Abstract:
A method includes defining a model matrix of size (n−k)×n, where n and k are positive integers, and where the model matrix includes a first sub-matrix corresponding to positions of data bits and a second sub-matrix corresponding to positions of parity bits. The second sub-matrix includes a multi-diagonal matrix with a triple diagonal structure. The triple diagonal structure includes a first and second central diagonals and a last row diagonal. Bits of the first and second central diagonal and the last row diagonal are equal to 1 and a remainder of bits in the multi-diagonal matrix are equal to 0. The method further includes: generating a compact matrix based on the model matrix; generating a parity matrix based on the compact matrix; determining the parity bits based on the parity matrix; and transmitting a codeword, based on the parity bits, over a channel and between communication devices.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a National Stage of International Application No. PCT/ES2010/070067, filed Feb. 5, 2010 and claims the benefit of Spanish Application P200900343, filed Feb. 6, 2009 and Spanish Application No. P201030066, filed Jan. 20, 2010. The disclosures of the above applications are incorporated herein by reference in their entirety. 
     FIELD 
     The invention presented here relates to data transmission, and more specifically, to the communication of data across noisy media, that is, communication media or channels that may introduce errors in the communication. 
     BACKGROUND 
     In communication environments it is common for a communication medium or external signals to introduce signal errors. Said errors must be detected and, if possible, corrected in reception so that correct data may be recovered. Several techniques exist for error detection and correction in the state of the art, including the codification and decodification of data based on low density parity check codes for correcting errors. 
     Low density parity check codes (LDPC) are error correction codes that are used in transmitting over noisy transmission channels. These codes introduce a certain redundancy in a message (a larger number of bits is sent than in the original message), but in such a way that at reception it is possible to detect whether there are errors in the message received and to correct the errors. 
     An LDPC code is a code whose parity matrix is not very dense, that is to say that the majority of its elements are zeros. This type of code was published for the first time at the beginning of the 1960s, by Robert G. Gallagher “Low Density Parity Check Codes,” M.I.T. Press, 1963, and was shown to have features very close to the known Shannon limit (theoretical maximum rate for data transmission). However, with the original definition of the codes and the technology of that time, an attainable implementation of adequate complexity was not possible. Recently, thanks to the evolution of integrated circuits and the invention of structured matrices, these codes are once again of great interest. 
     In the state of the art there are multiple methods for achieving codification and decodification of errors. Some methods are those published in patents U.S. Pat. No. 7,343,548B2 and U.S. Pat. No. 7,203,897B2, both entitled “Method and Apparatus for Encoding and Decoding Data,” each of which outlines methods for improving protection when confronted with errors in data transmission. The invention can also be related to standards IEEE802.16e and 802.11n, which present codification and decodification for reducing errors. In any case, the patents and standards mentioned use the double diagonal structure, which is known in the state of the art, while the structure presented in this document is new and allows an implementation with better features without increasing the level of complexity (thus at lower cost) of protection against errors in communicating data over noisy media. It is known in the state of the art that having columns with a Hamming weight equal to or less than 2 in the parity matrix restricts the features of the LDPCs. However, for reasons of complexity of implementation of the codifier, matrices with a double diagonal section H b1  have been used in the state of the art. The new structure presented in this document, which adds a third diagonal to the section H b1  of the binary model matrix, allows the total number of columns in the parity matrix with a Hamming weight less than or equal to 2 to be lower, and thus better features can be achieved. This third diagonal was selected in such a way that the increase in the complexity of implementation of the codifier is practicably negligible. 
     The documents previously presented do not interfere with the novelty nor the inventive superiority of the present invention. Although they are all based on utilization of the LDPC technology, which is known in the state of the art, the method and device of this document utilizes a type of quasi-cyclic code (Quasi-Cyclic Low Density Parity Check Code, or QC-LDPC), and applies a parity matrix with a different structure as the central point of the invention. 
     Throughout this document, a specific nomenclature will be employed to differentiate the elements utilized throughout the description of the invention. A bold capital letter (e.g., A) indicates that the element is a matrix; a bold small letter (e.g., a) indicates that the element is a vector, while a small non-bold letter (e.g., a) indicates that the element is a scalar value. On the other hand, scalar elements that comprise a matrix of the size M×N are indicated in the form a(i,j), where the tuple (i,j) is the position of said element within the matrix, with 0≦i≦M−1 being the row number and 0≦j≦N−1 the column number. The elements that comprise a vector of size M are annotated in the form a(i), with (i) being the position of the element in the vector (0≦i≦M−1). 
     Also, throughout the invention the term “cyclical rotation” will be used, which will be defined below. A cyclical rotation z on a vector a=[a(0), a(1), . . . , a(M−2), a(M−1)] consists in cyclically rotating its elements toward the right, obtaining the vector [a((M−z)% M), . . . , a((M−z−1)% M)] as the result, with % being the “module” operator. In the same way, a cyclical rotation z applied over a matrix A=[a(0), . . . , a(N−1)] operates on its columns, obtaining the matrix [a((N−z)% N), . . . , a((N−z−1)% N)] as the result. A cyclical rotation can also be defined in the opposite direction (toward the left) so that a cyclical rotation z toward the right is equivalent to a cyclical rotation M−z and N−z respectively for vector and matrix toward the left. 
     SUMMARY 
     In order to achieve the objectives and avoid the drawbacks indicated in above sections, the invention consists of a method and device for communicating data over noisy media. Concretely, the invention presents a method for codifying data used in transmission, its associated codification device, a method for decodifying and its associated decodification device. This group of inventions makes up a unique inventive concept, which will be described below. If the method or device is used in transmission, the equivalent must also be used in reception, and vice versa, so that the data sent can be recovered. 
     The method for codifying data is applied in transmission and generates parity bits on a block of data in such a way that a code word of N bits is generated from a word of K bits (N&gt;K) that includes protection against errors. Said procedure comprises multiple steps. First a factor b is selected, which is a natural number between 1 and k such that the division of N and K by the factor b will be natural numbers (n=N/b; k=K/b). Below we define a binary model matrix H 0 =[H a |H b ] of size (n−k)×n as the combination of a submatrix corresponding to the positions of the data bits H a  and a submatrix corresponding to the parity bits H b , where said second submatrix H b =[hb 0 |Hb 1 ] is composed of a column vector of n−k positions h b0  and a matrix H b1  having a triple diagonal structure, that is, where the elements of the two central diagonals h b1 (i, i), h b1 (i+1, i) 0≦i≦n−k−2 and the diagonal of the last row h b1 (n−k−1,0) are equal to 1, where n−k is the number of rows and columns of the matrix H b , and the rest of the elements are equal to zero. Afterward, the compact matrix H 1  is generated and from the compact matrix H 1 , the parity matrix H. From there, one takes a block of data and uses the parity matrix H on the block of data to determine the parity bits that correspond to said block. Finally, the parity bits are transmitted together with the block of data. 
     In one implementation of the method, it is possible to eliminate one or more elements of the code word before they are transmitted, reducing the redundancy in the transmission without seriously harming the capacity for protection against errors. This technique is called “puncturing.” In this case the word transmitted will have a smaller number of bits than the code word obtained with the initial procedure. 
     The data codification device comprises means for storing the compact matrix H 1  derived from a binary model matrix H 0 =[H a |H b ] formed as the combination of a submatrix corresponding to the position of the data bits H a  and a submatrix corresponding to the parity bits H b , where said second submatrix H b =[hb 0 |Hb 1 ] is composed of a column vector of n−k positions h b0  and a matrix H b1  having a triple diagonal structure, that is, where the elements of the two central diagonals h b1 (i, i), h b1 (i=1, i) 0≦i≦n−k−2 and the diagonal of the last row h b1 (n−k−1,0) are equal to 1, where n−k is the number of rows and columns of the matrix H b , and the rest of the elements are equal to zero; and of a microprocessor that takes the block of data, uses the compact matrix H 1  to generate the parity matrix H, applies the parity matrix H to the block of data to obtain the parity bits corresponding to the block and adds the parity bits to the block of data before they are transmitted. 
     In one concrete implementation of this device, one or more elements of the code word are eliminated after adding the parity bits to the block of data but prior to transmission by applying the puncturing technique. In this way the word transmitted will have a smaller number of bits than the code word originally generated. 
     On the other hand, the data decodification method operates at reception and estimates which is the block of data received from a signal vector received from the channel. From a received code word of N bits (which can have errors due to the channel noise) the K-bit data word that the transmitter wanted to send is obtained. This first takes a signal vector from the channel and the binary model matrix H 0 =[H a |H b ] which is a combination of a submatrix corresponding to the position of the data bits H a  and a submatrix corresponding to the parity bits H b , where said second submatrix H b =[hb 0 |Hb 1 ] is composed of a column vector of n−k positions h b0  and a matrix H b1  having a triple diagonal structure, that is, where the elements of the two central diagonals h b1 (i, i), h b1 (i+1, i) 0≦i≦n−k−2 and the diagonal of the last row h b1 (n−k−1,0) are equal to 1, where n−k is the number of rows and columns of the matrix H b , and the rest of the elements are equal to zero. Afterward, the compact matrix H 1  is generated and from it, the parity matrix H, and finally the data block is estimated from the vector signal received and the parity matrix H. 
     If the puncturing technique was used in transmission, the lost data must be recovered at reception. In this case, an indicator value is inserted in the positions eliminated in transmission by the puncturing technique prior to making the estimation of the data block from the signal vector received and the parity matrix. 
     The data decodification device comprises means for storing the compact matrix H 1  that is derived from the binary model matrix H 0 =[H a |H b ] which is a combination of a submatrix corresponding to the position of the data bits H a  and a submatrix corresponding to the parity bits H b , where said second submatrix H b =[hb 0 |Hb 1 ] is composed of a column vector of n−k positions h b0  and a triple diagonal structure H b1 , that is, one in which the elements of the two central diagonals h b1 (i, i), h b1 (i+1, i) 0≦i≦n−k−2 and the diagonal of the last row h b1 (n−k−1,0) are equal to 1, where n−k is the number of rows and columns of the matrix H b , and the rest of the elements are equal to zero; a microprocessor that generates the parity matrix H from the compact matrix H 1 , applies said parity matrix H to the signal vector received, and estimates the data block received. 
     If the transmitting device used the puncturing technique, the lost data must be recovered before error correction is done. Therefore, in this implementation and before applying the parity matrix H to the signal vector received, an indicator value is inserted in the positions eliminated in transmission by the puncturing technique. 
     In one implementation it is possible to use one of the following compact matrices H 1  to obtain 336-bit code words with a codification rate of 1/2. The matrix: 
     
       
         
           
             
               
                 
                   
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                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   12 
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   9 
                 
                 
                   11 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   2 
                 
                 
                   12 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   11 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   7 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   11 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   4 
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   2 
                 
                 
                   5 
                 
                 
                   4 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   6 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   9 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   9 
                 
                 
                   0 
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   12 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   1 
                 
                 
                   4 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   9 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     10 
                     ⁢ 
                       
                   
                 
                 
                   
                     11 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     3 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     4 
                     ⁢ 
                       
                   
                 
                 
                   
                     8 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
             
               
           
         
       
     
     In another implementation it is possible to use one of the following compact matrices H 1  to obtain 1920-bit code words with a codification rate of 1/2. The matrix: 
                       -   1     ⁢               52   ⁢                 -   1     ⁢               64   ⁢                 -   1     ⁢                 -   1     ⁢               60   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢               1   ⁢                 -   1     ⁢               0   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 10         -   1           -   1           -   1           -   1         79         -   1           -   1         79         -   1         78       51         -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1             9         -   1           -   1           -   1           -   1           -   1           -   1         75       29       72       8         -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1               -   1         52       16       63         -   1           -   1         65         -   1           -   1           -   1           -   1           -   1         40         -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1               -   1         24         -   1           -   1         47       1       39         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1             52         -   1           -   1           -   1           -   1           -   1           -   1           -   1         53       79       48         -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1               -   1         0         -   1           -   1         72         -   1         67       57         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1               -   1         7         -   1           -   1           -   1         2       50         -   1           -   1           -   1           -   1         15         -   1           -   1           -   1           -   1           -   1           -   1           -   1         0                 ⁢   0           -   1           -   1           -   1             15         -   1         19         -   1           -   1           -   1           -   1           -   1         75       51       43         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1           -   1             72         -   1           -   1           -   1         38         -   1           -   1           -   1         69         -   1         62         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1               -   1         19         -   1         41         -   1           -   1         1       41         -   1           -   1           -   1           -   1         0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0               41   ⁢                 -   1     ⁢               17   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢               15   ⁢                 -   1     ⁢               30   ⁢                 -   1     ⁢               40   ⁢               6   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                       ⁢   0   ⁢                         
or else this matrix:
 
     
       
         
           
             
               
                 
                   27 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   55 
                 
                 
                   19 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   30 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   70 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   47 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   62 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   41 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   44 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   59 
                 
                 
                   60 
                 
                 
                   25 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   16 
                 
                 
                   77 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     5 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   48 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   45 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   27 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   46 
                 
                 
                   19 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   63 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   55 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   48 
                 
                 
                   26 
                 
                 
                   10 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   42 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   21 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   58 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   41 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   78 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     7 
                   
                 
                 
                   52 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   29 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     9 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   37 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   35 
                 
                 
                   21 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   22 
                 
                 
                   72 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   47 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   35 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   35 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   70 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   46 
                 
                 
                   28 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   38 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     8 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   10 
                 
                 
                   58 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
             
               
           
         
       
     
     In another implementation it is possible to use one of the following compact matrices H 1  to obtain 8640-bit code words with a codification rate of 1/2. The matrix: 
                             ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1             -   1         297       106       328                 ⁢     -   1                     ⁢     -   1                     ⁢   99           -   1                   ⁢   0           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1             290                 ⁢   0         312                 ⁢     -   1                     ⁢   32           -   1         120                 ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1             -   1                   ⁢   0                   ⁢   0           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1             183                 ⁢   57                   ⁢     -   1                     ⁢     -   1           187       68                 ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1           260                 ⁢     -   1           81                 ⁢     -   1                     ⁢   0                   ⁢   0           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                       ⁢     -   1                     ⁢     -   1                     ⁢     -   1           323                 ⁢     -   1             -   1                   ⁢     -   1           137       354                 ⁢     -   1                     ⁢     -   1           162         -   1                   ⁢     -   1             -   1                   ⁢   0                   ⁢   0           -   1           -   1           -   1           -   1           -   1           -   1           -   1                       ⁢     -   1                     ⁢     -   1           228                 ⁢     -   1                     ⁢     -   1             -   1                   ⁢     -   1           224                 ⁢     -   1           114                 ⁢     -   1           245         -   1                   ⁢     -   1             -   1           -   1                   ⁢   0                   ⁢   0           -   1           -   1           -   1           -   1           -   1           -   1             113                 ⁢   98                   ⁢     -   1                     ⁢     -   1           120       23                 ⁢     -   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           138                 ⁢     -   1           187                 ⁢   45                   ⁢   62           -   1                   ⁢     -   1             -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1           -   1           -   1           -   1                       ⁢     -   1                     ⁢     -   1           142                 ⁢     -   1                     ⁢     -   1             -   1         347                 ⁢   67                   ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢   46           -   1                   ⁢     -   1             -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1           -   1           -   1             328       265                 ⁢     -   1                     ⁢   66         156       96                 ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1             -   1                   ⁢     -   1             -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1           -   1             212       184                 ⁢     -   1                     ⁢     -   1           102         -   1                   ⁢     -   1                     ⁢     -   1                     ⁢     -   1           120                 ⁢     -   1                     ⁢     -   1             -   1                   ⁢     -   1             -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1                       ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1             -   1                   ⁢     -   1                     ⁢   80                   ⁢   15                   ⁢     -   1           329       153                 ⁢   0                   ⁢     -   1             -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0             207       70                 ⁢     -   1                     ⁢   7         235         -   1                   ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1                     ⁢     -   1           81       185         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                     
or else the matrix:
 
     
       
         
           
             
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     34 
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                       
                     ⁢ 
                     1 
                   
                 
                 
                   
                     95 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     279 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   248 
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   134 
                 
                 
                   356 
                 
                 
                   275 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     51 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   27 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   22 
                 
                 
                   152 
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   57 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   124 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   290 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   281 
                 
                 
                   15 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   340 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   99 
                 
                 
                   336 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   33 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     163 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   46 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   306 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   86 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   185 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   24 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   94 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   223 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   225 
                 
                 
                   325 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   297 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     46 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   314 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   59 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   67 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   120 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   121 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   161 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   303 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   264 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   303 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   185 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   138 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     312 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     100 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     144 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     307 
                     ⁢ 
                       
                   
                 
                 
                   
                     33 
                     ⁢ 
                       
                   
                 
                 
                   
                     166 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     In another implementation it is possible to use the following compact matrix H 1  to obtain 1440-bit code words with a codification rate of 2/3. 
     
       
         
           
             
               
                 
                   
                     49 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     21 
                     ⁢ 
                       
                   
                 
                 
                   31 
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     57 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     19 
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   29 
                 
                 
                   
                     2 
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   19 
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   7 
                 
                 
                   22 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   37 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   32 
                 
                 
                   10 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   26 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   59 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   48 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   53 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   20 
                 
                 
                   50 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   16 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   49 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   28 
                 
                 
                   14 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   58 
                 
                 
                   23 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   15 
                 
                 
                   54 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   18 
                 
                 
                   49 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   55 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   58 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   9 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   26 
                 
                 
                   57 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   41 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   31 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   21 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   10 
                 
                 
                   49 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   59 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   7 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   30 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   18 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   48 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   7 
                 
                 
                   59 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   48 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   50 
                 
                 
                   18 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   11 
                 
                 
                   52 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   59 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   37 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   10 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     24 
                     ⁢ 
                       
                   
                 
                 
                   
                     16 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     53 
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     41 
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   38 
                 
                 
                   
                     51 
                     ⁢ 
                       
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   58 
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     59 
                     ⁢ 
                       
                   
                 
                 
                   
                     8 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     In another implementation it is possible to use the following compact matrix H 1  to obtain 6480-bit code words with a codification rate of 2/3: 
     
       
         
           
             
               
                 
                   
                     78 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     167 
                     ⁢ 
                       
                   
                 
                 
                   
                     237 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     3 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     266 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     1 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       n 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       02 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     153 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     212 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   1 
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
                 
                   
                       
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   83 
                 
                 
                   189 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   68 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   178 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   90 
                 
                 
                   205 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   13 
                 
                 
                   4 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   226 
                 
                 
                   147 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   46 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   76 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   116 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   211 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   112 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   118 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   92 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   214 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   236 
                 
                 
                   241 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   157 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   143 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   214 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   207 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   144 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   258 
                 
                 
                   264 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   53 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   114 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   172 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   82 
                 
                 
                   262 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   62 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   153 
                 
                 
                   120 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   199 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   126 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   61 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   183 
                 
                 
                   15 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   134 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   100 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   141 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   36 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   17 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   156 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   124 
                 
                 
                   162 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   57 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     196 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     187 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     73 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     80 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     139 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     57 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     236 
                     ⁢ 
                       
                   
                 
                 
                   
                     267 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     62 
                     ⁢ 
                       
                   
                 
                 
                   
                     256 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     In another implementation it is possible to use the following compact matrix H1 to obtain 1152-bit code words with a codification rate of 5/6: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     13 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       3 
                       ⁢ 
                         
                       ⁢ 
                       2 
                     
                     ⁢ 
                       
                   
                 
                 
                   47 
                 
                 
                   41 
                 
                 
                   24 
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     25 
                     ⁢ 
                       
                   
                 
                 
                   22 
                 
                 
                   40 
                 
                 
                   
                     1 
                     ⁢ 
                       
                   
                 
                 
                   
                     31 
                     ⁢ 
                       
                   
                 
                 
                   
                     8 
                     ⁢ 
                       
                   
                 
                 
                   15 
                 
                 
                   20 
                 
                 
                   15 
                 
                 
                   42 
                 
                 
                   
                     30 
                     ⁢ 
                       
                   
                 
                 
                   
                     13 
                     ⁢ 
                       
                   
                 
                 
                   
                     3 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   25 
                 
                 
                   46 
                 
                 
                   15 
                 
                 
                   43 
                 
                 
                   45 
                 
                 
                   29 
                 
                 
                   39 
                 
                 
                   47 
                 
                 
                   23 
                 
                 
                   38 
                 
                 
                   39 
                 
                 
                   12 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   21 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   38 
                 
                 
                   33 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   39 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   35 
                 
                 
                   45 
                 
                 
                   45 
                 
                 
                   38 
                 
                 
                   14 
                 
                 
                   16 
                 
                 
                   6 
                 
                 
                   11 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   18 
                 
                 
                   7 
                 
                 
                   41 
                 
                 
                   35 
                 
                 
                   17 
                 
                 
                   32 
                 
                 
                   45 
                 
                 
                   41 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   18 
                 
                 
                   17 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     9 
                     ⁢ 
                       
                   
                 
                 
                   
                     32 
                     ⁢ 
                       
                   
                 
                 
                   
                     6 
                     ⁢ 
                       
                   
                 
                 
                   22 
                 
                 
                   26 
                 
                 
                   31 
                 
                 
                   
                     9 
                     ⁢ 
                       
                   
                 
                 
                   
                     8 
                     ⁢ 
                       
                   
                 
                 
                   22 
                 
                 
                   32 
                 
                 
                   
                     40 
                     ⁢ 
                       
                   
                 
                 
                   
                     4 
                     ⁢ 
                       
                   
                 
                 
                   
                     18 
                     ⁢ 
                       
                   
                 
                 
                   40 
                 
                 
                   36 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     23 
                     ⁢ 
                       
                   
                 
                 
                   
                     31 
                     ⁢ 
                       
                   
                 
                 
                   
                     41 
                     ⁢ 
                       
                   
                 
                 
                   
                     39 
                     ⁢ 
                       
                   
                 
                 
                   
                     20 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     In another implementation it is possible to use the following compact matrix H 1  to obtain 5184-bit code words with a codification rate of 5/6: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     47 
                     ⁢ 
                       
                   
                 
                 
                   
                     146 
                     ⁢ 
                       
                   
                 
                 
                   
                     203 
                     ⁢ 
                       
                   
                 
                 
                   
                     184 
                     ⁢ 
                       
                   
                 
                 
                   
                     112 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     116 
                     ⁢ 
                       
                   
                 
                 
                   
                     103 
                     ⁢ 
                       
                   
                 
                 
                   
                     181 
                     ⁢ 
                       
                   
                 
                 
                   
                     3 
                     ⁢ 
                       
                   
                 
                 
                   
                     140 
                     ⁢ 
                       
                   
                 
                 
                   
                     38 
                     ⁢ 
                       
                   
                 
                 
                   
                     68 
                     ⁢ 
                       
                   
                 
                 
                   
                     91 
                     ⁢ 
                       
                   
                 
                 
                   
                     70 
                     ⁢ 
                       
                   
                 
                 
                   
                     191 
                     ⁢ 
                       
                   
                 
                 
                   
                     138 
                     ⁢ 
                       
                   
                 
                 
                   
                     62 
                     ⁢ 
                       
                   
                 
                 
                   
                     14 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   117 
                 
                 
                   203 
                 
                 
                   67 
                 
                 
                   194 
                 
                 
                   206 
                 
                 
                   133 
                 
                 
                   174 
                 
                 
                   212 
                 
                 
                   104 
                 
                 
                   171 
                 
                 
                   176 
                 
                 
                   56 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   96 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   167 
                 
                 
                   149 
                 
                 
                   4 
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   177 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   153 
                 
                 
                   206 
                 
                 
                   198 
                 
                 
                   173 
                 
                 
                   55 
                 
                 
                   72 
                 
                 
                   28 
                 
                 
                   53 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   82 
                 
                 
                   34 
                 
                 
                   186 
                 
                 
                   161 
                 
                 
                   80 
                 
                 
                   144 
                 
                 
                   204 
                 
                 
                   187 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   84 
                 
                 
                   77 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     44 
                     ⁢ 
                       
                   
                 
                 
                   
                     147 
                     ⁢ 
                       
                   
                 
                 
                   
                     27 
                     ⁢ 
                       
                   
                 
                 
                   
                     83 
                     ⁢ 
                       
                   
                 
                 
                   
                     118 
                     ⁢ 
                       
                   
                 
                 
                   
                     130 
                     ⁢ 
                       
                   
                 
                 
                   
                     41 
                     ⁢ 
                       
                   
                 
                 
                   
                     38 
                     ⁢ 
                       
                   
                 
                 
                   
                     100 
                     ⁢ 
                       
                   
                 
                 
                   
                     146 
                     ⁢ 
                       
                   
                 
                 
                   
                     183 
                     ⁢ 
                       
                   
                 
                 
                   
                     19 
                     ⁢ 
                       
                   
                 
                 
                   
                     85 
                     ⁢ 
                       
                   
                 
                 
                   
                     180 
                     ⁢ 
                       
                   
                 
                 
                   
                     163 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     106 
                     ⁢ 
                       
                   
                 
                 
                   
                     140 
                     ⁢ 
                       
                   
                 
                 
                   
                     185 
                     ⁢ 
                       
                   
                 
                 
                   
                     177 
                     ⁢ 
                       
                   
                 
                 
                   
                     94 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     One implementation in which the puncturing technique is used starts with an 1152-bit code word and codification rate of 5/6 and applies the following puncturing pattern: 
               pp   1152     (     16   /   18     )       ⁢     [     ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   720       ⁢           ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   36       ⁢           ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   360       ⁢           ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   36       ⁢     ]           
in order to obtain a 1080-bit code word and codification rate of 16/18.
 
     Another implementation in which the puncturing technique is used starts with a code word with 5184 bits and a codification rate of 5/6 and applies the following puncturing pattern 
               pp   5184     (     16   /   18     )       ⁢     [     ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   3240       ⁢           ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   162       ⁢           ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   972       ⁢           ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   162       ⁢           ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   648       ⁢     ]           
to obtain a code word of 4860 bits and codification rate of 16/18.
 
     Another implementation in which the puncturing technique is used starts with a code word with 1152 bits and a codification rate of 5/6 and applies the following puncturing pattern 
               pp   1152     (     20   /   21     )       ⁢     [     ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   720       ⁢           ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   48       ⁢           ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   240       ⁢           ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   96       ⁢           ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   48       ⁢     ]           
to obtain a 1008-bit code word and codification rate of 20/21.
 
     And last, a final implementation in which the puncturing technique is used starts with a 5184-bit code word and codification rate of 5/6 and applies the following puncturing pattern 
               pp   5184     (     20   /   21     )       ⁢     [     ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   216       ⁢           ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   4320       ⁢           ⁢       0   ⁢           ⁢   0   ⁢           ⁢   ⋯   ⁢           ⁢   0       ︸   432       ⁢           ⁢       1   ⁢           ⁢   1   ⁢           ⁢   ⋯   ⁢           ⁢   1       ︸   216       ⁢     ]           
to obtain a 4536-bit code word and codification rate of 20/21.
 
     Below, in order to facilitate a greater understanding of this descriptive document, is given the description, by way of illustration but not limitation, of one example implementation of the invention. This forms an integral part of the same document. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The present disclosure will become more fully understood from the detailed description and the accompanying drawings, wherein: 
         FIG. 1  is a block diagram of the codifier in the transmitter; 
         FIG. 2  is a block diagram of the decodifier in the transmitter; 
         FIG. 3  is a two-part graph of the matrix H of the example; and 
         FIG. 4  is a flow diagram of the construction of a structured LDPC code. 
     
    
    
     DESCRIPTION 
     Following is the description of one example of implementation of the invention, making reference to the numbering adopted in the figures. 
     The problem that the invention procedure is intended to solve, from a theoretical point of view, consists of success in optimizing correction of errors in data communication using low-cost hardware implementations and LDPC codes. 
     An LDPC code is a linear code that operates on blocks of data. The code is defined by its parity matrix H. In this implementation example the codes are binary codes, but it is possible to generalize the invention to codes on any Galois field GF(q), where q is ≧2. 
     In transmission, blocks of data are composed of K bits. Said data block is designated as u=[u(0), u(1), . . . , u(K−1)]. After applying the method of the invention, a word is generated of a linear code v=[v(0), v(1), . . . , v(N−1)] with N bits (where N&lt;K). Said code is generated through the product v=uG, where G is a binary matrix K×N, generator of the LDPC code. The set of possible codes generated is called set C, and the codification rate of the code will be R=K/N. 
     Therefore, it is possible to define a code C of codification R as the set of vectors vε C generated by all the possible 2 K  binary vectors by applying the generator matrix G to them. An equivalent definition would be that C is the vectorial space of size N included in the base composed of the K rows of the matrix G. Another alternative form of defining code C is through its parity matrix H, which is the form most used in the state of the art. This matrix, in size (N−K)×N, has as rows the base of the dual space C, and therefore GH T =0. Any vector of the code satisfies
 
vH T =0
 
(where “T” is the transposition operator).
 
     At the time these codes are utilized from a practical viewpoint, it is preferable to consider the codes as systematic codes, that is, those in which the bits of the code word are between the data bits. Without losing generality, this example is centered on the case where v=[u|p], where p=[p(0), p(1), . . . p(N−K−1)] is a vector composed of the parity bits, u the data block that is to be transmitted and v the code word actually transmitted (after including the LDPC code). 
     Below is shown an example of implementation in which the relation between the parity matrix and a code word can be observed. In this example, the code has a codification rate of R=1/2 and is defined by the following parity matrix: 
     
       
         
           
             H 
             = 
             
               [ 
               
                 
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     In this matrix, the left section corresponds to the K=5 data bits, while the right section corresponds to the N−K=5 parity bits. By applying the equation vH T =0 to the matrix H the following system of equations is obtained: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         u 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           3 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           3 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           4 
                           ) 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     
                       
                         u 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           4 
                           ) 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     
                       
                         u 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           4 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           3 
                           ) 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     
                       
                         u 
                         ⁡ 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           3 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           4 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           2 
                           ) 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     
                       
                         u 
                         ⁡ 
                         
                           ( 
                           3 
                           ) 
                         
                       
                       + 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           4 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           3 
                           ) 
                         
                       
                       + 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           4 
                           ) 
                         
                       
                     
                     = 
                     0 
                   
                 
               
             
               
           
         
       
     
     An LDPC code can also be represented in graphic form with a two-part graph called a Tanner graph. In a Tanner graph the vertices or nodes are classified in two separate groups or sets: the “variable nodes”, which represent the bits of the code word, and the “check nodes,” which represent the parity relationships. Between both sets of nodes are found the possible edges that define the parity equation. In the case of the code defined in the preceding example, its corresponding graph is represented in  FIG. 3 , where we find 10 variable nodes ( 14 ) and 5 check nodes ( 16 ), united by multiple edges ( 15 ). Each check node is united through edges to 6 variable nodes, as represented in the preceding system of equations. It can be observed that the graph has as many control and variable nodes as the corresponding parity matrix has rows and columns, and that an edge is found between the check node i and variable node j when the element h(i,j) of the matrix, that is, the one located in the row i=0, . . . , N−K−1 and column j=0, . . . , N−1 is non-zero. 
     On the other hand, cycles can be defined on LDPC codes, where a cycle of length  2   c  is defined as the path of  2   c  length edges that processes c check nodes and c variable nodes in the Tanner graph that represents the code before returning to the same beginning node. To optimize the features of the code, it can be demonstrated that it is of vital importance that the number of short cycles be the minimum possible. The cycle of minimum length is called girth. It is particularly desirable that the girth be greater than 4 in order avoid reducing the features of an iterative decodifier. 
     In the original description, R. Gallagher presented codes whose parity matrices were generated randomly. In the state of the art it is generally known that in order to obtain good features, near the Shannon limit, the size of the code must be relatively large, and as a consequence of that, the parity matrix must be large. The problem is that matrices that are large and generated randomly cause difficulty in the implementation of both the codifier and decodifier. One way to avoid this difficulty is to use matrices with a regular structure. Following are the necessary steps for generating a regular structure:
         1. First, one generates a binary model matrix H 0  of size (n−k)×n where n&lt;N, k&lt;K and R=k|n=K|N. If the Hamming weight of the columns and rows of H 0  is constant, the generated code is called regular LDPC. However, better features can be obtained if the matrix is irregular, that is, if the weights of the columns follow a statistical distribution dependent upon the codification rate and of the channel where the data transmission will finally be done.   2. Once the binary model matrix H 0  has been obtained, the compact matrix H 1  is generated, replacing each element of H 0  that is equal to “1” with a pseudo-random positive whole number 0≦x&lt;b (where b=N|n)and each element equal to “0” with the value −1.   3. To obtain the parity matrix H, the positive elements of H 1  are replaced by an identity submatrix rotated cyclically the number of times indicated by the value of the positive element of H 1  in question, and the elements equal to −1 are replaced by a null submatrix of the same size. The size of these submatrices will also be b×b.       

     The result is a parity matrix H of size (N−K)×N which defines an LDPC code with codification rate R=K|N. The grade of density (sparseness) of the matrix will depend upon the size of b. Generally, the larger b is, the better the features that are obtained by using an iterative decodifier. 
     If the cycles of the generated matrix (Tanner graph) turn out to be very short, step 2 (or even step 1 if necessary) should be applied for the purpose of improving these properties. 
     In order to facilitate the implementation of the codifier, it is necessary to generate the binary model matrix H 0  in a specific form. First, said matrix is divided into two parts H 0 =[H a |H b ] where the submatrix H a  corresponds to the positions of the data bits and H b  to the parity bits. The first submatrix is generated pseudo-randomly in the way described previously. However, the second section is usually deterministic. 
     This second section H b , according to the state of the art and intended to facilitate the design of an efficient codifier, takes one of the following two forms; the first form is 
               H   b     =       [       h     b   ⁢           ⁢   0       ❘     H     b   ⁢           ⁢   1         ]     =     [             h   b     ⁡     (   0   )           1       0                               0               h   b     ⁡     (   1   )           1       1       ⋱                                               0       1       ⋱       ⋱                                               ⋱       ⋱       1       0                                               ⋱       1       1               h   b     ⁡     (     n   -   k   -   1     )           0                               0       1         ]             
where the first section is a pseudo-random column vector having a Hamming weight greater than 2 and H b1  is a double-diagonal matrix whose elements h b1 (i, j) are equal to “1” when i=j, i=j+1 and equal to “0” in the remaining positions.
 
     The second way of generating H b  is totally double diagonal 
               H   b     =     [         1       0                                           0           1       1       ⋱                                               0       1       ⋱                                                           ⋱       ⋱       ⋱       ⋱                                               ⋱       1       1       0           0                               0       1       1         ]           
where the elements of the submatrix h b (i, j) are equal to “1” when i=j, i=j+1 and equal to zero in the remaining positions.
 
     Once this base matrix structure has been generated, the compact matrix H 1  is generated, in the form previously described with the sole exception that in the double diagonal part of H b , the “1”s are replaced by the same positive whole number and the “0”s by “−1.” Also, the final parity matrix is obtained by changing the positive wholes by identities rotated cyclically and the negatives by a null submatrix. The procedure can be observed graphically in  FIG. 4 , where block ( 17 ) generates the binary model matrix H 0 , block ( 18 ) generates the compact matrix H 1 , block ( 19 ) decides whether the cycles are long enough, and the process goes on to generate the parity matrix H with block  20 , or else if the cycles are short, the model matrix ( 21 ) or else the base matrix ( 22 ) is generated again. 
     The method and device of the invention together modify the structure of the parity matrix known in the state of the art in order to facilitate the final implementation of the codification and decodification and improve the features. For this, the proposed structure consists in that the section of the binary model H 0  corresponding to the parity bits has the following form: 
               H   b     =       [       h     b   ⁢           ⁢   0       ❘     H     b   ⁢           ⁢   1         ]     =     [             h     b   ⁢           ⁢   0       ⁡     (   0   )           1       0                               0               h     b   ⁢           ⁢   0       ⁡     (   1   )           1       1       ⋱                                               0       1       ⋱       ⋱                                               ⋱       ⋱       1       0                                               ⋱       1       1               h     b   ⁢           ⁢   0       ⁡     (     n   -   k   -   1     )           1                               0       1         ]             
where the structure H b1  is triple diagonal, that is, in addition to the elements of the two central diagonals h b1 (I, i), h b1 (i+1, i), the element of the diagonal of the last line h b1  (n−k−1,0) is also equal to “1.” The compact matrix is generated in the way described previously except that the element of the last row equal to “1” is replaced by a strictly positive whole w≧0.
 
     A binary model matrix H 0  for r=1/2 with n=24 and k=12 might have the following structure: 
     
       
         
           
             
               
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
               
               
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   1 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   1 
                 
               
             
               
           
         
       
     
     A compact matrix H 1  derived from the preceding by a block size N=336 and therefore having an expansion factor b=14 would be the following: 
                     13   ⁢                 -   1     ⁢               7   ⁢                 -   1     ⁢                 -   1     ⁢               1   ⁢                 -   1     ⁢               1   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢               0   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 5         -   1           -   1           -   1         11         -   1           -   1           -   1         4         -   1         4       6         -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1               -   1         13         -   1           -   1           -   1           -   1         11         -   1         10         -   1         9       13         -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1               -   1           -   1         13         -   1           -   1         6         -   1         10         -   1         5         -   1           -   1         4         -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1               -   1           -   1         8       8       2       11         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1               -   1         3         -   1           -   1           -   1           -   1           -   1           -   1         1         -   1         4       1         -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1               -   1           -   1         4       2         -   1         2       9         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1             2         -   1         13         -   1         4       9         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1         0                 ⁢   0           -   1           -   1           -   1               -   1         11         -   1           -   1           -   1           -   1           -   1           -   1         6       4       11       12         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1           -   1               -   1           -   1           -   1         10         -   1           -   1           -   1           -   1         1         -   1         13       13         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1               -   1           -   1         0         -   1           -   1         0       2       2         -   1           -   1           -   1           -   1         0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢               1   ⁢               2   ⁢               11   ⁢               2   ⁢               4   ⁢               12   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                       ⁢   0   ⁢                         
or as preferred, the following matrix can be used as an alternative:
 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     6 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     9 
                     ⁢ 
                       
                   
                 
                 
                   
                     6 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     2 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   12 
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   9 
                 
                 
                   11 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   2 
                 
                 
                   12 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   11 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   7 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   11 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   4 
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   2 
                 
                 
                   5 
                 
                 
                   4 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   6 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   9 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   3 
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   9 
                 
                 
                   0 
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   12 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   1 
                 
                 
                   4 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   9 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     10 
                     ⁢ 
                       
                   
                 
                 
                   
                     11 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     3 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     4 
                     ⁢ 
                       
                   
                 
                 
                   
                     8 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     For a different block size we can define a different compact matrix that can derive from the same binary model matrix or from another, different one. 
     To obtain code words of 1920 bits with a codification rate of 1/2, the following matrix can be used: 
                       -   1     ⁢               52   ⁢                 -   1     ⁢               64   ⁢                 -   1     ⁢                 -   1     ⁢               60   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢             1           -   1     ⁢               0   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 10         -   1           -   1           -   1           -   1         79         -   1           -   1         79         -   1         78       51         -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1             9         -   1           -   1           -   1           -   1           -   1           -   1         75       29       72       8         -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1               -   1         52       16       63         -   1           -   1         65         -   1           -   1           -   1           -   1           -   1         40         -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1           -   1               -   1         24         -   1           -   1         47       1       39         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1           -   1             52         -   1           -   1           -   1           -   1           -   1           -   1           -   1         53       79       48         -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1           -   1               -   1         0         -   1           -   1         72         -   1         67       57         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1         0       0         -   1           -   1           -   1           -   1               -   1         7         -   1           -   1           -   1         2       50         -   1           -   1           -   1           -   1         15         -   1           -   1           -   1           -   1           -   1           -   1           -   1         0                 ⁢   0           -   1           -   1           -   1             15         -   1         19         -   1           -   1           -   1           -   1           -   1         75       51       43         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1           -   1             72         -   1           -   1           -   1         38         -   1           -   1           -   1         69         -   1         62         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0           -   1               -   1         19         -   1         41         -   1           -   1         1       41         -   1           -   1         1         -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1           -   1                   ⁢   0                   ⁢   0               41   ⁢                 -   1     ⁢               17   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢               15   ⁢                 -   1     ⁢               30   ⁢                 -   1     ⁢               40   ⁢               6   ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                 -   1     ⁢                       ⁢   0   ⁢                         
or else, preferably, the matrix:
 
     
       
         
           
             
               
                 
                   
                     27 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     55 
                     ⁢ 
                       
                   
                 
                 
                   
                     19 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     30 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   1 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   70 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   47 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   62 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   41 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   44 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   59 
                 
                 
                   60 
                 
                 
                   25 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   16 
                 
                 
                   77 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   5 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   48 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   45 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   27 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   46 
                 
                 
                   19 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   63 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   55 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   48 
                 
                 
                   26 
                 
                 
                   10 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   42 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   21 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   58 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   41 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   78 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   7 
                 
                 
                   52 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   29 
                 
                 
                   9 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   37 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   35 
                 
                 
                   21 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   22 
                 
                 
                   72 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   47 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   35 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   13 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   35 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   70 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     46 
                     ⁢ 
                       
                   
                 
                 
                   
                     28 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     38 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     8 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     10 
                     ⁢ 
                       
                   
                 
                 
                   
                     58 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     A compact matrix for N=8640 bits with expansion factor 360 derived from a different binary model matrix would be the following: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     297 
                     ⁢ 
                       
                   
                 
                 
                   
                     106 
                     ⁢ 
                       
                   
                 
                 
                   
                     328 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     99 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   290 
                 
                 
                   0 
                 
                 
                   312 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   32 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   120 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   183 
                 
                 
                   57 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   187 
                 
                 
                   68 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   260 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   81 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   323 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   137 
                 
                 
                   354 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   162 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   228 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   224 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   114 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   245 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   113 
                 
                 
                   98 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   120 
                 
                 
                   23 
                 
                 
                   
                     - 
                     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 
                   
                 
                 
                   138 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   187 
                 
                 
                   45 
                 
                 
                   62 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   142 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   347 
                 
                 
                   67 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   46 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   328 
                 
                 
                   265 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   66 
                 
                 
                   156 
                 
                 
                   96 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   212 
                 
                 
                   184 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   102 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   120 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   80 
                 
                 
                   15 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   329 
                 
                 
                   153 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     207 
                     ⁢ 
                       
                   
                 
                 
                   
                     70 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     7 
                     ⁢ 
                       
                   
                 
                 
                   
                     235 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     81 
                     ⁢ 
                       
                   
                 
                 
                   
                     185 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     Another preferential alternative for obtaining code words of 8640 bits with a codification rate of 1/2 is the matrix: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     34 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     95 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     279 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     248 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     0 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   134 
                 
                 
                   356 
                 
                 
                   275 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   51 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   27 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   22 
                 
                 
                   152 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   57 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   124 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   290 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   281 
                 
                 
                   15 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   340 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   99 
                 
                 
                   336 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   33 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   163 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   46 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   306 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   86 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   185 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   24 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   94 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   223 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   225 
                 
                 
                   325 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   297 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   46 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   314 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   59 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   67 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   120 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   121 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   161 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   303 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   264 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
               
               
                 
                   
                     - 
                     1 
                   
                 
                 
                   303 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   8 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   185 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   138 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   0 
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     - 
                     1 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     312 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     100 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     144 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     307 
                     ⁢ 
                       
                   
                 
                 
                   
                     33 
                     ⁢ 
                       
                   
                 
                 
                   
                     166 
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                     ⁢ 
                       
                   
                 
                 
                   
                     
                         
                     
                     ⁢ 
                     0 
                     ⁢ 
                       
                   
                 
               
             
               
           
         
       
     
     The step from compact matrix to binary model is univocal, but for the opposite step, it is not; that is to say, different compact matrices can be obtained from a binary model matrix. The binary matrix is introduced as a step that facilitates the description of the invention. It is possible to develop a compact matrix directly without going through the binary model matrix. In that case, for that compact matrix, its corresponding binary model matrix can be obtained which, if it is triple diagonal, will be in accord with the method of the invention. 
     The error correction codes can be “punctured,” wherein the technique for puncturing consists in removing elements of the code word so that they will not be transmitted. In place of transmitting a code word v=[v(0), v(1), . . . , v(N−1)], a word w=[w(0), w(1), . . . , w(M−1)] will be transmitted, where M&lt;N. The puncturing must be done in a controlled way, so that the necessary redundancy is transmitted in order for the decodifier of the receptor to be able to evaluate the transmitted data. This puncturing is applied to both the data bits and the parity bits. A pattern of puncturing can be defined as the sequence of bits that are to be transmitted as punctured, where said bit pattern can be periodic or aperiodic. In case there is no regularity, said pattern can be described with a vector pp of N positions, indicating with a “1” the bits to be transmitted and with a “0” the bits to be eliminated (punctured). Thanks to the puncturing technique, data communication can be amplified, since less redundancy is being sent. If the Hamming weight of the pattern pp is M, the total system rate of error codification is R=K|M. 
     For example, if we have a code with R=5/6 and block size N=5184 and we want to perform a puncturing in order to increase the rate to R=16/18, we can use the following puncturing pattern, which will yield a block with N=4860 bits: 
     
       
         
           
             
               pp 
               5184 
               
                 ( 
                 
                   16 
                   / 
                   18 
                 
                 ) 
               
             
             ⁢ 
             
               [ 
             
             ⁢ 
             
               
                 1 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ⋯ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
               
                 ︸ 
                 3240 
               
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 0 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ⋯ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
               
               
                 ︸ 
                 162 
               
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 1 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ⋯ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
               
                 ︸ 
                 972 
               
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 0 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ⋯ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
               
               
                 ︸ 
                 162 
               
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 1 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ⋯ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
               
                 ︸ 
                 648 
               
             
             ⁢ 
             
               ] 
             
           
         
       
     
     In order to achieve implementation of LDPC codes, an electronic device is used, whether it be a program executed on a microprocessor or an FPGA or ASIC hardware implementation. Said device receives a block of data, calculates the parity bits, concatenates them with the information bits and submits them to the following phase of the transmitter to be adequately modulated and transmitted through the corresponding channel. The calculation of the parity bits may be done by means of the product of the generator matrix G or the solution of the system of equations presented previously. 
     The decodifier of LDPC codes is usually based on an iterative decodifier. A possible decodifier, among several state of the art options, consists of an estimator which, upon receiving the word corresponding to the transmitted code r=v+z, where Z is additive channel noise, makes a noise estimation {circumflex over (Z)} such that (r−{circumflex over (z)})H T =0. In the case where the system makes use of the puncturing technique, before the decodifier there will be a unit that inserts an indicator in the punctured positions. This indicator serves to direct the decodifier to estimate the proper value in those positions. 
       FIG. 1  shows the block diagram of a codifier, where ( 1 ) is the data block to be transmitted u=[u(0), u(1), . . . , u(K−1)], ( 2 ) is a memory that contains the representation of the parity matrix H or the generator matrix G, ( 3 ) is said matrix, ( 4 ) is the block that performs the codification algorithm, ( 5 ) is the linear code word obtained from the codification v=[v(0), v(1), . . . , v(N−1)], ( 6 ) is the block that does the puncturing, and ( 7 ) is the word obtained after puncturing w=[w(0), w(1), . . . , w(M−1)]. 
       FIG. 2  shows the block diagram for a decodifier, where ( 8 ) represents the received signal from the channel s=[s 0 , s 1 , . . . , s M−1 ], which will be similar to the word obtained after puncturing but after being affected by the channel noise, ( 9 ) represents the block that performs the “de-puncturing,” obtaining a word ( 10 ) r=[r(0), r(1), . . . , r(N−1)] with the number of bits in the LDPC code. ( 11 ) is the memory that contains either the parity matrix or the generator matrix in the receiver and transmits it ( 12 ) to the block that performs the decodifier algorithm ( 13 ). The output of this block will be the reconstituted data ( 14 ) û=[û(0), û(1), . . . , û(K−1)].