# Patent

## Title
Method and apparatus for encoding and decoding data

## Abstract

A structured parity-check matrix H is proposed, wherein H is an expansion of a base matrix Hb and wherein Hb comprises a section Hb1 and a section Hb2, and wherein Hb2 comprises a first part comprising a column hb having an odd weight greater than 2, and a second part comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. The expansion of the base matrix Hb uses identical submatrices for 1s in each column of the second part H′b2, and the expansion uses paired submatrices for an even number of 1s in hb.

## Background
FIELD OF THE INVENTION
The present invention relates generally to encoding and decoding data and in particular, to a method and apparatus for encoding and decoding data utilizing low-density parity-check (LDPC) codes.
BACKGROUND OF THE INVENTION
As described in U.S. patent application Ser. No. 10/839,995, which is incorporated by reference herein, a low-density parity-check (LDPC) code is a linear block code specified by a parity-check matrix H. In general, an LDPC code is defined over a Galois Field GF(q), q≧2. If q=2, the code is a binary code. All linear block codes can be described as the product of a k-bit information vector s1×k with a code generator matrix Gk×n to produce an n-bit codeword x1×n, where the code rate is r=k/n. The codeword x is transmitted through a noisy channel, and the received signal vector y is passed to the decoder to estimate the information vector S1×k.
Given an n-dimensional space, the rows of G span the k-dimensional codeword subspace C, and the rows of the parity-check matrix Hm×n span the m-dimensional dual space C⊥, where m=n−k. Since x=sG and GHT=0, it follows that xHT=0 for all codewords in subspace C, where “T” (or “T”) denotes matrix transpose. In the discussion of LDPC codes, this is generally written as

HxT=0T,  (1)

where 0 is a row vector of all zeros, and the codeword x=[s p]=[s0, s1, . . . , sk−1, p0, p1, . . . , pm−1], where p0, . . . , pm−1 are the parity-check bits; and s0, . . . , sk−1 are the systematic bits, equal to the information bits within the information vector.

For an LDPC code the density of non-zero entries in H is low, i.e., there are only a small percentage of 1's in H, allowing better error-correcting performance and simpler decoding than using a dense H. A parity-check matrix can be also described by a bipartite graph. The bipartite graph is not only a graphic description of the code but also a model for the decoder. In the bipartite graph, a codeword bit (therefore each column of H) is represented by a variable node on the left, and each parity-check equation (therefore each row of H) is represented by a check node on the right. Each variable node corresponds to a column of H and each check node corresponds to a row of H, with “variable node” and “column” of H referred to interchangeably, as are “check node” and “row” of H. The variable nodes are only connected to check nodes, and the check nodes are only connected to variable nodes. For a code with n codeword bits and m parity bits, variable node vi is connected to check node cj by an edge if codeword bit i participates in check equation j, i=0, 1, . . . , n−1, j=0, 1, . . . , m−1. In other words, variable node i is connected to check node j if entry hji of the parity-check matrix H is 1. Mirroring Equation (1), the variable nodes represent a valid codeword if all check nodes have even parity.
An example is shown below to illustrate the relationship between the parity-check matrix, the parity-check equations, and the bipartite graph. Let an n=12, rate−1⁄2 code be defined by




  
    
      
        
          H
          =
          
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                }
              
              
                }
                n
              
            
            ⁢
            m
          
        
        ,
      
    
    
      
        (
        2
        )
      
    
  




with the left side portion corresponding to k (=6) information bits s, the right side portion corresponding to m (=6) parity bits p. Applying (1), the H in (2) defines 6 parity-check equations as follows:





  
    
      
        {
        
          
            
              
                
                  
                    x
                    0
                  
                  +
                  
                    x
                    2
                  
                  +
                  
                    x
                    6
                  
                  +
                  
                    x
                    7
                  
                
                =
                0
              
            
          
          
            
              
                
                  
                    x
                    1
                  
                  +
                  
                    x
                    4
                  
                  +
                  
                    x
                    7
                  
                  +
                  
                    x
                    8
                  
                
                =
                0
              
            
          
          
            
              
                
                  
                    x
                    2
                  
                  +
                  
                    x
                    5
                  
                  +
                  
                    x
                    6
                  
                  +
                  
                    x
                    8
                  
                  +
                  
                    x
                    9
                  
                
                =
                0
              
            
          
          
            
              
                
                  
                    x
                    0
                  
                  +
                  
                    x
                    3
                  
                  +
                  
                    x
                    9
                  
                  +
                  
                    x
                    10
                  
                
                =
                0
              
            
          
          
            
              
                
                  
                    x
                    1
                  
                  +
                  
                    x
                    4
                  
                  +
                  
                    x
                    10
                  
                  +
                  
                    x
                    11
                  
                
                =
                0
              
            
          
          
            
              
                
                  
                    x
                    3
                  
                  +
                  
                    x
                    5
                  
                  +
                  
                    x
                    6
                  
                  +
                  
                    x
                    11
                  
                
                =
                0.
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




H also has the corresponding bipartite graph shown in FIG. 1.

As discussed above, the receiver obtains a contaminated version y of the transmitted codeword x. To decode y and determine the original information sequence s, an iterative decoding algorithm, such as belief propagation, is applied based on the bipartite graph. Soft information in the format of log-likelihood ratio (LLRs) of the codeword bits is passed between the bank of variable nodes and the bank of check nodes. The iteration is stopped either when all check equations are satisfied or a maximum allowed iteration limit is reached.
A structured LDPC code design starts with a small mb×nb base matrix Hb, makes z copies of Hb, and interconnects the z copies to form a large m×n H matrix, where m=mb×z, n=nb×z. Using the matrix representation, to build an H from Hb each 1 in Hb is replaced by a z×z permutation submatrix (square subblock), and each 0 in Hb is replaced by a z×z all-zero submatrix. This procedure essentially maps each edge of Hb to a vector edge of length z in H, each variable node of Hb to a vector variable node of length z in H, and each check node of Hb to a vector check node of length z in H. The benefits of vectorizing a small matrix Hb to build a large matrix H are:

    
    
        1. By using a different values of z, codes of rate kb/nb, where kb=nb−mb, can be designed for many different information sequence sizes k=z×kb from a single base matrix Hb.
        2. Memory requirements are greatly reduced. With a structured LDPC design, only the base matrix Hb and the permutation for its 1's need to be stored, which requires significantly less memory since Hb is typically much smaller than H and the permutation can be very simple.
        3. Encoding and decoding can be performed on groups of bits rather than by single bits. For example, a group of z messages can be fetched from memory, permuted, and passed between a vector variable node and a vector check node.
    
    


Although the structured LDPC design philosophy greatly reduces the implementation complexity, a technique does not exist for designing the base matrix and assigning the permutation matrices for a given target H size which results in a LDPC code that has good error-correcting performance and can be efficiently encoded and decoded. Therefore, a need exists for a method and apparatus for designing a structured H and a method and apparatus for encoding and decoding data utilizing the structured H matrix.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates the bipartite graph of an (12, 6) H matrix.
FIG. 2 illustrates the relationships between the base matrix Hb, the model matrix Hbm, and the final expanded matrix H.
FIG. 3 is a block diagram of an encoder.
FIG. 4 is a block diagram of a decoder.
FIG. 5 is a flow chart showing the operation of the encoder of FIG. 3.
FIG. 6 is a flow chart showing the operation of the decoder of FIG. 4.



DETAILED DESCRIPTION OF THE DRAWINGS
To address the above-mentioned need, a structured parity-check matrix H is proposed, wherein H is an expansion of a base matrix Hb and wherein Hb comprises a section Hb1 and a section Hb2, and wherein Hb2 comprises a first part comprising a column hb having an odd weight greater than 2, and a second part comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. The expansion of the base matrix Hb uses identical submatrices for 1s in each column of the second part H′b2, and the expansion uses paired submatrices for an even number of 1s in hb.
The present invention encompasses a method for operating a transmitter that generates parity-check bits p=(p0, . . . , pm−1) based on a current symbol set s=(s0, . . . , sk−1). The method comprises the steps of receiving the current symbol set s=(s0, . . . , sk−1), and using a matrix H to determine the parity-check bits. The parity-check bits are transmitted along with the current symbol set. Matrix H is an expansion of a base matrix Hb where Hb comprises a section Hb1 and a section Hb2, and wherein Hb2 comprises a first part comprising a column hb having an odd weight greater than 2, and a second part H′b2 comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. The expansion of the base matrix Hb uses identical submatrices for 1s in each column of the second part H′b2, and wherein the expansion uses paired submatrices for an even number of 1s in hb.
The present invention additionally encompasses a method for operating a receiver that estimates a current symbol set s=(s0, . . . , sk−1). The method comprises the steps of receiving a received signal vector y=(y0 . . . yn−1) and using a matrix H to estimate the current symbol set s=(s0, . . . , sk−1). Matrix H is an expansion of a base matrix Hb and wherein Hb comprises a section Hb1 and a section Hb2, with Hb2 comprising a first part comprising a column hb having an odd weight greater than 2, and a second part H′b2 comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. The expansion of the base matrix Hb uses identical submatrices for 1s in each column of the second part H′b2, and wherein the expansion uses paired submatrices for an even number of 1s in hb.
The present invention additionally encompasses an apparatus comprising storage means for storing a matrix H, a microprocessor using a matrix H to determine parity-check bits, wherein H is an expansion of a base matrix Hb and Hb comprises a section Hb1 and a section Hb2, with Hb2 comprising a first part comprising a column hb having an odd weight greater than 2, and a second part H′b2 comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. The expansion of the base matrix Hb uses identical submatrices for 1s in each column of the second part H′b2, and wherein the expansion uses paired submatrices for an even number of 1s in hb.
The present invention encompasses an apparatus comprising storage means for storing a matrix H, a receiver for receiving a signal vector y=(y0 . . . yn−1), and a microprocessor using a matrix H to determine a current symbol set (s0, . . . , sk−1). Matrix H is an expansion of a base matrix Hb with Hb comprising a section Hb1 and a section Hb2, and wherein Hb2 comprises a first part comprising a column hb having an odd weight greater than 2. Hb2 comprises a second part H′b2 having matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. Two identical submatrices are used to expand is in every column of H′b2, and paired submatrices are used to expand an even number of 1s in hb.
Turning now to the drawings, wherein like numerals designate like components, FIG. 3 is a block diagram of encoder 300 in accordance with a first embodiment of the present invention. As shown, encoder 300 comprises microprocessor 301 and lookup table 303. In the first embodiment of the present invention, microprocessor 301 comprises a digital signal processor (DSP), such as, but not limited to MSC8300 and DSP56300 DSPs. Additionally, lookup table 303 serves as storage means to store a matrix, and comprises read-only memory; however, one of ordinary skill in the art will recognize that other forms of memory (e.g., random-access memory, magnetic storage memory, etc.) may be utilized as well. In a second embodiment, the functionality of the microprocessor 301 and the lookup table 303 can be incorporated into an application specific integrated circuit (ASIC) or field programmable gate array (FPGA). In particular, the lookup table 303 can be implemented in a form of memory corresponding to the existence or non-existence of signal paths in a circuit.
As discussed above, encoded data is generally output as a plurality of parity-check bits in addition to the systematic bits, where together the parity-check and systematic bits form a codeword x. In the first embodiment of the present invention, a parity-check matrix H is stored in lookup table 303, and is accessed by microprocessor 301 to solve Equation (1). In particular, microprocessor 301 determines appropriate values for the parity-check bits p=(p0, . . . , pm−1) based on the current symbol set s=(s0, . . . , sk−1) and the parity-check matrix H. The parity-check bits and the symbol set are then passed to a transmitter and transmitted to a receiver.
FIG. 4 is a block diagram of decoder 400 in accordance with one embodiment of the present invention. As shown, decoder 400 comprises microprocessor 401 and lookup table 403. In a first embodiment of the present invention, microprocessor 401 comprises a digital signal processor (DSP), such as, but not limited to MSC8300 and DSP56300 DSPs. Additionally, lookup table 403 acts as storage means for storing matrix H, and comprises read-only memory. However, one of ordinary skill in the art will recognize that other forms of memory (e.g., random-access memory, magnetic storage memory, etc.) may be utilized as well. In a second embodiment, the functionality of the microprocessor 401 and the lookup table 403 can be incorporated into an application specific integrated circuit (ASIC) or field programmable gate array (FPGA). In particular, the lookup table 403 can be implemented in a form of memory corresponding to the existence or non-existence of signal paths in a circuit.
The received signal vector (received via a receiver) y=(y0, . . . , yn−1) corresponds to the codeword x transmitted through a noisy channel, where the encoded data x, as discussed above, is a codeword vector. In the first embodiment of the present invention, a parity-check matrix H is stored in lookup table 403, and is accessed by microprocessor 401 to decode y and estimate the current symbol set s (i.e., the current symbol set (s0, . . . , sk−1)). In particular, microprocessor 401 estimates the current symbol set (s0, . . . , sk−1) based on the received signal vector y=(y0, . . . , yn−1) and the parity-check matrix H.
As is well known in the art, there are many ways the decoder 400 can use the parity-check matrix H in the microprocessor 401 for decoding. One such way is to perform a vector-matrix multiplication with H to determine a likely error pattern. Another such way is to use H to construct a bipartite graph where the edges in the graph correspond to 1's in H, and to iteratively process y on the bipartite graph.
For a structured LDPC, the z×z submatrix may be a permutation matrix, a sum of permutation matrices, or any type of binary matrix. Since a permutation matrix P has a single 1 in each row and a single 1 in each column, the weight distribution of the expanded matrix H is the same as the base matrix Hb if the permutation submatrix is used. Therefore, the weight distribution of Hb is chosen as close to the desired final weight distribution as possible. The following description is illustrative of the case where the entries of Hb are replaced by permutation matrices, though any matrices may be used. If a permutation submatrix Pz×z of a vector edge has a 1 at (row, column) entry (p(i), i), then the i-th edge within the vector edge is permuted to the p(i)-th position before the vector edge is connected to the vector check node. In other words, this permutation makes the i-th variable node within the related vector variable node connected to the p(i)-th check node within the related vector check node.
The permutations comprising H can be very simple without compromising performance, such as simple cyclic shifts and/or bit-reversals. For instance, a simple circular right shift can be used. With this constraint, each H matrix can be uniquely represented by a mb×nb model matrix Hbm, which can be obtained by

    
    
        replacing each 0 in Hb by −1 to denote a z×z all-zero submatrix (i.e., a null submatrix), and
        replacing each hi,j=1 in Hb by a circular shift size p(i,j) where p(i,j) is non-negative.

Note that the model matrix can also be referred to as a matrix prototype. Additionally note that the z×z all-zero submatrix (null submatrix) may also be represented using ‘−’ or by any other designator.

    
    


Since circular left shift (x mod z) times is equivalent to circular right shift ((z−x) mod z) times, it is adequate to discuss circular right shift and refer it as a circular shift for brevity. As discussed previously, there is a one-to-one mapping between H and Hbm. Therefore, Hbm is a shorthand representation of H if z is given. Notationally, the matrix prototype is distinguished from the base matrix by the subscript ‘bm’, and the expanded matrix is distinguished by removing the subscript ‘bm’. The relationship between the three matrices is illustrated in FIG. 2. Using the structure, the code has error-correcting performance similar to a random H of size m×n, while encoding and decoding are performed based on a much smaller Hbm.
For example, the matrix of Equation (2) may be used as a base matrix Hb to build a matrix prototype Hbm as follows:




  
    
      
        
          H
          bm
        
        =
        
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      2
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      2
                    
                    
                      2
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      2
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      1
                    
                    
                      0
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      
                        -
                        1
                      
                    
                    
                      0
                    
                  
                
                ]
              
              }
            
            
              }
              
                n
                b
              
            
          
          ⁢
          
            
              m
              b
            
            .
          
        
      
    
    
      
        (
        4
        )
      
    
  




When z=3, Hbm is converted to a (mb×z)×(nb×z) matrix H by replacing each −1 with a 3×3 all-zero submatrix and each i with submatrix Pi, i=0, 1, 2, where





  
    
      
        
          
            P
            0
          
          =
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
        
        ,
      
    
    
      
        
          
            P
            1
          
          =
          
            [
            
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
              
                
                  1
                
                
                  0
                
                
                  0
                
              
            
            ]
          
        
        ,
      
    
    
      
        
          P
          2
        
        =
        
          
            [
            
              
                
                  0
                
                
                  0
                
                
                  1
                
              
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
            
            ]
          
          .
        
      
    
  




Note that P0 is the z×z identity matrix. The cyclic-permutation matrix Pi is obtained from the z×z identity matrix by cyclically shifting the columns to the right by i elements, i>0.

Given a vector q=[q0, q1, q2], qP0=[q0, q1, q2], qP1=[q2, q0, q1], qP2=[q1, q2, q0]. In other words, qPi results in circular right shift of the vector q. On the other hand, PiqT, results in circular upper shift of qT, or equivalently circular left shift of q. Similar rules apply when a z×z matrix Q is used: QPi results in circular right shift of the columns of Q, PiQ results in circular upper shift of the rows of Q.
Base Matrix H
For an LDPC code without vectorization, an H matrix with a modified staircase structure for the parity part of H leads to efficient encoding without compromising performance. In general, assuming x=[s p]=[s0, s1, . . . , sk−1, p0, p1, . . . , pm−1], an m-by-n H matrix can be divided into two submatrices,

H=[H1 H2],  (5)

where H2 has a modified staircase structure, and H1 can be any binary matrix of size m-by-k. This same structure can be used to build the base matrix Hb in a structured LDPC design. Similarly, using the modified staircase structure, Hb can be partitioned into two sections, where Hb1 corresponds to the systematic bits s, Hb2 corresponds to the parity-check bits p:





  
    
      
        
          H
          b
        
        =
        
          
            [
            
              
                
                  
                    
                      (
                      
                        H
                        
                          b
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      )
                    
                    
                      
                        m
                        b
                      
                      ×
                      
                        k
                        b
                      
                    
                  
                
                
                  
                    
                      (
                      
                        H
                        
                          b
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                    
                      
                        m
                        b
                      
                      ×
                      
                        m
                        b
                      
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        6
        )
      
    
  




Section Hb2 can be further partitioned into two sections, where vector hb has odd weight wh, and H′b2 has a staircase structure:





  
    
      
        
          
            
              
                H
                
                  b
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
              =
              
                [
                
                  
                    
                      
                        h
                        b
                      
                    
                    
                      
                        H
                        
                          b
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                        ′
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
                
                  [
                  
                    
                      
                        
                          
                            h
                            b
                          
                          ⁡
                          
                            (
                            0
                            )
                          
                        
                      
                      
                        1
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        
                          
                            h
                            b
                          
                          ⁡
                          
                            (
                            1
                            )
                          
                        
                      
                      
                        1
                      
                      
                        1
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        .
                      
                      
                        
                          
                        
                      
                      
                        1
                      
                      
                        ⋰
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        .
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        ⋰
                      
                      
                        1
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        .
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        
                          
                            h
                            b
                          
                          ⁡
                          
                            (
                            
                              
                                m
                                b
                              
                              -
                              1
                            
                            )
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        1
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




Section Hb1 can be built randomly. Preferably the entire matrix Hb has a weight distribution as close to the desired weight distribution as possible.

Shift Sizes

To convert the base matrix Hb to the mb×nb matrix prototype Hbm (which expands to H), the circular shift sizes p(i,j) need to be determined for each 1 in Hb. The shift sizes can be first specified for the H2. After the shift sizes for the H2 section are determined, the H1 section shift sizes can be determined to achieve overall good performance of H. The H1 portion of the base matrix and the shift sizes of the H1 portion of the base matrix (section Hbm1) can be assigned in many different ways. For example, random values for the shift sizes may be selected and accepted if the shift sizes do not cause significant performance degradation. Performance degradation may result from the introduction of excessive numbers of short-length cycles or low-weight codewords. Other techniques available in the LDPC art may also be used.
The circular shift sizes p(i,j) for a given target H size should be specified to allow efficient encoding without compromising decoding performance. To facilitate encoding, the shifts may be assigned such that all but one of the shift matrices corresponding to 1's in hb cancel when added together, and all vector rows of H′b2 cancel when summed up. This translates to assigning shift sizes to hb in pairs except for one entry, and assigning the same shift size to both 1's in each column of H′b2. For example, if hb=[1 0 0 1 0 0 1]T, it is acceptable to have hbm=[3 −1 −1 3 −1 −1 4]T as the corresponding column in the matrix prototype since shift size 3 is assigned in pairs. Since all of the non-zero entries (both 1's) in each column H′b2 are assigned the same shift sizes, any shift size option is equivalent to the shift size of 0 (i.e., identity submatrices) plus a permutation of the bits within the vector column. Thus, all shift sizes of H′b2 can be assigned 0 for convenience, i.e., each 1 in H′b2 is replaced by a z×z identity submatrix when expanding to H.
Due to existence of cycles, the shift sizes of hbm should be assigned carefully. Rules can be applied to avoid forming short cycles or low weight codewords. One property that can be used to avoid cycles is:
If 2c edges form a cycle of length 2c in base matrix Hb, then the corresponding 2c vector edges form z cycles of length 2c in the expanded matrix H if and only if




  
    
      
        
          
            
              ∑
              
                
                  i
                  =
                  
                    2
                    ⁢
                    j
                  
                
                
                  
                    j
                    =
                    0
                  
                  ,
                  
                    
                  
                  ⁢
                  ...
                  ⁢
                  
                    
                  
                  ,
                  
                    c
                    -
                    1
                  
                
              
            
            ⁢
            
              p
              ⁡
              
                (
                i
                )
              
            
          
          =
          
            
              ∑
              
                
                  i
                  =
                  
                    
                      2
                      ⁢
                      j
                    
                    +
                    1
                  
                
                
                  
                    j
                    =
                    0
                  
                  ,
                  
                    
                  
                  ⁢
                  ...
                  ⁢
                  
                    
                  
                  ,
                  
                    c
                    -
                    1
                  
                
              
            
            ⁢
            
              
                p
                ⁡
                
                  (
                  i
                  )
                
              
              ⁢
              
                
              
              ⁢
              mod
              ⁢
              
                
              
              ⁢
              z
            
          
        
        ,
      
    
    
      
        (
        8
        )
      
    
  




where z is the expansion factor, p(i) is the circular shift size of edge i in the matrix prototype Hbm, and edge 0, 1, 2, . . . , 2c−1 (in this order) form a cycle in Hb.

Due to the structure of Hb2, cycles exist between hb and H′b2. Thus any two identical shift sizes in hbm would result in replicating the cycle z times in the expanded matrix H according to the above property. However, if these two shifts are located far apart, then the cycles have long length, and have little effect on iterative decoding. Therefore, in a preferred embodiment, when hb of the base matrix has three 1s, to maximize the cycle length, two is that are assigned equal shift sizes can be located at the top and the bottom of hbm (as far apart as possible), while leaving one 1 in the middle of hb with an unpaired shift size. For instance, hbm=[3 −1 3 −1 −1 −1 4]T would result in z cycles of length 6 between h and H′2, while hbm=[3 −1 4 −1 −1 −1 3]T would result in z cycles of length 14 between h and H′2, where h and H′2 are expanded from hb and H′b2.
In summary, the Hb2 section is mapped to the matrix prototype




  
    
      
        
          
            
              
                H
                
                  bm
                  ⁢
                  
                    
                  
                  ⁢
                  2
                
              
              =
              
                [
                
                  
                    
                      
                        h
                        bm
                      
                    
                    
                      
                        H
                        
                          bm
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                        ′
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
                =
                
                  [
                  
                    
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              0
                              ,
                              
                                k
                                b
                              
                            
                            )
                          
                        
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              0
                              ,
                              
                                
                                  k
                                  b
                                
                                +
                                1
                              
                            
                            )
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              1
                              ,
                              
                                k
                                b
                              
                            
                            )
                          
                        
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              1
                              ,
                              
                                
                                  k
                                  b
                                
                                +
                                1
                              
                            
                            )
                          
                        
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              1
                              ,
                              
                                
                                  k
                                  b
                                
                                +
                                2
                              
                            
                            )
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                      
                        
                          
                        
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              2
                              ,
                              
                                
                                  k
                                  b
                                
                                +
                                2
                              
                            
                            )
                          
                        
                      
                      
                        ⋰
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        ⋰
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              
                                
                                  m
                                  b
                                
                                -
                                3
                              
                              ,
                              
                                
                                  n
                                  b
                                
                                -
                                2
                              
                            
                            )
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              
                                
                                  m
                                  b
                                
                                -
                                2
                              
                              ,
                              
                                
                                  n
                                  b
                                
                                -
                                2
                              
                            
                            )
                          
                        
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              
                                
                                  m
                                  b
                                
                                -
                                2
                              
                              ,
                              
                                
                                  n
                                  b
                                
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              
                                
                                  m
                                  b
                                
                                -
                                1
                              
                              ,
                              
                                k
                                b
                              
                            
                            )
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          p
                          ⁡
                          
                            (
                            
                              
                                
                                  m
                                  b
                                
                                -
                                1
                              
                              ,
                              
                                
                                  n
                                  b
                                
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                  
                  ]
                
              
              ,
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




where kb=nb−mb, there are wh (odd, wh>=3) nonnegative entries in hbm, and the −1 entries in H′bm2 are left blank for brevity. All p(i,kb) values appear an even number of times in hbm except for one, which may be mapped to any non-zero submatrix. Therefore, all wh shifts could be given the same value (e.g., 0), since wh is odd. For H′bm2, p(i,j)=p(i+1,j), j=kb+1, kb+2, . . . , nb−1, i=j−kb−1. In the preferred embodiment, assuming wh=3, one example has hbm=[0−1 . . . −1 ph−1 . . . −1 . . . 0]T, ph mod z≠0, and p(i,j)=p(i+1,j)=0, j=kb+1, kb+2, . . . , nb−1, i=j−kb−1 in the H′bm2 portion.

Although the discussion above focused on using submatrices that are circular shifts of the identity matrix, in general, any other submatrices may be used (and be represented in an equivalent to the base matrix prototype). In order to facilitate encoding, the constraints are then:

    
    
        1. In every column of H′bm2, the two non-zero submatrices are identical;
        2. The wh (odd, wh>=3) non-zero submatrices of hbm are paired (i.e., one submatrix is identical to another submatrix), except for one submatrix, which can be any non-zero matrix.

Encoding

    
    


Encoding is the process of determining the parity sequence p given an information sequence s. To encode the structured LDPC code, each operation is performed over a group of z bits instead of a single bit. Alternatively, vector operations do not have to be used, and the equations below are implemented in equivalent scalar form. To encode, s is divided into kb=nb−mb groups of z bits. Let this grouped s be denoted u,

u=[u(0)u(1) . . . u(kb−1)],  (10)

where each element of u is a column vector as follows

u(i)=[sizsiz+1 . . . s(i+1)z−1]T.  (11)

Using the matrix prototype Hbm, the parity sequence p is determined in groups of z. Let the grouped p be denoted v,

v=[v(0)v(1) . . . v(mb−1)],  (12)

where each element of v is a column vector as follows

v(i)=[pizpiz+1 . . . p(i+1)z−1]T.  (13)

Encoding proceeds in two steps, (a) initialization, which determines v(0), and (b) recursion, which determines v(i+1) from v(i), 0≦i≦mb−2.

An expression for v(0) can be derived by summing over the rows of Equation (1) to obtain




  
    
      
        
          
            P
            
              p
              ⁡
              
                (
                
                  x
                  ,
                  
                    k
                    b
                  
                
                )
              
            
          
          ⁢
          
            v
            ⁡
            
              (
              0
              )
            
          
        
        =
        
          
            ∑
            
              j
              =
              0
            
            
              
                k
                b
              
              -
              1
            
          
          ⁢
          
            
              ∑
              
                i
                =
                0
              
              
                
                  m
                  b
                
                -
                1
              
            
            ⁢
            
              
                P
                
                  p
                  ⁡
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                
              
              ⁢
              
                u
                ⁡
                
                  (
                  j
                  )
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




where x is the row index of hbm where the entry is nonnegative and is used an odd number of times. In the preferred embodiment, the top and bottom entries of hbm are paired, thus 1≦x≦mb−2. Equation (14) is solved for v(0) by multiplying both sides by Pp(x,kb)−1. For the special case considered here where p(x,kb) represents a circular shift, Pp(x,kb)−1=Pz−p(x,kb). In other words, v(0) is obtained by





  
    
      
        
          v
          ⁡
          
            (
            0
            )
          
        
        =
        
          
            P
            
              z
              -
              
                p
                ⁡
                
                  (
                  
                    x
                    ,
                    
                      k
                      b
                    
                  
                  )
                
              
            
          
          ⁢
          
            
              ∑
              
                j
                =
                0
              
              
                
                  k
                  b
                
                -
                1
              
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  
                    m
                    b
                  
                  -
                  1
                
              
              ⁢
              
                
                  P
                  
                    p
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                
                ⁢
                
                  
                    u
                    ⁡
                    
                      (
                      j
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




In general, the recursions expressed in Equations (16) and (17) can be derived by considering the structure of H′b2,




  
    
      
        
          
            
              
                
                  
                    P
                    
                      p
                      ⁡
                      
                        (
                        
                          i
                          ,
                          
                            
                              k
                              b
                            
                            +
                            1
                          
                        
                        )
                      
                    
                  
                  ⁢
                  
                    v
                    ⁡
                    
                      (
                      1
                      )
                    
                  
                
                =
                
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        
                          k
                          b
                        
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        P
                        
                          p
                          ⁡
                          
                            (
                            
                              i
                              ,
                              j
                            
                            )
                          
                        
                      
                      ⁢
                      
                        u
                        ⁡
                        
                          (
                          j
                          )
                        
                      
                    
                  
                  +
                  
                    
                      P
                      
                        p
                        ⁡
                        
                          (
                          
                            i
                            ,
                            
                              k
                              b
                            
                          
                          )
                        
                      
                    
                    ⁢
                    
                      v
                      ⁡
                      
                        (
                        0
                        )
                      
                    
                  
                
              
              ,
            
          
          
            
              
                i
                =
                0
              
              ,
            
          
        
      
    
    
      
        (
        16
        )
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        
          
            
              P
              
                p
                ⁡
                
                  (
                  
                    i
                    ,
                    
                      
                        k
                        b
                      
                      +
                      i
                      +
                      1
                    
                  
                  )
                
              
            
            ⁢
            
              v
              ⁡
              
                (
                
                  i
                  +
                  1
                
                )
              
            
          
          =
          
            
              
                P
                
                  p
                  ⁡
                  
                    (
                    
                      i
                      ,
                      
                        
                          k
                          b
                        
                        +
                        i
                      
                    
                    )
                  
                
              
              ⁢
              
                v
                ⁡
                
                  (
                  i
                  )
                
              
            
            +
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  
                    k
                    b
                  
                  -
                  1
                
              
              ⁢
              
                
                  P
                  
                    p
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                
                ⁢
                
                  u
                  ⁡
                  
                    (
                    j
                    )
                  
                
              
            
            +
            
              
                P
                
                  p
                  ⁡
                  
                    (
                    
                      i
                      ,
                      
                        k
                        b
                      
                    
                    )
                  
                
              
              ⁢
              
                v
                ⁡
                
                  (
                  0
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        
          i
          =
          1
        
        ,
        ...
        ⁢
        
          
        
        ,
        
          
            m
            b
          
          -
          2
        
        ,
      
    
    
      
        
      
    
  
  
    
      where
    
    
      
        
      
    
  
  
    
      
        
          P
          
            -
            1
          
        
        ≡
        
          
            0
            
              z
              ×
              z
            
          
          .
        
      
    
    
      
        (
        18
        )
      
    
  




Thus all parity bits not in v(0) are determined by iteratively evaluating Equations (16) and (17) for 0≦i≦mb−2.
In a preferred embodiment where the shifts sizes of the 1's in H′b2 are all zero, Equations (16) and (17) can be simplified to Equations (19) and (20),




  
    
      
        
          
            
              
                
                  v
                  ⁡
                  
                    (
                    1
                    )
                  
                
                =
                
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        
                          k
                          b
                        
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        P
                        
                          p
                          ⁡
                          
                            (
                            
                              i
                              ,
                              j
                            
                            )
                          
                        
                      
                      ⁢
                      
                        u
                        ⁡
                        
                          (
                          j
                          )
                        
                      
                    
                  
                  +
                  
                    
                      P
                      
                        p
                        ⁡
                        
                          (
                          
                            i
                            ,
                            
                              k
                              b
                            
                          
                          )
                        
                      
                    
                    ⁢
                    
                      v
                      ⁡
                      
                        (
                        0
                        )
                      
                    
                  
                
              
              ,
            
          
          
            
              
                i
                =
                0
              
              ,
            
          
        
      
    
    
      
        (
        19
        )
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        
          
            v
            ⁡
            
              (
              
                i
                +
                1
              
              )
            
          
          =
          
            
              v
              ⁡
              
                (
                i
                )
              
            
            +
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  
                    k
                    b
                  
                  -
                  1
                
              
              ⁢
              
                
                  P
                  
                    p
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                
                ⁢
                
                  u
                  ⁡
                  
                    (
                    j
                    )
                  
                
              
            
            +
            
              
                P
                
                  p
                  ⁡
                  
                    (
                    
                      i
                      ,
                      
                        k
                        b
                      
                    
                    )
                  
                
              
              ⁢
              
                v
                ⁡
                
                  (
                  0
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          i
          =
          1
        
        ,
        ...
        ⁢
        
          
        
        ,
        
          
            m
            b
          
          -
          2.
        
      
    
    
      
        
      
    
  




Thus, as in the general case, all parity bits not in v(0) are determined by iteratively evaluating Equation (19) and (20) for 0≦i≦mb−2.
Equations (14), (19), and (20) describe the encoding algorithm. These equations also have a straightforward interpretation in terms of standard digital logic architectures. First, since the non-negative elements p(i,j) of Hbm represent circular shift sizes of a vector, all products of the form Pp(i,j)u(j) can be implemented by a size-z barrel shifter. A circular shift size of zero may not need to be barrel-shifted. Since a barrel shifter that implements all possible circular shifts must provide connections from each input bit to all output bits, the speed with which it can be run depends upon z. For a given z, complexity can be reduced and speed increased by allowing only a proper subset of all possible circular shifts. For instance, Hbm could be constructed with only even circular shift sizes. The summations in Equations (14), (19), and (20) represent vector-wise XOR (exclusive OR) operations that are gated (i.e., do not update) when p(i,j)=−1. In the preferred embodiment of the present invention H is represented by any or no column or row-permuted version of the matrix prototype Hbm which is equal to:




  
    
      40
    
    
      -
    
    
      -
    
    
      -
    
    
      22
    
    
      -
    
    
      49
    
    
      23
    
    
      43
    
    
      -
    
    
      -
    
    
      -
    
    
      1
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      50
    
    
      1
    
    
      -
    
    
      -
    
    
      48
    
    
      35
    
    
      -
    
    
      -
    
    
      13
    
    
      -
    
    
      30
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      39
    
    
      50
    
    
      -
    
    
      -
    
    
      4
    
    
      -
    
    
      2
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      49
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      33
    
    
      -
    
    
      -
    
    
      38
    
    
      37
    
    
      -
    
    
      -
    
    
      4
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      45
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      22
    
    
      -
    
    
      -
    
    
      20
    
    
      42
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      51
    
    
      -
    
    
      -
    
    
      48
    
    
      35
    
    
      -
    
    
      -
    
    
      -
    
    
      44
    
    
      -
    
    
      18
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      47
    
    
      11
    
    
      -
    
    
      -
    
    
      -
    
    
      17
    
    
      -
    
    
      -
    
    
      51
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      5
    
    
      -
    
    
      25
    
    
      -
    
    
      6
    
    
      -
    
    
      45
    
    
      -
    
    
      13
    
    
      40
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      33
    
    
      -
    
    
      -
    
    
      34
    
    
      24
    
    
      -
    
    
      -
    
    
      -
    
    
      23
    
    
      -
    
    
      -
    
    
      46
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
  
  
    
      1
    
    
      -
    
    
      27
    
    
      -
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      38
    
    
      -
    
    
      44
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
  
  
    
      -
    
    
      18
    
    
      -
    
    
      -
    
    
      23
    
    
      -
    
    
      -
    
    
      8
    
    
      0
    
    
      35
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
  
  
    
      49
    
    
      -
    
    
      17
    
    
      -
    
    
      30
    
    
      -
    
    
      -
    
    
      -
    
    
      34
    
    
      -
    
    
      -
    
    
      19
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0.
    
  




Note that using the ‘−’ is equivalent to putting a ‘−1’ or any other designator for the z×z all-zero submatrix.

An example of a column-permuted version of the matrix prototype Hbm wherein the high weight columns appear on the left hand side is shown below.





  
    
      43
    
    
      22
    
    
      40
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      23
    
    
      49
    
    
      -
    
    
      -
    
    
      -
    
    
      1
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      13
    
    
      48
    
    
      50
    
    
      1
    
    
      -
    
    
      30
    
    
      -
    
    
      -
    
    
      -
    
    
      35
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      4
    
    
      39
    
    
      50
    
    
      49
    
    
      -
    
    
      -
    
    
      -
    
    
      2
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      1
    
    
      37
    
    
      33
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      4
    
    
      -
    
    
      -
    
    
      38
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      20
    
    
      0
    
    
      45
    
    
      -
    
    
      -
    
    
      -
    
    
      42
    
    
      -
    
    
      -
    
    
      22
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      44
    
    
      35
    
    
      51
    
    
      -
    
    
      -
    
    
      18
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      48
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      51
    
    
      -
    
    
      47
    
    
      11
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      17
    
    
      -
    
    
      -
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      13
    
    
      6
    
    
      5
    
    
      -
    
    
      -
    
    
      -
    
    
      40
    
    
      -
    
    
      45
    
    
      -
    
    
      -
    
    
      25
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      23
    
    
      24
    
    
      33
    
    
      -
    
    
      46
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      34
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
  
  
    
      38
    
    
      1
    
    
      1
    
    
      -
    
    
      -
    
    
      44
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      27
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
  
  
    
      0
    
    
      23
    
    
      -
    
    
      18
    
    
      -
    
    
      -
    
    
      35
    
    
      8
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
  
  
    
      34
    
    
      30
    
    
      49
    
    
      -
    
    
      19
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      17
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0.
    
  




The matrix prototype Hbm may be expanded with z=54 to define a rate 1⁄2 code with n=1296. Following are two more examples of model matrices that may be expanded to defined rate 1⁄2 codes for different values of n.

A rate 1⁄2 code with n=1944 may be defined by expanding any or no column or row-permuted version of the following matrix prototype Hbm and using z=81:





  
    
      -
    
    
      -
    
    
      -
    
    
      18
    
    
      32
    
    
      -
    
    
      10
    
    
      4
    
    
      -
    
    
      -
    
    
      -
    
    
      19
    
    
      1
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      -
    
    
      47
    
    
      -
    
    
      47
    
    
      -
    
    
      28
    
    
      8
    
    
      -
    
    
      34
    
    
      20
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      71
    
    
      -
    
    
      1
    
    
      -
    
    
      -
    
    
      22
    
    
      37
    
    
      -
    
    
      -
    
    
      -
    
    
      25
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      8
    
    
      -
    
    
      -
    
    
      6
    
    
      47
    
    
      -
    
    
      -
    
    
      53
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      23
    
    
      45
    
    
      -
    
    
      -
    
    
      60
    
    
      22
    
    
      -
    
    
      -
    
    
      51
    
    
      74
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      78
    
    
      -
    
    
      28
    
    
      -
    
    
      56
    
    
      -
    
    
      32
    
    
      -
    
    
      -
    
    
      -
    
    
      26
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      -
    
    
      67
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      15
    
    
      49
    
    
      8
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      41
    
    
      -
    
    
      -
    
    
      20
    
    
      -
    
    
      45
    
    
      -
    
    
      48
    
    
      -
    
    
      -
    
    
      -
    
    
      48
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      14
    
    
      44
    
    
      -
    
    
      -
    
    
      -
    
    
      9
    
    
      -
    
    
      71
    
    
      -
    
    
      41
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      72
    
    
      -
    
    
      -
    
    
      3
    
    
      -
    
    
      -
    
    
      16
    
    
      -
    
    
      -
    
    
      66
    
    
      9
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
  
  
    
      28
    
    
      -
    
    
      -
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      42
    
    
      -
    
    
      41
    
    
      -
    
    
      62
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
  
  
    
      -
    
    
      -
    
    
      76
    
    
      19
    
    
      -
    
    
      -
    
    
      -
    
    
      21
    
    
      34
    
    
      -
    
    
      -
    
    
      32
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
  




A rate 1⁄2 code with n=648 may be defined by expanding any or no column or row-permuted version of the following matrix prototype Hbm and using z=27:





  
    
      24
    
    
      -
    
    
      -
    
    
      18
    
    
      20
    
    
      -
    
    
      -
    
    
      23
    
    
      -
    
    
      -
    
    
      -
    
    
      13
    
    
      1
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      21
    
    
      -
    
    
      17
    
    
      -
    
    
      -
    
    
      -
    
    
      26
    
    
      -
    
    
      17
    
    
      -
    
    
      24
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      -
    
    
      16
    
    
      10
    
    
      0
    
    
      -
    
    
      8
    
    
      -
    
    
      -
    
    
      5
    
    
      20
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      25
    
    
      19
    
    
      -
    
    
      -
    
    
      10
    
    
      25
    
    
      -
    
    
      -
    
    
      -
    
    
      19
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      17
    
    
      -
    
    
      -
    
    
      24
    
    
      -
    
    
      1
    
    
      -
    
    
      26
    
    
      -
    
    
      -
    
    
      -
    
    
      23
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      15
    
    
      -
    
    
      3
    
    
      14
    
    
      -
    
    
      15
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      -
    
    
      11
    
    
      -
    
    
      23
    
    
      -
    
    
      19
    
    
      -
    
    
      -
    
    
      -
    
    
      21
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      24
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      24
    
    
      6
    
    
      -
    
    
      16
    
    
      20
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
  
  
    
      -
    
    
      -
    
    
      11
    
    
      16
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      12
    
    
      14
    
    
      12
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
    
      -
    
  
  
    
      5
    
    
      -
    
    
      -
    
    
      0
    
    
      -
    
    
      -
    
    
      -
    
    
      10
    
    
      23
    
    
      -
    
    
      -
    
    
      18
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
    
      -
    
  
  
    
      -
    
    
      26
    
    
      -
    
    
      14
    
    
      -
    
    
      -
    
    
      19
    
    
      2
    
    
      -
    
    
      -
    
    
      7
    
    
      7
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
    
      0
    
  
  
    
      -
    
    
      -
    
    
      17
    
    
      23
    
    
      -
    
    
      -
    
    
      -
    
    
      13
    
    
      1
    
    
      -
    
    
      -
    
    
      8
    
    
      1
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      -
    
    
      0
    
  




To implement the summations in Equations (14), (19), and (20) the entries p(i,j) of Hbm, 0≦i≦kb, 0≦j≦mb−1, can be stored in a read-only memory (ROM) of width ┌log2 z┐+1 bits. The grouped information sequence can be stored in a size-z memory, which can be read out in sequential order. As each information vector u(j) is read out, the corresponding entry from the Hbm ROM can be read out, which instructs the barrel shifter of the necessary circular shift. After the circular shift, a register containing a partial summation is updated. For Equation (14), after each inner summation is completed, the result can be used to update another register containing the outer summation. When the outer summation is complete, it can be circularly shifted by z−p(x,kb).

Assuming the barrel shifting can be implemented in a single clock cycle, encoding can be accomplished in approximately (kb+1)mb clock cycles. This number can be reduced at the expense of mb−1 extra z-wide registers by computing and storing the summations of Equation (19) and (20), using results that become available as Equation (14) is being computed.
Extending the Matrix
The code extension procedure can be applied to the structured code to reach a lower-rate code. Progressively lower-rate code can be used in successive transmissions of an incremental redundancy (IR) procedure. Specifically, if the matrix prototype of the 1sty transmission is

Hbm(1)=[Hbm1(1) Hbm2(1)],  (21)

then the matrix prototype for the 2nd transmission may use





  
    
      
        
          
            H
            bm
            
              (
              2
              )
            
          
          =
          
            [
            
              
                
                  
                    
                      
                        
                          H
                          
                            bm
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          
                            (
                            1
                            )
                          
                        
                      
                      
                        
                          H
                          
                            bm
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                          
                            (
                            2
                            )
                          
                        
                      
                    
                  
                
                
                  0
                
              
              
                
                  
                    H
                    
                      bm
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                    
                      (
                      2
                      )
                    
                  
                
                
                  
                    H
                    
                      bm
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                    
                      (
                      2
                      )
                    
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        22
        )
      
    
  




etc., where for each transmission i, submatrix Hbm2(i) has the format in (9) and has size mb(i)×mb(i). The first transmission may send nb(1)=kb+mb(1) groups of bits, [u(0), u(1), . . . , u(kb−1), v(1)(0), v(1)(1), . . . , v(1)(mb(1)−1)], each group having size z. The decoding after the 1sty transmission is performed using received signals of [u(0), u(1), . . . , u(kb−1), v(1)(0), v(1)(1), . . . , v(1)(mb(1)−1)] and (21). The 2nd transmission may send another mb(2) groups of bits of size z, [v(2)(0), v(2)(1), . . . , v(2)(mb(2)−1)], where m2=mb(2)z, and the bits of the first transmission and the second transmission together, [u(0), u(1), . . . , u(kb−1), v(1)(0), v(1)(1), . . . , v(1)(mb(1)−1), v(2)(0), v(2)(1), . . . , v(2)(mb(2)−1)],

are a codeword corresponding to (22). Therefore, the decoding after the second transmission is performed based on (22) and the combined received signal from the 1sty transmission and the 2nd transmission. This procedure may be repeated for more transmissions. The decoding after the 2nd transmission is based on a code of rate kb/nb(2)=kb/(nb(1)+mb(2)), which is lower than that of 1sty transmission. This procedure may be repeated for more transmissions, with each additional transmission contributing to a stronger, lower-rate code.

FIG. 5 is a flow chart showing the operation of encoder 300, and in particular, microprocessor 301. The logic flow begins at step 501 where a current symbol set (s0, . . . , sk−1) is received by microprocessor 301. At step 503, values of parity-check bits are determined based on the current symbol set and H. In particular, the parity-check bits (p0, . . . , pm−1) are determined as described above, with H being an expansion of a base matrix Hb. As discussed, Hb comprises a section Hb1 and a section Hb2, and wherein Hb2 comprises a first part comprising a column hb having an odd weight greater than 2, and a second part comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere. In addition, the expansion of the base matrix Hb (to produce H) uses identical submatrices for the 1s in each column of the second part H′b2, and wherein the expansion uses paired submatrices for an even number of 1s in hb. At step 505 the current symbol set and the parity-check bits are transmitted via over-the-air transmission.
FIG. 6 is a flow chart showing the operation of decoder 400, and in particular, microprocessor 401. The logic flow begins at step 601 where the received signal vector y=(y0, . . . , yn−1) is received. At step 603, estimates of the current symbol set s (i.e., the current symbol set (s0, . . . , sk−1)) are determined based on H. As discussed, H is an expansion of a base matrix Hb and wherein Hb comprises a section Hb1 and a section Hb2, and wherein Hb2 comprises a first part comprising a column hb having an odd weight greater than 2, and a second part comprising matrix elements for row i, column j equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere.
While the invention has been particularly shown and described with reference to a particular embodiment, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention. For example, while the invention has been shown with the ordering of si and pi within x defined, one of ordinary skill in the art will recognize that other ordering of the bits within x can occur since the codeword bits can be collected in any order as long as the columns of H are reordered accordingly. Additionally, while the above-description has been particularly shown and described with reference to binary codes (i.e., codes defined over the Galois Field GF(2)), one of ordinary skill in the art will recognize that an arbitrary GF may be utilized as well. Although the examples given above are shown in one format, other formats are possible which allow similar encoding and code modification procedure. For example, the rows of H may be permuted without affecting the value of the parity-check bits. In another example, the modified staircase structure may be used for a subset of the parity-check bits. In yet another example, additional steps may be performed when expanding the base matrix to the expanded matrix. The matrix H may also be used in any type of decoder that relies upon a parity-check matrix. It is intended that such changes come within the scope of the following claims.

## Claims

1. A method for operating a transmitter that generates parity-check bits p=(p0, . . . , pm−1) based on a current symbol set s=(s0, . . . , sk−1), the method comprising the steps of:
receiving the current symbol set s=(s0, . . . , Sk−1);
using a matrix H to determine the parity-check bits wherein H is represented by Hbm or a column or row permutation of the matrix Hbm, where Hbm=





   
  
    
  
  ⁢
  
    
      
        
          40
        
        
          -
        
        
          -
        
        
          -
        
        
          22
        
        
          -
        
        
          49
        
        
          23
        
        
          43
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          50
        
        
          1
        
        
          -
        
        
          -
        
        
          48
        
        
          35
        
        
          -
        
        
          -
        
        
          13
        
        
          -
        
        
          30
        
        
          -
        
      
      
        
          39
        
        
          50
        
        
          -
        
        
          -
        
        
          4
        
        
          -
        
        
          2
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          49
        
      
      
        
          33
        
        
          -
        
        
          -
        
        
          38
        
        
          37
        
        
          -
        
        
          -
        
        
          4
        
        
          1
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          45
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          22
        
        
          -
        
        
          -
        
        
          20
        
        
          42
        
        
          -
        
        
          -
        
      
      
        
          51
        
        
          -
        
        
          -
        
        
          48
        
        
          35
        
        
          -
        
        
          -
        
        
          -
        
        
          44
        
        
          -
        
        
          18
        
        
          -
        
      
      
        
          47
        
        
          11
        
        
          -
        
        
          -
        
        
          -
        
        
          17
        
        
          -
        
        
          -
        
        
          51
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          5
        
        
          -
        
        
          25
        
        
          -
        
        
          6
        
        
          -
        
        
          45
        
        
          -
        
        
          13
        
        
          40
        
        
          -
        
        
          -
        
      
      
        
          33
        
        
          -
        
        
          -
        
        
          34
        
        
          24
        
        
          -
        
        
          -
        
        
          -
        
        
          23
        
        
          -
        
        
          -
        
        
          46
        
      
      
        
          1
        
        
          -
        
        
          27
        
        
          -
        
        
          1
        
        
          -
        
        
          -
        
        
          -
        
        
          38
        
        
          -
        
        
          44
        
        
          -
        
      
      
        
          -
        
        
          18
        
        
          -
        
        
          -
        
        
          23
        
        
          -
        
        
          -
        
        
          8
        
        
          0
        
        
          35
        
        
          -
        
        
          -
        
      
      
        
          49
        
        
          -
        
        
          17
        
        
          -
        
        
          30
        
        
          -
        
        
          -
        
        
          -
        
        
          34
        
        
          -
        
        
          -
        
        
          19
        
      
    
    ⁢
    
      

    
    ⁢
    
      
        
          1
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
        
          -
        
      
      
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          0
        
        
          0
        
      
      
        
          1
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          -
        
        
          
            0
            ;
          
        
      
    
  




and transmitting the parity-check bits along with the current symbol set.

2. The method of claim 1 wherein:
H is an expansion of a base matrix Hb with Hb comprising a section Hb1 and a section Hb2, with Hb2 comprising a first part having a column hb having an odd weight greater than 2, and a second part H′b2 comprising matrix elements for row i, column j equal to
1 for i=j,
1 for i=j+1,
0 elsewhere;

wherein the expansion of the base matrix Hb uses identical submatrices for 1s in each column of the second part H′b2, and wherein the expansion uses paired submatrices for an even number of 1s in hb.

3. The method of claim 1 wherein the parity-check matrix H can be partitioned into square subblocks (submatrices) of size z×z, where subblock size is z=54 bits.

4. The method of claim 3 wherein the submatrices are either cyclic-permutations of the identity matrix or null submatrices.

5. The method of claim 3 wherein the cyclic-permutation matrix Pi is obtained from the z×z identity matrix by cyclically shifting the columns to the right by i elements.

6. The method of claim 1 wherein the step of using matrix H to determine the parity-check bits p=(p0, . . . , pm−1) comprises the step of determining




  
    
      v
      ⁡
      
        (
        0
        )
      
    
    =
    
      
        P
        
          z
          -
          
            p
            ⁡
            
              (
              
                x
                ,
                
                  k
                  b
                
              
              )
            
          
        
      
      ⁢
      
        
          ∑
          
            j
            =
            0
          
          
            
              k
              b
            
            -
            1
          
        
        ⁢
        
          
            ∑
            
              i
              =
              0
            
            
              
                m
                b
              
              -
              1
            
          
          ⁢
          
            
              P
              
                p
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
            ⁢
            
              u
              ⁡
              
                (
                j
                )
              
            
          
        
      
    
  
  ,




where u=[u(0) u(1) . . . u(kb−1)] is the current symbol set s grouped into kb groups of z bits, v=[v(0) v(1) . . . v(mb−1)] is the parity-check bits p grouped into mb groups of z bits, p(i,j) is a shift size, and multiplication by Px circularly shifts a vector by x positions.

7. The method of claim 1 wherein the step of using matrix H to determine the parity-check bits (p0, . . . , pm−1) comprises the step of determining




  
    v
    ⁡
    
      (
      0
      )
    
  
  =
  
    
      P
      
        z
        -
        
          p
          ⁡
          
            (
            
              x
              ,
              
                k
                b
              
            
            )
          
        
      
    
    ⁢
    
      
        ∑
        
          j
          =
          0
        
        
          
            k
            b
          
          -
          1
        
      
      ⁢
      
        
          ∑
          
            i
            =
            0
          
          
            
              m
              b
            
            -
            1
          
        
        ⁢
        
          
            P
            
              p
              ⁡
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
          
          ⁢
          
            u
            ⁡
            
              (
              j
              )
            
          
        
      
    
  




and determining v(1), v(2), . . . , v(mb−1) recursively via




  
    
      v
      ⁡
      
        (
        1
        )
      
    
    =
    
      
        
          ∑
          
            j
            =
            0
          
          
            
              k
              b
            
            -
            1
          
        
        ⁢
        
          
            P
            
              p
              ⁡
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
          
          ⁢
          
            u
            ⁡
            
              (
              j
              )
            
          
        
      
      +
      
        
          P
          
            p
            ⁡
            
              (
              
                i
                ,
                
                  k
                  b
                
              
              )
            
          
        
        ⁢
        
          v
          ⁡
          
            (
            0
            )
          
        
      
    
  
  ,
  
    i
    =
    0
  
  ,
  
    

  
  ⁢
  
    
      v
      ⁡
      
        (
        
          i
          +
          1
        
        )
      
    
    =
    
      
        v
        ⁡
        
          (
          i
          )
        
      
      +
      
        
          ∑
          
            j
            =
            0
          
          
            
              k
              b
            
            -
            1
          
        
        ⁢
        
          
            P
            
              p
              ⁡
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
          
          ⁢
          
            u
            ⁡
            
              (
              j
              )
            
          
        
      
      +
      
        
          P
          
            p
            ⁡
            
              (
              
                i
                ,
                
                  k
                  b
                
              
              )
            
          
        
        ⁢
        
          v
          ⁡
          
            (
            0
            )
          
        
      
    
  
  ,
  
    i
    =
    1
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      m
      b
    
    -
    2
  
  ,




where u=[u(0) u(1) . . . u(kb−1)] is the current symbol set s grouped into kb groups of z bits, v=[v(0) v(1) . . . v(mb−1)] is the parity-check bits p grouped into mb groups of z bits, p(i,j) is a shift size, and multiplication by Px circularly shifts a vector by x positions.

8. The method of claim 1 wherein the step of using matrix H to determine the parity-check bits (p0, . . . , pm−1) comprises the step of determining




  
    v
    ⁡
    
      (
      0
      )
    
  
  =
  
    
      P
      
        z
        -
        
          p
          ⁡
          
            (
            
              x
              ,
              
                k
                b
              
            
            )
          
        
      
    
    ⁢
    
      
        ∑
        
          j
          =
          0
        
        
          
            k
            b
          
          -
          1
        
      
      ⁢
      
        
          ∑
          
            i
            =
            0
          
          
            
              m
              b
            
            -
            1
          
        
        ⁢
        
          
            P
            
              p
              ⁡
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
          
          ⁢
          
            u
            ⁡
            
              (
              j
              )
            
          
        
      
    
  




and determining v(1), v(2), . . . , v(mb−1) recursively, where u=[u(0) u(1) . . . u(kb−1)] is the current symbol set s grouped into kb groups of z bits, v=[v(0) v(1) . . . v(mb−1)] is the parity-check bits p grouped into mb groups of z bits, p(i,j) is a shift size, and multiplication by Px circularly shifts a vector by x positions.

9. The method of claim 1 further comprising the steps of:
determining additional parity-check bits need to be transmitted; and
transmitting the additional parity-check bits (pm, . . . , pm+m2−1) based on the current symbol set s=(s0, . . . , sk−1) and p=(p0, . . . , pm−1), where m2=mb(2)z.

10. The method of claim 9 further comprising the step of using a matrix Prototype Hbm(2) to determine the additional parity-check bits wherein Hbm(2) is based on Hbm.

11. A method for operating a receiver that estimates a current symbol set s=(s0, . . . , sk−1), the method comprising the steps of:
receiving a received signal vector y=(y0. . . yn−1); and
using a matrix H to estimate the current symbol set s=(s0, . . . , sk−1), wherein matrix H is represented by Hbm or a column or row permutation of the matrix Hbm, where Hbm=





  
    
      
        40
      
      
        -
      
      
        -
      
      
        -
      
      
        22
      
      
        -
      
      
        49
      
      
        23
      
      
        43
      
      
        -
      
      
        -
      
      
        -
      
      
        1
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        50
      
      
        1
      
      
        -
      
      
        -
      
      
        48
      
      
        35
      
      
        -
      
      
        -
      
      
        13
      
      
        -
      
      
        30
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        39
      
      
        50
      
      
        -
      
      
        -
      
      
        4
      
      
        -
      
      
        2
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        49
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        33
      
      
        -
      
      
        -
      
      
        38
      
      
        37
      
      
        -
      
      
        -
      
      
        4
      
      
        1
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        45
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        22
      
      
        -
      
      
        -
      
      
        20
      
      
        42
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        51
      
      
        -
      
      
        -
      
      
        48
      
      
        35
      
      
        -
      
      
        -
      
      
        -
      
      
        44
      
      
        -
      
      
        18
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        47
      
      
        11
      
      
        -
      
      
        -
      
      
        -
      
      
        17
      
      
        -
      
      
        -
      
      
        51
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        5
      
      
        -
      
      
        25
      
      
        -
      
      
        6
      
      
        -
      
      
        45
      
      
        -
      
      
        13
      
      
        40
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        33
      
      
        -
      
      
        -
      
      
        34
      
      
        24
      
      
        -
      
      
        -
      
      
        -
      
      
        23
      
      
        -
      
      
        -
      
      
        46
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
    
    
      
        1
      
      
        -
      
      
        27
      
      
        -
      
      
        1
      
      
        -
      
      
        -
      
      
        -
      
      
        38
      
      
        -
      
      
        44
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
    
    
      
        -
      
      
        18
      
      
        -
      
      
        -
      
      
        23
      
      
        -
      
      
        -
      
      
        8
      
      
        0
      
      
        35
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
    
    
      
        49
      
      
        -
      
      
        17
      
      
        -
      
      
        30
      
      
        -
      
      
        -
      
      
        -
      
      
        34
      
      
        -
      
      
        -
      
      
        19
      
      
        1
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0.

12. The method of claim 11 wherein the parity-check matrix H can be partitioned into square subblocks (submatrices) of size z×z, where subblock size is z=54 bits.

13. The method of claim 12 wherein the submatrices are either cyclic-permutations of the identity matrix or null submatrices.

14. The method of claim 13 wherein the cyclic-permutation matrix Pi is obtained from the z×z identity matrix by cyclically shifting the columns to the right by i elements.

15. An apparatus comprising:
storage means for storing a matrix H;
a microprocessor using a matrix H to determine parity-check bits; and
a transmitter for transmitting the parity-check bits;
wherein matrix H is represented by Hbm or a column or row permutation of the matrix Hbm, where Hbm=





  
    
      
        40
      
      
        -
      
      
        -
      
      
        -
      
      
        22
      
      
        -
      
      
        49
      
      
        23
      
      
        43
      
      
        -
      
      
        -
      
      
        -
      
      
        1
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        50
      
      
        1
      
      
        -
      
      
        -
      
      
        48
      
      
        35
      
      
        -
      
      
        -
      
      
        13
      
      
        -
      
      
        30
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        39
      
      
        50
      
      
        -
      
      
        -
      
      
        4
      
      
        -
      
      
        2
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        49
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        33
      
      
        -
      
      
        -
      
      
        38
      
      
        37
      
      
        -
      
      
        -
      
      
        4
      
      
        1
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        45
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        22
      
      
        -
      
      
        -
      
      
        20
      
      
        42
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        51
      
      
        -
      
      
        -
      
      
        48
      
      
        35
      
      
        -
      
      
        -
      
      
        -
      
      
        44
      
      
        -
      
      
        18
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        47
      
      
        11
      
      
        -
      
      
        -
      
      
        -
      
      
        17
      
      
        -
      
      
        -
      
      
        51
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        5
      
      
        -
      
      
        25
      
      
        -
      
      
        6
      
      
        -
      
      
        45
      
      
        -
      
      
        13
      
      
        40
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        33
      
      
        -
      
      
        -
      
      
        34
      
      
        24
      
      
        -
      
      
        -
      
      
        -
      
      
        23
      
      
        -
      
      
        -
      
      
        46
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
    
    
      
        1
      
      
        -
      
      
        27
      
      
        -
      
      
        1
      
      
        -
      
      
        -
      
      
        -
      
      
        38
      
      
        -
      
      
        44
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
    
    
      
        -
      
      
        18
      
      
        -
      
      
        -
      
      
        23
      
      
        -
      
      
        -
      
      
        8
      
      
        0
      
      
        35
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
    
    
      
        49
      
      
        -
      
      
        17
      
      
        -
      
      
        30
      
      
        -
      
      
        -
      
      
        -
      
      
        34
      
      
        -
      
      
        -
      
      
        19
      
      
        1
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0.

16. An apparatus comprising:
storage means for storing a matrix H;
a receiver for receiving a signal vector y=(y0. . . yn−1); and
a microprocessor using a matrix H to determine a current symbol set (s0, . . . , sk−1), wherein matrix H is represented by Hbm or a column or row permutation of the matrix Hbm, where Hbm=





  
    
      
        40
      
      
        -
      
      
        -
      
      
        -
      
      
        22
      
      
        -
      
      
        49
      
      
        23
      
      
        43
      
      
        -
      
      
        -
      
      
        -
      
      
        1
      
      
        0
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
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        50
      
      
        1
      
      
        -
      
      
        -
      
      
        48
      
      
        35
      
      
        -
      
      
        -
      
      
        13
      
      
        -
      
      
        30
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
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        -
      
      
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        39
      
      
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        4
      
      
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        2
      
      
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        -
      
      
        -
      
      
        -
      
      
        49
      
      
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        0
      
      
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        -
      
    
    
      
        33
      
      
        -
      
      
        -
      
      
        38
      
      
        37
      
      
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        -
      
      
        4
      
      
        1
      
      
        -
      
      
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        -
      
    
    
      
        45
      
      
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        22
      
      
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        20
      
      
        42
      
      
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        -
      
      
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        51
      
      
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        48
      
      
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        44
      
      
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        18
      
      
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        -
      
      
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        -
      
      
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        0
      
      
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        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
    
    
      
        47
      
      
        11
      
      
        -
      
      
        -
      
      
        -
      
      
        17
      
      
        -
      
      
        -
      
      
        51
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
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        -
      
      
        -
      
      
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        0
      
      
        0
      
      
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        -
      
      
        -
      
      
        -
      
    
    
      
        5
      
      
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        25
      
      
        -
      
      
        6
      
      
        -
      
      
        45
      
      
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        13
      
      
        40
      
      
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        33
      
      
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        23
      
      
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        46
      
      
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        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0
      
      
        0
      
      
        -
      
      
        -
      
    
    
      
        1
      
      
        -
      
      
        27
      
      
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        1
      
      
        -
      
      
        -
      
      
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        38
      
      
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        44
      
      
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        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
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        18
      
      
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        23
      
      
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        8
      
      
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        35
      
      
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        -
      
      
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        -
      
      
        -
      
      
        -
      
      
        -
      
      
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        -
      
      
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        0
      
    
    
      
        49
      
      
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        17
      
      
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        30
      
      
        -
      
      
        -
      
      
        -
      
      
        34
      
      
        -
      
      
        -
      
      
        19
      
      
        1
      
      
        -
      
      
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        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        -
      
      
        0.