 
               
               
                   
                   
                   
                 60 
                 85 
                 55 
                 58 
                   
                   
                   
                   
                   
                   
                 82 
               
               
                   
                 37 
                 82 
                 91 
                 9 
                 36 
               
               
                 46 
                   
                 48 
                 14 
                 72 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 17 
               
               
                 71 
                 16 
                 21 
                   
                   
                   
                 78 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
               
               
                 45 
                 33 
                 39 
                   
                 61 
                   
                   
                   
                   
                 4 
               
               
                   
                 75 
                 28 
                   
                 46 
                   
                   
                   
                   
                   
                   
                 93 
               
               
                 13 
                 93 
                 92 
                 31 
                   
                   
                   
                   
                 16 
               
               
                 42 
                   
                 74 
                 45 
                 52 
                   
                   
                   
                 53 
               
               
                   
                   
                   
                   
                   
                 65 
                 76 
                 91 
                   
                   
                   
                 55 
                   
                   
                 34 
               
               
                 78 
                 34 
                 41 
                   
                 48 
                   
                 27 
               
               
                 72 
                 83 
                   
                 24 
                 53 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 2 
               
               
                 54 
                 40 
                   
                 7 
                 73 
                   
                   
                 87 
               
               
                   
                 20 
                 54 
                   
                 7 
                 14 
                   
                   
                   
                   
                   
                   
                 60 
                   
                   
                   
                 1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                 0 
                 0 
               
               
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
               
               
                   
                   
               
            
           
         
       
     
     Other embodiments of parity check matrices are illustrated in Table 19, Table 20, and Table 21. Table 19 to Table 21 represent the exponent matrix of each parity check matrix. For convenience of design, the numbers of columns in mother matrices are equally 37. Code rates of 32/37, 24/37, and 16/37 are set respectively for Table 19 to Table 21. For lifting, Z is set to 12, 24, 36, 48, 60, 72, 84, and 96, which means support of a total of 8 lengths. 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 19 
               
               
                   
               
             
            
               
                 43 
                 15 
                 3 
                 66 
                 59 
                 47 
                 39 
                 34 
                   
                 86 
                 95 
                 37 
                 13 
                   
                 32 
                 82 
                 24 
                 80 
                   
                 56 
               
               
                 36 
                 62 
                 65 
                 43 
                 44 
                 93 
                 21 
                   
                 90 
                 45 
                 43 
                   
                   
                 24 
                   
                 25 
                   
                   
                 72 
               
               
                 62 
                 30 
                 20 
                 36 
                 51 
                   
                   
                 11 
                 33 
                   
                   
                 59 
                   
                 43 
                 29 
                   
                 27 
                   
                 61 
               
               
                 50 
                 15 
                 16 
                 24 
                 62 
                 81 
                 51 
                 39 
                 86 
                 4 
                   
                   
                 36 
                 46 
                   
                   
                   
                 0 
                 27 
                 77 
               
               
                 72 
                 0 
                 89 
                 86 
                 70 
                   
                   
                   
                   
                   
                 49 
                 64 
                 30 
                   
                 64 
                 81 
                 25 
                 39 
                   
                 0 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 62 
                 18 
                   
                   
                 77 
                 33 
                   
                   
                 41 
                   
                   
                   
                 1 
                 0 
                   
                   
                   
               
               
                   
                   
                   
                 75 
                 57 
                 33 
                   
                 67 
                 10 
                 46 
                 26 
                 36 
                 60 
                   
                 0 
                 0 
               
               
                   
                 4 
                   
                 95 
                 31 
                 13 
                   
                 76 
                 93 
                   
                 7 
                 42 
                 2 
                 0 
                   
                 0 
                 0 
               
               
                   
                 34 
                 72 
                 24 
                 50 
                   
                 52 
                   
                   
                   
                   
                 76 
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                 64 
                   
                   
                   
                 42 
                 34 
                 33 
                 11 
                 64 
                   
                 89 
                 1 
                   
                   
                   
                 0 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 20 
               
               
                   
               
             
            
               
                 39 
                 65 
                 34 
                 37 
                   
                 38 
                 39 
                   
                 36 
                 42 
                 28 
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 95 
                 26 
                 32 
                   
                   
                   
                   
                 13 
                   
                   
                   
                   
                   
                 13 
                   
                 29 
                 13 
               
               
                 36 
                 82 
                 48 
                 81 
                   
                   
                   
                   
                   
                   
                   
                   
                 92 
                   
                 86 
                   
                   
                 89 
                 92 
                 93 
               
               
                 71 
                 88 
                 65 
                 17 
                 17 
                   
                   
                 77 
                   
                   
                   
                 93 
               
               
                 87 
                 23 
                 78 
                 50 
                   
                   
                 19 
                   
                 55 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 10 
                 25 
               
               
                 86 
                   
                 87 
                 55 
                   
                   
                   
                   
                 81 
                 32 
                   
                 77 
                 80 
                   
                   
                   
                   
                 52 
               
               
                 9 
                 58 
                 25 
                 87 
                   
                 82 
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                   
                   
                   
                   
                 88 
               
               
                 84 
                 32 
                 53 
                 24 
                 91 
                   
                   
                 56 
                   
                   
                 81 
                   
                   
                 75 
                 61 
               
               
                 58 
                 40 
                 48 
                 61 
                   
                   
                   
                   
                   
                   
                   
                   
                 84 
                 95 
                   
                   
                 31 
               
               
                 31 
                 50 
                   
                 93 
                 20 
                   
                   
                   
                   
                   
                 7 
                   
                   
                   
                   
                   
                   
                 49 
               
               
                 41 
                 77 
                 51 
                 37 
                 57 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 75 
               
               
                 62 
                 23 
                 46 
                 45 
                   
                 29 
                 16 
                   
                   
                 35 
                   
                 41 
               
               
                 85 
                 36 
                 60 
                 77 
                 27 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 64 
                 90 
                   
                 24 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                   
                   
                   
                   
                 1 
                 0 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                   
                   
                 0 
                 93 
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                 94 
                 48 
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                 75 
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                 50 
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                 24 
                 0 
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                 85 
                 44 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                 74 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                 28 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                 48 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                 1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 21 
               
               
                   
               
             
            
               
                 50 
                 51 
                 94 
                 93 
                   
                   
                   
                 38 
                   
                   
                   
                   
                   
                   
                   
                   
                 1 
                 0 
                   
                   
               
               
                 23 
                   
                 37 
                 62 
                   
                   
                 69 
                 39 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                 90 
                   
                   
                 19 
                   
                   
                   
                   
                 28 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                 93 
                   
                 19 
                 75 
                   
                   
                   
                   
                   
                 37 
                 23 
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
               
               
                 32 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 47 
                 25 
                   
                 41 
                 10 
               
               
                 89 
                 81 
                 41 
                 83 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 36 
               
               
                 81 
                   
                 3 
                   
                   
                   
                   
                   
                   
                   
                   
                 21 
               
               
                 72 
                 48 
                   
                   
                 7 
                   
                   
                   
                   
                   
                   
                   
                   
                 92 
                 14 
               
               
                 58 
                 49 
                 86 
                   
                 57 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 89 
               
               
                 32 
                 90 
                   
                 22 
                   
                 44 
               
               
                 86 
                   
                 75 
                   
                 59 
                 11 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
               
               
                 24 
                 7 
                   
                   
                 53 
                   
                 32 
                 89 
               
               
                 3 
                 12 
                 62 
                 79 
                   
                   
                 41 
               
               
                 85 
                   
                 70 
                 5 
                   
                 55 
                   
                   
                 81 
               
               
                 68 
                   
                   
                   
                 16 
                 69 
                   
                   
                 74 
                   
                 5 
               
               
                 52 
                 39 
                 7 
                 4 
                   
                   
                   
                   
                   
                 21 
               
               
                 33 
                 41 
                 14 
                   
                   
                   
                   
                   
                   
                   
                 88 
                   
                   
                 58 
               
               
                 27 
                 93 
                   
                 80 
                   
                   
                   
                   
                   
                   
                   
                   
                 19 
                   
                   
                 60 
               
               
                 24 
                 50 
                   
                 82 
                 3 
                   
                   
                   
                   
                   
                   
                   
                 82 
               
               
                 45 
                 49 
                 16 
                 54 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 56 
               
               
                 7 
                   
                   
                   
                   
                 50 
                   
                   
                   
                 3 
                   
                 81 
                   
                   
                   
                   
                 1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                 0 
                 0 
               
               
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
                 0 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 0 
               
               
                   
                   
               
            
           
         
       
     
     In the case where LDPC encoding is performed using the parity check matrices illustrated in Table 19 to Table 21, if information word bits corresponding to the first column block in a partial matrix corresponding to an information word are punctured, prior to transmission, the code rates of Table 19 to Table 21 are finally  8 / 9 ,  2 / 3 , and  4 / 9 , respectively, which are the same as the code rates of Table 16 to Table 18. Since an LDPC code has improved performance through appropriate puncturing, LDPC encoding may be performed using Table 19 to Table 21 for performance improvement. 
       FIGS.  13 A and  13 B  illustrate a parity check matrix with ID=6 and R=1/3 in Table 15 according to various embodiments of the present disclosure. 
     Referring to  FIGS.  13 A and  13 B , the exponent matrix of the parity check matrix is illustrated. A small empty block represents a Z×Z zero matrix. For lifting, Z is set to 12, 24, 36, 48, 60, 72, 84, and 96, which means support of a total of 8 lengths. For reference, the 37 th  to last column blocks illustrated in  FIGS.  13 A and  13 B  are characterized by a degree of 1. For convenience of description, the column blocks are partially omitted in the tables. Column blocks with a degree of 1 include identity matrices. 
     Since the parity check matrix to which single parity check codes are concatenated is easily extended, it is advantageous in applying an incremental redundancy (IR) scheme. The IR scheme is very important to support hybrid automatic repeat request (HARQ). Therefore, an IR scheme with excellent performance increases the efficiency of an HARQ system. As LDPC codes based on the parity check matrices are transmitted by generating a new parity using a part extended to the single parity check codes, an efficient IR scheme with excellent performance may be applied. 
     Regarding the parity check matrix illustrated in  FIGS.  13 A and  13 B , a partial matrix with the top four row blocks by 36 column blocks of the parity check matrix is identical to the parity check matrix of Table 35. For example, it is noted that the parity check matrix illustrated in  FIGS.  13 A and  13 B  is extended from the parity check matrix of Table 35 by concatenating a plurality of single parity check codes to the parity check matrix of Table 35. 
     Another embodiment of a parity check matrix designed according to the design method of the present disclosure is illustrated in  FIGS.  14 A and  14 B . 
       FIGS.  14 A and  15 A  illustrate exponent matrices of parity check matrices according to various embodiments of the present disclosure. 
     The parity check matrix illustrated in  FIG.  14 A  is divided into parts  1410 ,  1420 ,  1430 , and  1440 .  FIGS.  14 B to  14 E  are enlarged views of the parts  1410 ,  1420 ,  1430 , and  1440 . Similarly, parts denoted by  1510 ,  1520 ,  1530 , and  1540  in  FIG.  15 A  are illustrated respectively in  15 B,  154 C,  154 D, and  154 E. The diagonal elements of the diagonal matrices illustrated in  FIGS.  14 E and  15 E  are filled with zeroes. 
       FIGS.  15 B,  154 C,  154 D, and  15 E  are enlarged views of the parts  1510 ,  1520 ,  1530 , and  1540  divided from the parity check matrix of  FIG.  15 A  according to various embodiments of the present disclosure. 
     Referring to  FIGS.  14 A and  15 A , a small empty block represents a Z×Z zero matrix, and the exponent matrices are designed in consideration of Equation 8 to Equation 15. 
     For reference, all of the 37 th  to last column blocks  1420  and  1440  illustrated in  FIG.  14 A  and the 39 th  to last column blocks  1520  and  1540  illustrated in  FIG.  15 A  have a degree of 1. For convenience of description, the column blocks with a degree of 1 include identity matrices, for convenience of description. 
     Regarding the parity check matrix of  FIG.  14 A , the partial matrix  1410  including the top 4 row blocks by 36 column blocks of the whole parity check matrix does not have a column block with a degree of 1. For example, it may be noted that the parity check matrix of  FIG.  14 A  is extended by concatenating a plurality of single parity check codes to a small QC-LDPC code corresponding to the partial matrix  1410 . 
     Regarding the parity check matrix of  FIG.  15 A , the partial matrix  1510  including the top 6 row blocks by 38 column blocks of the whole parity check matrix does not have a column block of a degree of 1. For example, it may be noted that the parity check matrix of  FIG.  15 A  is extended by concatenating a plurality of single parity check codes to a small QC-LDPC code corresponding to the partial matrix  1510 . 
     The parity check matrix of  FIG.  15 A  is extended to support R=32/38 to R=32/98. If the parity check matrix is continuously extended by use of a plurality of single parity check codes, a low code rate may be supported readily. 
       FIGS.  16 A,  16 B,  16 C, and  16 D  illustrate a parity check matrix (an exponent matrix) designed in consideration of lifting according to various embodiments of the present disclosure. 
     Referring to  FIG.  16 A , a low code rate may be supported by concatenating a single parity check code  1610  illustrated in  FIGS.  16 B and  16 C , and a single parity check code  1620  in  FIG.  16 D  to the parity check matrix of  FIG.  15 A . In  FIGS.  16 B,  16 C, and  16 D , the number of rows is 64 and thus up to R=32/162 may be supported with the parity check matrix of  FIG.  16 A . For reference, reference numerals  1630  and  1640  denote zero matrices in  FIG.  16 A . For reference,  FIG.  16 C  is connected to  FIG.  16 B , and reference numeral  160  of  FIG.  16 A  denotes a combination of  FIGS.  16 B and  16 C . For reference,  FIG.  16 D  illustrates the extended parity check matrix shown as rotated to the right at 90 degrees. 
     Since a parity check matrix to which single parity check codes are concatenated is easily extended, it is advantageous in applying an IR scheme. The IR scheme is very important to support HARQ. Therefore, an IR scheme with excellent performance increases the efficiency of an HARQ system. As LDPC codes based on the foregoing parity check matrices are transmitted by generating a new parity using a part extended to the single parity check codes, an efficient IR scheme with excellent performance may be applied. 
     While parity check matrices of various lengths, that is, QC-LDPC codes may be generated by applying lifting proposed by the present disclosure to exponent matrices designed according to the designing method proposed by the present disclosure, appropriate application of shortening or puncturing may enable application of an LDPC encoding scheme supporting various information word lengths and code rates. In other words, if lifting, shortening, or puncturing is appropriately applied to the exponent matrix of  FIG.  14 A  or  FIG.  15 A , IR or HARQ are readily supported, thereby increasing system flexibility. 
     A method for designing an LDPC code suitable for using the lifting method of the present disclosure will be described below. 
     In general, a QC-LDPC code has a special cycle property according to the characteristics of the mother matrix and exponent matrix of a parity check matrix. In the following cited reference [Myung2005], a couple of examples in which a cycle property is determined according to a mother matrix and an exponent matrix are described. 
     Reference [Myung2005] S. Myung, K. Yang, and J. Kim, “Quasi-Cyclic LDPC Codes for Fast Encoding,” IEEE Transactions on Information Theory. vol. 51, No. 8, pp. 2894-2901, August 2005. 
     The cycle property of a QC-LDPC code disclosed in [Myung2005] will be described briefly. 
     To describe the cycle property of the simplest QC-LDPC code, four circulant permutation matrices with a 4-cycle in a mother matrix are assumed, as in Equation 46. The size of each circulant permutation matrix is assumed to be Z×Z. 
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           p 
                           
                             a 
                             1 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           p 
                           
                             a 
                             2 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋱ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           p 
                           
                             a 
                             4 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           p 
                           
                             a 
                             S 
                           
                         
                       
                     
                   
                   ] 
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   46 
                 
               
             
           
         
       
     
     According to [Myung2005], if there is a minimum positive integer r satisfying Equation 47, a cycle of length 4r exists on the Tanner graph of a parity check matrix corresponding to Equation 46. 
         r ·( a   1   −a   2   +a   3   −a   4 )∝0(modZ)  Equation 47
 
       FIGS.  17 A and  17 B  illustrate a cycle property of a quasi-cyclic LDPC (QC-LDPC) code according to various embodiments of the present disclosure. 
     Referring to  FIG.  17 A , for example, if Z=6, a 1 =a 2 =0, and a 3 =a 4 =1, a 1 −a 2 +a 3 − a 4=0. Thus, a 4-cycle is readily derived on the Tanner graph. 
     Referring to  FIG.  17 B , if Z=6, a 1 =a 2 =0, a 3 =3, and a 4 =1, r·(a 1 −a 2 +a 3 −a 4 )∝3·2∝0 (mod 6). Thus, a 12-cycle is readily derived on the Tanner graph. 
     In this manner, the cycle property of a QC-LDPC code may be defined from the relationship between the exponents of QC permutation matrices of the parity check matrix. 
     Because the lifting method of the present disclosure may cause use of the same exponent matrix for different Z values in some cases, an exponent matrix should be selected carefully. For example, even though the same a 1 =a 2 =0, a 3 =3, and a 4 =1 are used in Equation 46, if Z=4, r·(a 1 −a 2 +a 3 −a 4 )∝2·2∝0 (mod 4), resulting in an 8-cycle. For example, if the same exponent matrix is used in consideration of different Z values, a change in cycle property should be considered. 
     However, it is very difficult to select an exponent matrix satisfying a cycle property by calculating all r values by modulo-Z in Equation 47, while changing the exponent of a QC permutation matrix from many cycles in a mother matrix. In this context, the present disclosure proposes a method for fast determining an exponent matrix in a simple manner, as follows. 
     For this purpose, if a circulant permutation matrix size is Z×Z in Equation 46, an extended Tanner graph with 8(Z−1)+2 variable nodes and 8(Z−1)+2 check nodes corresponding to Equation 46 will be described with reference to  FIG.  18   . 
     In general, one Z×Z QC matrix corresponds to Z variable nodes and Z check notes on a Tanner graph. Therefore, QC permutation matrices with a 4-cycle in a mother matrix as described in Equation 46 correspond to 2Z variable nodes and 2Z check nodes. However, since variable and check nodes corresponding to one QC-permutation matrix are extended by 4(Z−1)+1 nodes ranging from the −2(Z−1) th  to 2(Z−1) th  nodes, the extended Tanner graph of  FIG.  18    includes 8(Z−1)+2 variable nodes and 8(Z−1)+2 check nodes. 
     For convenience of description, it is assumed that a cycle starts from a 0 th  check node of check node group 1 on the extended Tanner graph. A cycle property is determined for the circulant permutation matrices of Equation 46 according to the exponent of each circulant permutation matrix, as illustrated in  FIG.  18   . If (a 1 −a 2 +a 3 −a 4 )=0, the circulant permutation matrices form a 4-cycle as indicated by dotted lines in  FIG.  18   . Otherwise, a cycle larger than the 4-cycle may be achieved. 
     A method for designing parity check matrices (or exponent matrices) for QC-LDPC codes with the same exponent matrix without a 4-cycle for Z, Z+1, Z+2, . . . , Z+m (m&gt;1) for convenience of description will be described in brief. An extended Tanner graph for the largest Z value, Z+m is considered. For example, the extended Tanner graph includes 8(Z+m−1)+2 variable nodes and 8(Z+m−1)+2 check nodes, and covers all extended Tanner graphs for Z, Z+1, . . . , Z+m−1. If the exponents a 1 , a 2 , a 3 , and a 4  are changed while checking whether −Z&lt;(a 1 −a 2 +a 3 −a 4 )&lt;Z is satisfied for all circulant permutation matrix combinations, such as Equation 46, for Z, Z+1, Z+2, Z+3, . . . , Z+m, a design without a 4-cycle is possible without the need for performing a modulo operation or calculating r for Z, Z+1, Z+2, Z+3, . . . , by Equation 47. For example, it may be concluded that the use of an extended Tanner graph obviates the need for performing a modulo operation or calculating r for all of Z, Z+1, . . . , Z+m, and facilitates design of a parity check matrix without a short cycle. 
     The above method for designing a parity check matrix for an LDPC code may be applied to any of Z 1 , Z 2 , . . . , Z max  to be supported. Once only one exponent matrix (or sequence) corresponding to Z max  is stored in a system, all exponent matrices (or sequences) corresponding to Z 1 , Z 2 , . . . , Z max  may be generated and applied to LDPC encoding. 
     Now, a detailed description will be given of the rate matcher  340  of the transmitter  300 . 
     Input bits of the rate matcher  340  are output bits of the LDPC encoder  330 , C=(i 0 , i 1 , i 2 , . . . , i Kldpc−1 , p 0 , p 1 , p2, . . . , p Nldpx−Kldpc−1 ). i k , (0≤k&lt;K ldpc ) represents the input bits of the LDPC encoder  330 , and p k (0≤k&lt;N ldpc −K ldpc ) represents LDPC parity bits. The rate matcher  340  includes the interleaver  341  and the puncturer/repeater/zero remover  342 . 
       FIGS.  9 A and  9 B  illustrate structures of interleavers according to various embodiments of the present disclosure. 
     Referring to  FIG.  9 A , the interleaver  341  interleaves i k . As illustrated in  FIG.  9 B , the interleaver  341  may interleave both i and p. 
     p k  is interleaved in the following interleaving method. 
     Step 1) The number of columns in a block interleaver is set to C subblock   LDPC =N parity_b  based on N parity_b  of Table 3. 
     Step 2) The number of rows in the block interleaver is set to the size Z of a circulant permutation matrix of a parity check matrix. 
     Step 3) Parity bits of an LDPC code, p k (k=0, 1, . . . , N parity−1 ) are input, in an ascending order of row indexes starting from the first row of the first column, as in Equation 48. 
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           p 
                           0 
                         
                       
                       
                         
                           p 
                           
                             ( 
                             
                               R 
                               subblock 
                               LDPC 
                             
                             ) 
                           
                         
                       
                       
                         
                           p 
                           
                             ( 
                             
                               2 
                               ⁢ 
                                  
                               
                                 XR 
                                 subblock 
                                 LDPC 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           p 
                           
                             
                               ( 
                               
                                 
                                   C 
                                   subblock 
                                   TC 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                             ⁢ 
                             
                               XR 
                               subblock 
                               LDPC 
                             
                           
                         
                       
                     
                     
                       
                         
                           p 
                           1 
                         
                       
                       
                         
                           p 
                           
                             ( 
                             
                               
                                 R 
                                 subblock 
                                 LDPC 
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           p 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                    
                                 
                                   XR 
                                   subblock 
                                   LDPC 
                                 
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           p 
                           
                             
                               
                                 ( 
                                 
                                   
                                     C 
                                     subblock 
                                     TC 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 XR 
                                 subblock 
                                 LDPC 
                               
                             
                             + 
                             1 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋱ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           p 
                           
                             ( 
                             
                               
                                 R 
                                 subblock 
                                 LDPC 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           p 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                    
                                 
                                   XR 
                                   subblock 
                                   LDPC 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           p 
                           
                             ( 
                             
                               
                                 3 
                                 ⁢ 
                                    
                                 
                                   XR 
                                   subblock 
                                   LDPC 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           p 
                           
                             
                               
                                 ( 
                                 
                                   C 
                                   subblock 
                                   TC 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 XR 
                                 subblock 
                                 LDPC 
                               
                             
                             - 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   48 
                 
               
             
           
         
       
     
     For p x : 
     Step 4) p k  arranged as illustrated in Equation 48 is interleaved column-wise based on an inter-column permutation pattern so that the positions of columns may be changed. Table 22 illustrates inter-column permutation patterns for a sub-block interleaver. 
     
       
         
           
               
               
               
             
               
                 TABLE 22 
               
               
                   
               
               
                   
                 Number of 
                   
               
               
                 Code 
                 columns 
                 Inter-column permutation pattern 
               
               
                 Rate 
                 C subblock   LDPC   
                 &lt; P(0), P(1), . . . ,P(C subblock   LDPC  − 1) &gt; 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 ¼ 
                 24 
                 &lt; 0, 2, 4, 8, 10, 12, 14, 16, 18, 20, 22, 
               
               
                   
                   
                 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 &gt; 
               
               
                 ½ 
                 16 
                 &lt; 0, 2, 4, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15 &gt; 
               
               
                 ¾ 
                 8 
                 &lt; 0, 2, 4, 1, 3, 5, 7 &gt; 
               
               
                 ⅞ 
                 4 
                 &lt; 0, 2, 1, 3 &gt; 
               
               
                   
               
            
           
         
       
     
     The parameters of Table 22 may be changed according to a system, for example, to Table 23. 
     
       
         
           
               
               
               
               
             
               
                 TABLE 23 
               
               
                   
               
               
                   
                   
                 Number of 
                   
               
               
                   
                 Code 
                 Columns 
                 Inter-column permutation pattern 
               
               
                 ID 
                 Rate 
                 C subblock   LDPC   
                     P(0), P(1), . . . ,P(C subblock   LDPC  − 1)    
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 0 
                  8/9 
                 5 
                 0 4 2 3 1 
               
               
                 1 
                 ⅔ 
                 13 
                 0 12 8 4 10 6 2 11 9 8 5 3 1 
               
               
                 2 
                  4/9 
                 21 
                 0 20 16 12 8 4 18 14 10 6 2 19 17 15 13 
               
               
                   
                   
                   
                 11 9 7 5 3 1 
               
               
                 3 
                  8/9 
                 4 
                 0 2 3 1 
               
               
                 4 
                 ⅔ 
                 12 
                 0 8 4 10 62 11 9 7 5 3 1 
               
               
                 5 
                  4/9 
                 20 
                 0 16 12 8 4 18 14 10 6 2 19 17 15 13 11 
               
               
                   
                   
                   
                 9 7 5 3 1 
               
               
                 6 
                 ⅓ 
                 64 
                 0 2 3 1 4 5 6 7 8 9 10 11 12 13 14 15 16 
               
               
                   
                   
                   
                 17 18 19 20 22 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 67 68 69 60 61 62 63 
               
               
                   
               
            
           
         
       
     
     In the above inter-column permutation patterns, the columns are arranged in a reverse order of puncturing. For example, for an ID of  5 , the bits of the first column out of 20 columns are first punctured. If consecutive parity blocks are punctured, it may affect performance. Therefore, the bits of the first column are first punctured, and the bits of the third column are punctured in the second place. In this manner, the order of puncturing odd-numbered columns ending with the 19 th  column is determined and an order of puncturing even-numbered blocks is determined in such a manner that the punctured even-numbered blocks may be spaced from each other by four blocks. A similar method is used for other code rat 
     For example, P(j) of  P(j)   jϵ{0,1 . . . c     sublock       LDPC     −1 } in Table 22 represents a pre-permutation index of a j th  permuted column. After the column-wise permutation, the inter-column permutated (R subblock   LDPC ×C subblock   LDPC ) matrix may be represented as Equation 49. 
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           y 
                           0 
                         
                       
                       
                         
                           y 
                           
                             ( 
                             
                               R 
                               subblock 
                               LDPC 
                             
                             ) 
                           
                         
                       
                       
                         
                           y 
                           
                             ( 
                             
                               2 
                               ⁢ 
                                  
                               
                                 XR 
                                 subblock 
                                 LDPC 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           y 
                           
                             
                               ( 
                               
                                 
                                   C 
                                   subblock 
                                   TC 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                             ⁢ 
                             
                               XR 
                               subblock 
                               LDPC 
                             
                           
                         
                       
                     
                     
                       
                         
                           y 
                           1 
                         
                       
                       
                         
                           y 
                           
                             ( 
                             
                               
                                 R 
                                 subblock 
                                 LDPC 
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           y 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                    
                                 
                                   XR 
                                   subblock 
                                   LDPC 
                                 
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           y 
                           
                             
                               
                                 ( 
                                 
                                   
                                     C 
                                     subblock 
                                     TC 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 XR 
                                 subblock 
                                 LDPC 
                               
                             
                             + 
                             1 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋱ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           y 
                           
                             ( 
                             
                               
                                 R 
                                 subblock 
                                 LDPC 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           y 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                    
                                 
                                   XR 
                                   subblock 
                                   LDPC 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         
                           y 
                           
                             ( 
                             
                               
                                 3 
                                 ⁢ 
                                    
                                 
                                   XR 
                                   subblock 
                                   LDPC 
                                 
                               
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           y 
                           
                             
                               
                                 ( 
                                 
                                   C 
                                   subblock 
                                   TC 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 XR 
                                 subblock 
                                 LDPC 
                               
                             
                             - 
                             1 
                           
                         
                       
                     
                   
                   ] 
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   49 
                 
               
             
           
         
       
     
     Step 5) The values of the inter-column permutated (R subblock   LDPC ×C subblock   LDPC ) matrix described in Equation 49 are output row by row, starting from the first row of the first column, while increasing the indexes of columns. 
     The resulting subblock interleaved bits are (v 0 , v 1 , v 2 , . . . , v Nparity −1). 
     The interleaver  341  of the rate matcher  340  receives C=(i 0 , i 1 , i 2 , . . . , i Kldpc−1 , p 0 , p 1 , p 2 , . . . , p Nldpc−Kldpc−1 ), and block-interleaves p k , thus outputting C′=(i 0 , i 1 , i 2 , . . . , i Kldpc−1 , v 0 , v 1 , v 2 , . . . , v Nldpc−Kldpc−1 ). 
     The parity bits are interleaved on a Z bit basis, Z being the circulant permutation matrix size of the parity check matrix. Thus, the input bits of the LDPC code i 0 , i 1 , i 2 , . . . , i Kldpc−1  and the parity bits of the LDPC code p 0 , p 1 , p 2 , . . . , p Nkdpc−Kldpc−1  may be interleaved on a Z bit basis, Z being the circulant permutation matrix size of the parity check matrix. 
     Because similar encoding or decoding characteristics may result on a Z bit basis, Z being the circulant permutation matrix size of the parity check matrix, Z unit-based interleaving may optimize encoding or decoding performance. 
     The output bits of the interleaver  341  in the rate matcher  340  of the transmitter  330 , i 0 , i 1 , i 2 , . . . , i Klpc−1 , v 0 , v 1 , v 2 , . . . , v Nldpc−Kldpc−1  are input to the puncturer/repeater/zero remover  342 . 
     The puncturer/repeater/zero remover  342  performs puncturing/repetition according to the size of bits to be transmitted and removes the &lt;Null&gt; bits input by the zero padder  320 . 
     The puncturing refers to non-transmission of some bits except for the &lt;Null&gt; bits among the outputs bits of the interleaver  341 , i 0 , i 1 , i 2 , . . . , i Klpc−1 , v 0 , v 1 , v 2 , . . . , v Nldpc−Kldpc−1 , and the repetition refers to repeating some bits except for the &lt;Null&gt; bits among the outputs bits of the interleaver  341 , i 0 , i 1 , i 2 , . . . , i Klpc-1 , v 0 , v 1 , v 2 , . . . , v Nldpc−Kldpc−1 . 
     The number of codeword bits to be transmitted may be controlled by puncturing and repetition. 
     More specifically, the puncturer/repeater/zero remover  342  operates in the following manner. 
     K w =N ldpc  bits are input to a circular buffer. 
     W k =i k  for k=0, . . . , K ldpc    
     W k     ldpc     +k =V k  for k=0, . . . , N parity    
     If E bits are transmitted at this transmission, for HARQ, transmission bits are determined as follows. A maximum allowed transmission number is M DL_HARQ . 
     Set k 0 =0, k 0 =i·E−1 for 1≤i≤M DL_HARQ  (if incremental redundancy is used), 
     set k 0 =0 for 1≤i≤M DL_HARQ  (if chase combining is used) 
     Set k=0 and j=0 
     while {k&lt;E}
         if w (K     0     +j) mod N cb ≠&lt;NULL&gt;
           e k =w (k     0     +j)  mod N cb  
               k=k+1   
               
               

     end if 
     j=j+1 
     end while 
     Further, if E bits are transmitted at this transmission, for HARQ, transmission bits are determined as follows. 
     The indexes k 0  of initially transmitted bits may be determined by Equation 50. 
     
       
         
           
             
               
                 
                   
                     k 
                     0 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           ⌈ 
                           
                             
                               N 
                               
                                 c 
                                 ⁢ 
                                 b 
                               
                             
                             4 
                           
                           ⌉ 
                         
                         · 
                         
                           rv 
                           idx 
                         
                       
                       + 
                       Z 
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   50 
                 
               
             
           
         
       
     
     In Equation 50, Z is the circulant permutation matrix size of a parity check matrix, rv idx  is an integer selected from {0, 1, 2, 3}, and N cb  is the number of bits that can be processed in a receiver, in consideration of a buffer size in the receiver. For example, N cb  may be equal to or less than the number of codeword bits, N ldpc . Considering Z in Equation 50 implies that transmission bits are selected from among information bits except for Z bits. 
     Thus, if bits except for Z bits are transmitted, Equation 50 may be expressed as Equation 51. 
     
       
         
           
             
               
                 
                   
                     k 
                     0 
                   
                   = 
                   
                     ( 
                     
                       
                         ⌈ 
                         
                           
                             N 
                             
                               c 
                               ⁢ 
                               b 
                             
                           
                           4 
                         
                         ⌉ 
                       
                       · 
                       
                         rv 
                         idx 
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   51 
                 
               
             
           
         
       
     
     In the above case, rv idx  is an integer and four values of {0, 1, 2, 3} are available as rv idx . If rv idx  is {0, 1, 2, . . . , M−1}, the indexes k 0  of initially transmitted bits may be determined by Equation 52. 
     
       
         
           
             
               
                 
                   
                     k 
                     0 
                   
                   = 
                   
                     ( 
                     
                       
                         ⌈ 
                         
                           
                             N 
                             
                               c 
                               ⁢ 
                               b 
                             
                           
                           M 
                         
                         ⌉ 
                       
                       · 
                       
                         rv 
                         idx 
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   52 
                 
               
             
           
         
       
     
     Bits are transmitted by dividing the number Ncb of bits storable in the buffer of the receiver by M. In order not to transmit X bits at rv0, k0 may be determined by Equation 53. 
     
       
         
           
             
               
                 
                   
                     k 
                     0 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           ⌈ 
                           
                             
                               N 
                               
                                 c 
                                 ⁢ 
                                 b 
                               
                             
                             M 
                           
                           ⌉ 
                         
                         · 
                         
                           rv 
                           idx 
                         
                       
                       + 
                       X 
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                       
                   53 
                 
               
             
           
         
       
     
     Set k=0 and j=0 
     while {k&lt;E}
         if w (k     0     +j)mod N     cb     ≠&lt;NULL&gt; 
           e k =w (k     0     +j)mod N     cb    
               k=k+1   
               
               

     end if 
     j=j+1 
     end while 
     Transmission bits e k  (0≤k&lt;E) are selected from among the interleaved bits w k  (0≤k&lt;N cb ) except for &lt;NULL&gt; values. If E is larger than N cb , transmission bits are repeatedly selected. 
     The modulator  350  modulates a bit stream received from the rate matcher  340  and transmits the modulated bit stream to a receiver (for example, the receiver  400  in  FIG.  4   ). 
     Specifically, the modulator  350  may demultiplex bits received from the rate matcher  340  and map the demultiplexed bits to a constellation. 
     For example, the modulator  350  converts serial bits received from the rate matcher  340  to parallel bits, and form cells each including a predetermined number of bits. The number of bits per cell may be equal to the number of bits that form a modulation symbol mapped to the constellation. 
     Subsequently, the modulator  350  may map the demultiplexed bits to the constellation. For example, the modulator  350  may modulate the demultiplexed bits in any of various modulation schemes, such as QPSK, 16-QAM, 64-QAM, 256-QAM, 1024-QAM, and 4096-QAM, and map the modulated bits to constellation points. In this case, since cells are formed with the demultiplexed bits, each cell including the number of bits per modulation symbol, each cell may be mapped sequentially to a constellation point. 
     The modulator  350  may modulate the signals mapped to the constellation and transmit the modulated signals to the receiver  400 . For example, the modulator  350  may map the signals mapped to the constellation to an OFDM frame and transmit the OFDM frame on an allocated channel to the receiver  400 . 
     Meanwhile, the transmitter  300  may pre-store various parameters used in encoding, interleaving, and modulation. Parameters for encoding may be information about a code rate, codeword length, and parity check matrix of an LDPC code. An interleaving parameter may be information about an interleaving rule, and a modulation parameter may be information about a modulation scheme. A puncturing parameter may be information about a puncturing length. A repetition parameter may be information about a repetition length. The information about a parity check matrix may be information about the exponents of circulant permutation matrices given by Equation 3 and Equation 4, if a parity check matrix of the present disclosure is used. 
     In this case, the components of the transmitter  300  may operate using these parameters. 
     While not shown, the transmitter  300  may further include a controller (not shown) for controlling operations of the transmitter  300 . 
       FIG.  6    is a block diagram of an encoder according to an embodiment of the present disclosure. 
     Referring to  FIG.  6   , an encoder  600  may perform LDPC encoding and includes the LDPC encoder  610 . The LDPC encoder  610  may generate an LDPC codeword by LDPC-encoding input bits based on a parity check matrix. 
     The parity check matrix may have the same structure as a parity check matrix described by Equation 3 and Equation 4. 
     In this case, the LDPC encoder  610  may perform LDPC encoding using a parity check matrix defined differently according to a code rate (i.e., the code rate of an LDPC code). 
     For example, if the code rate is 7/8, the LDPC encoder  610  may perform LDPC encoding using a parity check matrix as defined by Table 6. If the code rate is 3/4, the LDPC encoder  610  may perform LDPC encoding using a parity check matrix as defined by Table 5. If the code rate is 1/2, the LDPC encoder  610  may perform LDPC encoding using a parity check matrix as defined by Table 8. If the code rate is 1/4, the LDPC encoder  610  may perform LDPC encoding using a parity check matrix as defined by Table 7. 
     A specific method for performing LDPC encoding has been described below, and thus will not be described herein to avoid redundancy. 
     The encoder  600  may further include a memory (not shown) for pre-storing information about the code rates, codeword lengths, and parity check matrices of LDPC codes, and the LDPC encoder  610  may perform LDPC encoding using this information. Information about a parity check matrix may include information about the exponents of circulant matrices, when a parity check matrix proposed by the present disclosure is used. 
     Now, a detailed description will be given of an operation of a receiver with reference to  FIG.  4   . 
     The demodulator  410  demodulates a signal received from the transmitter  300 . Specifically, the demodulator  410 , which is a counterpart of the modulator  350  of the transmitter  300 , may generate values corresponding to bits transmitted by the transmitter  300  by demodulating a signal received from the transmitter  300 . 
     For this purpose, the receiver  400  may pre-store information about modulation schemes according to modes of the transmitter  300 . Accordingly, the demodulator  410  may generate values corresponding to LDPC codeword bits by demodulating a signal received from the transmitter  300  according to a mode. 
     The values corresponding to the bits transmitted by the transmitter  300  may be LLRs. 
     Specifically, the LLR of a bit transmitted by the transmitter  300  may be a value obtained by performing a log operation on a ratio between the probability of 0 and the probability of 1 for the bit. The LLR may also be the value of the bit itself. The LLR may be a representative value of a range to which the probability of the transmitted bit being 0 or 1 belongs. 
     The demodulator  410  may include a multiplexer (MUX) for multiplexing the LLRs. Specifically, the MUX is a counterpart of a bit DEMUX (not shown) of the transmitter  300  and may perform an operation corresponding to the bit DEMUX. 
     For this purpose, the receiver  400  may pre-store information about parameters used for demultiplexing and block interleaving of the transmitter  300 . Accordingly, the MUX may multiplex LLRs corresponding to a cell word on a bit basis by performing the demultiplexing and block interleaving of the bit DEMUX in a reverse order. 
     The rate dematcher  420  may insert LLRs in LLRs received from the demodulator  410 . In this case, the rate dematcher  420  may insert predetermined LLRs in the LLRs received from the demodulator  410 . 
     Specifically, the rate dematcher  420 , which is a counterpart of the rate matcher  340  of the transmitter  300 , may perform operations corresponding to the interleaver  341  and the puncturer/repeater/zero remove  342 . 
     The rate dematcher  420  deinterleaves in correspondence with the interleaver  341  of the transmitter  300 . LLRs corresponding to zero bits may be inserted at the positions of the zero bits added to the LDPC codeword in the output values of the deinterleaver  424  by the LLR inserter  422 . In this case, the LLRs corresponding to the padded zero bits, that is, shortened zero bits may be ∞ or −∞. However, ∞ or −∞ is a theoretical value, and may be the maximum or minimum value of the LLRs used in the receiver  400 . 
     For this purpose, the receiver  400  may pre-store information about a parameter used for padding zero bits in the transmitter  300 . Therefore, the rate dematcher  420  may determine the positions of padded zero bits in the LDPC code and insert LLRs corresponding to shortened zero bits at the positions. 
     The LLR inserter  422  of the rate dematcher  420  may insert LLRs corresponding to puncturing bits at the positions of the puncturing bits in the LDPC codeword. In this case, the LLRs corresponding to the punctured bits may be zeroes. 
     For this purpose, the receiver  400  may pre-store information about a parameter used for puncturing in the transmitter  300 . Therefore, the LLR inserter  422  may insert corresponding LLRs at the positions of punctured parity bits. 
     The LLR combiner  423  may combine, that is, sum the LLRs output from the LL inserter  422  and the demodulator  410 . Specifically, the LLR combiner  423 , which is a counterpart of the puncturer/repeater/zero remover  342  of the transmitter  300 , may perform an operation corresponding to the repeater  342 . First, the LLR combiner  423  may combine LLRs corresponding to repeated bits with other LLRs. The other LLRs may be LLRs of bits based on which the repeated bits are generated, that is, LLRs of LDPC parity bits selected for repetition. 
     For example, as described before, the transmitter  300  selects bits from among LDPC parity bits, repeats the selected bits between LDPC information word bits and LDPC parity bits, and transmits them to the receiver  400 . 
     Therefore, the LLRs of the LDPC parity bits may include the LLRs of repeated LDPC parity bits and the LLRs of non-repeated LDPC parity bits, that is, LDPC parity bits generated by encoding. Accordingly, the LLR combiner  423  may combine the LLRs of the same LDPC parity bits. 
     For this purpose, the receiver  400  may pre-store information about a parameter used for repetition in the transmitter  300 . Thus, the LLR combiner  423  may determine the LLRs of repeated LDPC parity bits and combine the LLRs with the LLRs of LDPC parity bits based on which the repeated LDPC parity bits are produced. 
     Further, the LLR combiner  423  may combine the LLRs of retransmission bits or IR bits with other LLRs. The other LLRs may be the LLRs of bits selected for generation of LDPC codeword bits, based on which the retransmission bits or the IR bits are generated. 
     For example, as described before, if a negative acknowledgement (NACK) is generated in HARQ, the transmitter  300  may transmit all or part of codeword bits to the receiver  400 . 
     Therefore, the LLR combiner  423  may combine the LLRs of the retransmission bits or the IR bits with the LLRs of LDPC codeword bits received in a previous frame. 
     For this purpose, the receiver  400  may pre-store information about a parameter used for generation of the retransmission bits or the IR bits in the transmitter  300 . Thus, the LLR combiner  423  may determine the LLRs of the retransmission bits or the IR bits and combine the LLRs with the LLRs of LDPC parity bits based on which the retransmission bits or the IR bits are produced. 
     The deinterleaver  424  may deinterleave LLRs received from the LLR combiner  423 . 
     Specifically, the deinterleaver  424 , which is a counterpart of the interleaver  341  of the transmitter  300 , may perform an operation corresponding to the interleaver  341 . 
     For this purpose, the receive  400  may pre-store information about a parameter used for interleaving in the transmitter  300 . Thus, the deinterleaver  424  may deinterleave the LLRs of the LDPC codeword bits by reversely performing interleaving performed in the interleaver  341 . 
     The LDPC decoder  4300  may perform LDPC decoding based on the LLRs received from the rate dematcher  420 . 
     Specifically, the LDPC decoder  430 , which is a counterpart of the LDPC encoder  330  of the transmitter  300 , may perform an operation corresponding to the LDPC encoder  330 . 
     For this purpose, the receiver  400  may pre-store information about a parameter used for LDPC encoding according to a mode in the transmitter  300 . Thus, the LDPC decoder  430  may perform LDPC decoding based on the LLRs received from the rate dematcher  420  according to a mode. 
     For example, the LDPC decoder  430  may perform LDPC decoding based on the LLRs received from the rate dematcher  420  in an iterative decoding scheme based on a sum-product algorithm, and output error-corrected bits according to the LDPC decoding. 
     The zero remover  440  may remove zero bits in the bits received from the LDPC decoder  430 . 
     Specifically, the zero remover  440 , which is a counterpart of the zero padder  320  in the transmitter  300 , may perform an operation corresponding to the zero padder  320 . 
     For this purpose, the receiver  400  may pre-store information about a parameter used for padding zero bits in the transmitter  300 . Thus, the zero remover  440  may remove zero bits padded by the zero padder  320  in the bits received from the LDPC decoder  430 . 
     The desegmenter  450 , which is a counterpart of the segmenter  310  in the transmitter  300 , may perform an operation corresponding to the segmenter  310 . 
     For this purpose, the receiver  400  may pre-store information about a parameter used for segmentation in the transmitter  300 . Thus, the desegmenter  450  may recover pre-segmentation bits by combining segments of the bits received from the zero remover  440 , that is, input bits of a variable length. 
       FIG.  10    is a block diagram of a decoder according to an embodiment of the present disclosure. 
     Referring to  FIG.  10   , a decoder  1000  may include an LDPC decoder  1010 . The LDPC decoder  1010  performs LDPC decoding on an LDPC codeword based on a parity check matrix. 
     For example, the LDPC decoder  1010  may generate information word bits by performing LDPC decoding by passing the LLRs of LDPC codeword bits in an iterative decoding algorithm. 
     An LLR is a channel value corresponding to an LDPC codeword bit, which may be expressed in various manners. 
     For example, an LLR may be represented as a value obtained by performing a log operation on the ratio between the probability of 0 and the probability of 1 for a bit transmitted on a channel by a transmitter. The LLR may be a bit value decided by hard decision, and may be a representative value of a range to which the probability of the transmitted bit being 0 or 1 belongs. 
     In this case, the transmitter may generate an LDPC codeword using the LDPC encoder  610  illustrated in  FIG.  6   . 
     The parity check matrix used for the LDPC decoding may have the same structure as a parity check matrix described by Equation 3 and Equation 4. 
     In this case, the LDPC decoder  1010  may perform LDPC decoding using a parity check matrix defined differently according to a code rate (i.e., the code rate of an LDPC code). 
     For example, if the code rate is 7/8, the LDPC decoder  1010  may perform LDPC decoding using a parity check matrix as defined by Table 6. If the code rate is 3/4, the LDPC decoder  1010  may perform LDPC decoding using a parity check matrix as defined by Table 5. If the code rate is 1/2, the LDPC decoder  1010  may perform LDPC decoding using a parity check matrix as defined Table 8. If the code rate is 1/4, the LDPC decoder  1010  may perform LDPC decoding a parity check matrix as defined by [Table 7]. 
       FIG.  11    is a block diagram of an LDPC decoder according to an embodiment of the present disclosure. 
     Referring to  FIG.  11   , the LDPC decoder  1010  may perform LDPC decoding using an iterative decoding algorithm. In this case, the LDPC decoder  1010  may be configured in the structure illustrated in  FIG.  11   . The detailed structure illustrated in  FIG.  11    in that the iterative decoding algorithm is well known. 
     Referring to  FIG.  11   , a decoder  1100  includes an input processor  1101 , a memory  1102 , a variable node operator  1104 , a controller  1106 , a check node operator  1108 , and an output processor  1110 . 
     The input processor  1101  stores input values. Specifically, the input processor  1101  may store LLRs of a signal received on a radio channel. 
     The controller  1104  determines the number of values input to the variable node operator  1104 , an address of the memory  1102 , the number of values input to the check node operator  1108 , an address of the memory  1102 , and so on based on a block size (that is, a codeword length) of the signal received on the radio channel, and a parity check matrix corresponding to a code rate. 
     According to an embodiment of the present disclosure, decoding may be performed based on a parity check matrix with the indexes of rows having is in column 0 of an i th  column group as defined in Table 6 to Table 9. 
     The memory  1102  stores input data and output data of the variable node operator  1104  and the check node operator  1108 . 
     The variable node operator  1104  receives data from the memory  1102  according to the information about the addresses and number of input data, received from the controller  1106 , and performs variable node computation. The variable node operator  1104  stores variable node computation results in the memory  1102  based on the information about the addresses and number of output data, received from the controller  1106 . The variable node operator  1104  also provides the variable node calculation results to the output processor  1110  based on data received from the input processor  1101  and the memory  1102 . Herein, the variable node computation has been described before with reference to  FIG.  5   . 
     The check node operator  1108  receives data from the memory  1102  according to information about the addresses and number of input data, received from the controller  1106 , and performs check node computation. The check node operator  1108  stores variable node computation results in the memory  1102  based on information about the addresses and number of output data, received from the controller  1106 . Herein, the check node computation has been described before with reference to  FIG.  5   . 
     The output processor  1110  hard-decides whether information word bits of a codeword transmitted by the transmitter are 0s or is based on data received from the variable node operator  1104 , and outputs the hard-decision values. The output values of the output processor  1110  are final decoded values. In this case, the hard decision may be made based on the sum of all message values input to one variable node (an initial message value and all message values received from check nodes). 
     Meanwhile, the decoder  1100  may further include a memory (not shown) for pre-storing information about code rates, codeword lengths, and parity check matrices of LDPC codes, and the LDPC decoder  1010  may perform LDPC decoding using this information. However, the information may be received from the transmitter. 
       FIG.  12    illustrates a structure of a transport block according to an embodiment of the present disclosure. 
     Referring to  FIG.  12   , &lt;Null&gt; bits may be added to make the lengths of segments equal. 
     &lt;Null&gt; bits may be added to match the information length of an LDPC code. Since the same exponent matrix is produced for different Z values in the present disclosure, the increase of implementation complexity of a parity check matrix may be overcome. Even though encoding is performed using different encoders for Z=a and Z=b, the same result may be achieved. 
       FIG.  19    is a flowchart illustrating an LDPC encoding method based on a sequence according to an embodiment of the present disclosure. 
     Referring to  FIG.  19   , the transmitter/receiver reads a stored exponent matrix (or sequence) corresponding to a parity check matrix in operation  1910 . In operation  1920 , the transmitter/receiver determines a block size Z corresponding to the size of a circulant permutation matrix included in a parity check matrix. The parity-check matrices in operations  1910  and  1920  may be the same or differ from each other. The sequence of operations  1910  and  1920  may be changed. 
     Subsequently, the transmitter may determine an appropriate integer based on the determined block size in a predetermined method in operation  1930 . Operation  1930  may be performed in various methods. For example, if the determined block size is Z, the integer may be determined by k=└log 2  Z┘. In another embodiment of the present disclosure, the transmitter may determine a range or set including the determined block size, when needed. A representative integer of the range or set may be the integer determined in operation  1930 . Although the representative integer may be a minimum value, a maximum value, an intermediate value, or an average value of the values of the range or set, any value is available as far as the value is an integer uniquely representing the range or set. 
     In operation  1940 , the transmitter converts the sequence read in operation  1910  based on the integer determined in operation  1930 . In operation  1950 , the transmitter performs LDPC encoding based on the converted sequence. 
     The sequence conversion in operations  1930  and  1940  is characterized by conversion to the same sequence for at least two different block sizes among block sizes determined in operation  1920 . This characteristic may be obtained in various manners. In an embodiment, if a rule of determining the same integer is applied to the at least two different block sizes in operation  1930 , the characteristic may be easily obtained. 
       FIG.  20    is a block diagram of a transmitter for performing LDPC encoding based on a sequence according to an embodiment of the present disclosure. 
     Referring to  FIG.  20   , the transmitter includes an LDPC encoder  2010 , a memory  2020 , a controller  2030 , and a converter  2040 . 
     The memory  2020  reads a sequence corresponding to a parity check matrix. 
     The controller  2030  provides information about block sizes to the converter  2040  and controls conversion of the sequence. 
     Even though the converter  2040  receives the information about different block sizes from the controller  2030 , there is always a case in which the same sequence is out for the input of a sequence from the memory  202 . 
     The LDPC encoder  2010  performs LDPC encoding based on the converted sequence. 
     Obviously, the receiver may include a controller for receiving an LDPC codeword produced by LDPC encoding based on a sequence converted in the manner illustrated in  FIGS.  19  and  20   , and decoding the received codeword. 
     As is apparent from the foregoing description, the present disclosure can support an LDPC code of a variable length and a variable code rate. 
     While the present disclosure has been shown and described with reference to various embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the present disclosure as defined by the appended claims and their equivalents.