Method and apparatus for encoding and decoding data

[0001] 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.

[0002] 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 s_1×k with a code generator matrix G_k×n to produce an n-bit codeword x_1×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 s_1×k.

[0003] Given an n-dimensional space, the rows of G span the k-dimensional codeword subspace C, and the rows of the parity-check matrix H_m×n span the m-dimensional dual space C^⊥, where m=n−k. Since x=sG and GH^T=0, it follows that xH^T=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
HX^T=0^T,  (1)
where 0 is a row vector of all zeros, and the codeword x=[s p]=[s₀, s₁, . . . ,s_k−1, p₀, p₁, . . . , p_m−1], where p₀, . . . , p_m−1 are the parity-check bits; and s₀, . . . , s_k−1 are the systematic bits, equal to the information bits within the information vector.

[0004] 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 v_i is connected to check node c_j 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 h_ji 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.

[0005] 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

[0006] $\begin{array}{cc}H=\underset{\underset{n}{︸}}{\left[\begin{array}{cccccccccccc}1& 0& 1& 0& 0& 0& 1& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 1& 0& 0& 1& 1& 0& 0& 0\\ 0& 0& 1& 0& 0& 1& 1& 0& 1& 1& 0& 0\\ 1& 0& 0& 1& 0& 0& 0& 0& 0& 1& 1& 0\\ 0& 1& 0& 0& 1& 0& 0& 0& 0& 0& 1& 1\\ 0& 0& 0& 1& 0& 1& 1& 0& 0& 0& 0& 1\end{array}\right]\}}m,& \left(2\right)\end{array}$
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:

[0007] $\begin{array}{cc}\{\begin{array}{c}{x}_0+x_2+x_6+x_7=0\\ x_1+x_4+x_7+x_8=0\\ x_2+x_5+x_6+x_8+x_9=0\\ x_0+x_3+x_9+x_{10}=0\\ x_1+x_4+x_{10}+x_{11}=0\\ x_3+x_5+x_6+x_{11}=0\end{array}.& \left(3\right)\end{array}$
H also has the corresponding bipartite graph shown in FIG. 1.

[0008] 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.

[0009] A structured LDPC code design starts with a small m_b×n_b base matrix H_b, makes z copies of H_b, and interconnects the z copies to form a large m×n H matrix, where m=m_b×z, n=n_b×z. Using the matrix representation, to build an H from H_b each 1 in H_b is replaced by a z×z permutation submatrix, and each 0 in H_b is replaced by a z×z all-zero submatrix. This procedure essentially maps each edge of H_b to a vector edge of length z in H, each variable node of H_b to a vector variable node of length z in H, and each check node of H_b to a vector check node of length z in H. The benefits of vectorizing a small matrix H_b to build a large matrix H are:

• • [0010] 1. By using a different values of z, codes of rate k_b/n_b, where k_b=n_b−m_b, can be designed for many different information sequence sizes k=z×k_b from a single base matrix H_b.
  • [0011] 2. Memory requirements are greatly reduced. With a structured LDPC design, only the base matrix H_b and the permutation for its 1's need to be stored, which requires significantly less memory since H_b is typically much smaller than H and the permutation can be very simple.
  • [0012] 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.

[0013] 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.

[0020] To address the above-mentioned need, a structured parity-check matrix H is proposed, wherein H is an expansion of a base matrix H_b and wherein H_b comprises a section H_b1 and a section H_b2, and wherein H_b2 comprises a first part comprising a column h_b 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 H_b 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 h_b.

[0021] The present invention encompasses a method for operating a transmitter that generates parity-check bits p=(p₀, . . . , p_m−1) based on a current symbol set s=(s₀, . . . , s_k−1). The method comprises the steps of receiving the current symbol set s=(s₀, . . . , s_k−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 H_b where H_b comprises a section H_b1 and a section H_b2, and wherein H_b2 comprises a first part comprising a column h_b 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 H_b 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 h_b.

[0022] The present invention additionally encompasses a method for operating a receiver that estimates a current symbol set s=(s₀, . . . , s_k−1). The method comprises the steps of receiving a received signal vector y=(y₀ . . . y_n−1) and using a matrix H to estimate the current symbol set s=(s₀, . . . , s_k−1). Matrix H is an expansion of a base matrix H_b and wherein H_b comprises a section H_b1 and a section H_b2, with H_b2 comprising a first part comprising a column h_b 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 H_b uses identical submatrices for is in each column of the second part H′_b2, and wherein the expansion uses paired submatrices for an even number of 1s in h_b.

[0023] 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 H_b and H_b comprises a section H_b1 and a section H_b2, with H_b2 comprising a first part comprising a column h_b 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 H_b 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 h_b.

[0024] The present invention encompasses an apparatus comprising storage means for storing a matrix H, a receiver for receiving a signal vector y=(y₀ . . . y_n−1), and a microprocessor using a matrix H to determine a current symbol set (s₀, . . . , s_k−1). Matrix H is an expansion of a base matrix H_b with H_b comprising a section H_b1 and a section H_b2, and wherein H_b2 comprises a first part comprising a column h_b having an odd weight greater than 2. H_b2 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 1s in every column of H′_b2, and paired submatrices are used to expand an even number of 1s in h_b.

[0025] 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.

[0026] 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, . . . , p_m−1) based on the current symbol set s=(s₀, . . . , s_k−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.

[0027] 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.

[0028] The received signal vector (received via a receiver) y=(y₀, . . . , y_n−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, . . . , s_k−1)). In particular, microprocessor 401 estimates the current symbol set (s₀, . . . , s_k−1) based on the received signal vector y=(y₀, . . . , y_n−1) and the parity-check matrix H.

[0029] 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.

[0030] 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 H_b if the permutation submatrix is used. Therefore, the weight distribution of H_b is chosen as close to the desired final weight distribution as possible. The following description is illustrative of the case where the entries of H_b are replaced by permutation matrices, though any matrices may be used. If a permutation submatrix P_z×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.

[0031] 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 m_b×n_b model matrix H_bm, which can be obtained by

• • [0032] replacing each 0 in H_b by −1 to denote a z×z all-zero submatrix, and
  • [0033] replacing each h_i,j=1 in H_b by a circular shift size p(i,j) where p(i,j) is non-negative.

[0034] 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 H_bm. Therefore, H_bm is a shorthand representation of H if z is given. Notationally, the model matrix 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 H_bm.

[0035] For example, the matrix of Equation (2) may be used as a base matrix H_b to build a model matrix H_bm as follows:

[0036] $\begin{array}{cc}{H}_{\mathrm{bm}}=\underset{\underset{n_b}{︸}}{\left[\begin{array}{cccccccccccc}1& -1& 0& -1& -1& -1& 0& 0& -1& -1& -1& -1\\ -1& 2& -1& -1& 0& -1& -1& 0& 0& -1& -1& -1\\ -1& -1& 1& -1& -1& 2& 2& -1& 0& 0& -1& -1\\ 2& -1& -1& 1& -1& -1& -1& -1& -1& 0& 0& -1\\ -1& 1& -1& -1& 0& -1& -1& -1& -1& -1& 0& 0\\ -1& -1& -1& 0& -1& 1& 0& -1& -1& -1& -1& 0\end{array}\right]\}}{m}_b.& \left(4\right)\end{array}$
When z=3, H_bm is converted to a (m_b×z)×(n_b×z) matrix H by replacing each −1 with a 3×3 all-zero submatrix and each i with submatrix P_i, i=0, 1, 2, where

[0037] $P_0=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],P_1=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right],P_2=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right].$
Note that P₀ is the identity matrix, and the columns of P_i, i>0, are the columns of P₀ circular right shifted i times.

[0038] Given a vector q=[q₀, q₁, q₂], qP₀=[q₀, q₁, q₂], qP₁=[q₂, q₀, q₁], qP₂=[q₁, q₂, q₀]. In other words, q P_i results in circular right shift of the vector q. On the other hand, P_iq^T, results in circular upper shift of q^T, or equivalently circular left shift of q. Similar rules apply when a z×z matrix Q is used: QP_i results in circular right shift of the columns of Q, P_iQ results in circular upper shift of the rows of Q.

Base Matrix H

[0039] 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, . . . , s_k−1, p₀, p₁, . . . ,p_m−1], an m-by-n H matrix can be divided into two submatrices,
H=[H₁H₂],  (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 H_b in a structured LDPC design. Similarly, using the modified staircase structure, H_b can be partitioned into two sections, where H_b1 corresponds to the systematic bits s, H_b2 corresponds to the parity-check bits p:
H_b=[(H_b1)_mb×kb(H_b2)_mb×mb].  (6)
Section H_b2 can be further partitioned into two sections, where vector h_b has odd weight w_h, and H′_b2 has a staircase structure:

[0040] $\begin{array}{cc}\begin{array}{c}{H}_{\mathrm{b2}}=\left[\begin{array}{cc}{h}_b& H_{\mathrm{b2}}^{\prime }\end{array}\right]\\ =\left[\begin{array}{cccccc}{h}_b\left(0\right)& 1& & & & \\ h_b\left(1\right)& 1& 1& & 0& \\ .& & 1& ⋰& & \\ .& & & ⋰& 1& \\ .& & 0& & 1& 1\\ h_b\left(m_b-1\right)& & & & & 1\end{array}\right].\end{array}& \left(7\right)\end{array}$
Section H_b1 can be built randomly. Preferably the entire matrix H_b has a weight distribution as close to the desired weight distribution as possible.
Shift Sizes

[0041] To convert the base matrix H_b to the m_b×n_b model matrix H_bm (which expands to H), the circular shift sizes p(i,j) need to be determined for each 1 in H_b. 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 H_bm1) 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.

[0042] 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 h_b cancel when added together, and all vector rows of H′_b2 cancel when summed up. This translates to assigning shift sizes to h_b 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 h_b=[1 0 0 1 0 0 1]^T, it is acceptable to have h_bm=[3−1−1 3−1−1 4]^T as the corresponding column in the model matrix 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.

[0043] Due to existence of cycles, the shift sizes of h_bm 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 H_b, then the corresponding 2c vector edges form z cycles of length 2c in the expanded matrix H if and only if

[0044] $\begin{array}{cc}\sum _{\underset{j=0,\dots ,c-1}{i=2j}}p\left(i\right)=\sum _{\underset{j=0,\dots ,c-1}{i=2j+1}}p\left(i\right)\mathrm{mod}z,& \left(8\right)\end{array}$
where z is the expansion factor, p(i) is the circular shift size of edge i in the model matrix H_bm, and edge 0, 1, 2, . . . , 2c−1 (in this order) form a cycle in H_b.

[0045] Due to the structure of H_b2, cycles exist between h_b and H′_b2. Thus any two identical shift sizes in h_bm 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 h_b of the base matrix has three 1s, to maximize the cycle length, two 1s that are assigned equal shift sizes can be located at the top and the bottom of h_bm (as far apart as possible), while leaving one 1 in the middle of h_b with an unpaired shift size. For instance, h_bm=[3−1 3−1−1−1 4]^T would result in z cycles of length 6 between h and H′₂, while h_bm=[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 h_b and H′_b2.

[0046] In summary, the H_b2 section is mapped to the model matrix

[0047] $\begin{array}{cc}\begin{array}{c}{H}_{\mathrm{bm2}}=\left[\begin{array}{cc}{h}_{\mathrm{bm}}& H_{\mathrm{bm2}}^{\prime }\end{array}\right]\\ =\left[\begin{array}{cccccc}p\left(0,k_b\right)& p\left(0,k_b+1\right)& & & & \\ p\left(1,k_b\right)& p\left(1,k_b+1\right)& p\left(1,k_b+2\right)& & & \\ ⋮& & p\left(2,k_b+2\right)& ⋰& & \\ & & & ⋰& p\left(m_b-3,n_b-2\right)& \\ & & & & p\left(m_b-2,n_b-2\right)& p\left(m_b-2,n_b-1\right)\\ p\left(m_b-1,k_b\right)& & & & & p\left(m_b-1,n_b-1\right)\end{array}\right],\end{array}& \left(9\right)\end{array}$
where k_b=n_b−m_b, there are w_h (odd, w_h>=3) nonnegative entries in h_bm, and the −1 entries in H′_bm2 are left blank for brevity. All p(i,k_b) values appear an even number of times in h_bm except for one, which may be mapped to any non-zero submatrix. Therefore, all w_h shifts could be given the same value (e.g., 0), since w_h is odd. For H′_bm2, p(i,j)=p(i+1,j), j=k_b+1,k_b+2, . . . , n_b−1, i=j−k_b−1. In the preferred embodiment, assuming w_h=3, one example has h_bm=[0−1 . . . −1 p_h−1 . . . −1 . . . 0]^T, p_h mod z ≠0, and p(i,j)=p(i+1,j)=0, j=k_b+1,k_b+2, . . . , n_b−1, i=j−k_b−1 in the H′_bm2 portion.

[0048] 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 model matrix). In order to facilitate encoding, the constraints are then:

• • [0049] 1. In every column of H′_bm2, the two non-zero submatrices are identical;
  • [0050] 2. The w_h (odd, w_h>=3) non-zero submatrices of h_bm are paired (i.e., one submatrix is identical to another submatrix), except for one submatrix, which can be any non-zero matrix.
    Encoding

[0051] 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 k_b=n_b−m_b groups of z bits. Let this grouped s be denoted u,
u=[u(0) u(1) . . . u(k_b−1)],  (10)
where each element of u is a column vector as follows
u(i)=[s_iz s_iz+1 . . . s_(i+1)z−1]^T.  (11)

[0052] Using the model matrix H_bm, the parity sequence p is determined in groups of z. Let the grouped p be denoted v,
v=[v(0) v(1) . . . v(m_b−1)],  (12)
where each element of v is a column vector as follows
v(i)=[p_iz p_iz+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≦m_b−2.

[0053] An expression for v(0) can be derived by summing over the rows of Equation (1) to obtain

[0054] $\begin{array}{cc}{P}_{p\left(x,k_b\right)}v\left(0\right)=\sum _{j=0}^{k_b-1}\sum _{i=0}^{m_b-1}{P}_{p\left(i,j\right)}u\left(j\right)& \left(14\right)\end{array}$
where x is the row index of h_bm where the entry is nonnegative and is used an odd number of times. In the preferred embodiment, the top and bottom entries of h_bm are paired, thus 1≦x≦m_b−2. Equation (14) is solved for v(0) by multiplying both sides by P_p(x,kb)⁻¹. For the special case considered here where p(x,k_b) represents a circular shift, P_p(x,kb)⁻¹=P_z−p(x,kb). In other words, v(0) is obtained by

[0055] $\begin{array}{cc}v\left(0\right)=P_{z-p\left(x,k_b\right)}\sum _{j=0}^{k_b-1}\sum _{i=0}^{m_b-1}{P}_{p\left(i,j\right)}u\left(j\right).& \left(15\right)\end{array}$

[0056] In general, the recursions expressed in Equations (16) and (17) can be derived by considering the structure of H′_b2,

[0057] $\begin{array}{cc}{P}_{p\left(i,k_b+1\right)}v\left(1\right)=\sum _{j=0}^{k_b-1}{P}_{p\left(i,j\right)}u\left(j\right)+P_{p\left(i,k_b\right)}v\left(0\right),i=0,& \left(16\right)\end{array}$
and

[0058] $\begin{array}{cc}\begin{array}{c}{P}_{p\left(i,k_b+i+1\right)}v\left(i+1\right)=P_{p\left(i,k_b+i\right)}v\left(i\right)+\sum _{j=0}^{k_b-1}{P}_{p\left(i,j\right)}u\left(j\right)+P_{p\left(i,k_b\right)}v\left(0\right),\\ i=1,\dots ,m_b-2,\end{array}& \left(17\right)\end{array}$
where
P₋₁≡0_z×z.  (18)

[0059] Thus all parity bits not in v(0) are determined by iteratively evaluating Equations (16) and (17) for 0≦i≦m_b−2.

[0060] 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),

[0061] $\begin{array}{cc}\begin{array}{cc}v\left(1\right)=\sum _{j=0}^{k_b-1}{P}_{p\left(i,j\right)}u\left(j\right)+P_{p\left(i,k_b\right)}v\left(0\right),& i=0,\end{array}& \left(19\right)\end{array}$
and

[0062] $\begin{array}{cc}\begin{array}{c}v\left(i+1\right)=v\left(i\right)+\sum _{j=0}^{k_b-1}{P}_{p\left(i,j\right)}u\left(j\right)+P_{p\left(i,k_b\right)}v\left(0\right),\\ i=1,\dots ,m_b-2.\end{array}& \left(20\right)\end{array}$

[0063] 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≦m_b−2.

[0064] 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 H_bm represent circular shift sizes of a vector, all products of the form P_p(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, H_bm 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.

[0065] To implement the summations in Equations (14), (19), and (20) the entries p(i,j) of H_bm, 0≦i≦k_b, 0≦j≦m_b−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 H_bm 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,k_b).

[0066] Assuming the barrel shifting can be implemented in a single clock cycle, encoding can be accomplished in approximately (k_b+1)m_b clock cycles. This number can be reduced at the expense of m_b−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

[0067] 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 model matrix of the 1^st transmission is

[0068] $\begin{array}{cc}{H}_{\mathrm{bm}}^{\left(1\right)}=\left[\begin{array}{cc}{H}_{\mathrm{bm1}}^{\left(1\right)}& H_{\mathrm{bm2}}^{\left(1\right)}\end{array}\right],& \left(21\right)\end{array}$
then the model matrix for the 2^nd transmission may use

[0069] $\begin{array}{cc}{H}_{\mathrm{bm}}^{\left(2\right)}=\left[\begin{array}{cc}\begin{array}{cc}{H}_{\mathrm{bm1}}^{\left(1\right)}& H_{\mathrm{bm2}}^{\left(1\right)}\end{array}& 0\\ H_{\mathrm{bm1}}^{\left(2\right)}& H_{\mathrm{bm2}}^{\left(2\right)}\end{array}\right],& \left(22\right)\end{array}$
etc., where for each transmission i, submatrix H_bm2⁽ⁱ⁾ has the format in (9) and has size m_b⁽ⁱ⁾×m_b⁽ⁱ⁾. The first transmission may send n_b⁽¹⁾=k_b+m_b⁽¹⁾ groups of bits, └u(0),u(1), . . . , u(k_b−1), v⁽¹⁾(0), v⁽¹⁾(1), . . . , v⁽¹⁾(m_b⁽¹⁾−1)┘, each group having size z. The decoding after the 1^st transmission is performed using received signals of └u(0),u(1), . . . , u(k_b−1), v⁽¹⁾(0), v⁽¹⁾(1), . . . , v⁽¹⁾(m_b⁽¹⁾−1)┘ and (21). The 2^nd transmission may send another m_b⁽²⁾ groups of bits of size z, └v⁽²⁾(0), v⁽²⁾(1), . . . , v⁽²⁾(m_b⁽²⁾−1)┘, where m₂=m_b⁽²⁾z, and the bits of the first transmission and the second transmission together, └u(0), u(1), . . . , u(k_b−1), v⁽¹⁾(0), v⁽¹⁾(1), . . . , v⁽¹⁾(m_b⁽¹⁾−1), v⁽²⁾(0), v⁽²⁾(1), . . . , v⁽²⁾(m_b⁽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 1^st transmission and the 2^nd transmission. This procedure may be repeated for more transmissions. The decoding after the 2^nd transmission is based on a code of rate k_b/n_b⁽²⁾=k_b/(n_b⁽¹⁾+m_b⁽²⁾), which is lower than that of 1^st transmission. This procedure may be repeated for more transmissions, with each additional transmission contributing to a stronger, lower-rate code.

[0070] 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, . . . , s_k−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, . . . , p_m−1) are determined as described above, with H being an expansion of a base matrix H_b. As discussed, H_b comprises a section H_b, and a section H_b2, and wherein H_b2 comprises a first part comprising a column h_b 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 H_b (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 h_b. At step 505 the current symbol set and the parity-check bits are transmitted via over-the-air transmission.

[0071] 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, . . . , y₋₁) is received. At step 603, estimates of the current symbol set s (i.e., the current symbol set (s₀, . . . , s_k−1)) are determined based on H. As discussed, H is an expansion of a base matrix H_b and wherein H_b comprises a section H_b1 and a section H_b2, and wherein H_b2 comprises a first part comprising a column h_b 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.

[0072] 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 s_i and p_i 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.