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arXiv:cs/0607024 · IEEE Transactions on Information Theory

Results on Parity-Check Matrices With Optimal Stopping And/Or Dead-End Set Enumerators

Jos H. Weber, Khaled Abdel-Ghaffar

Published Feb 29, 2008• 39 citations
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Abstract

The performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying Tanner graph, or, equivalently, the parity-check matrix representing the code. In this correspondence, we introduce the notion of dead-end sets to explicitly demonstrate this dependency. The choice of the parity-check matrix entails a tradeoff between performance and complexity. We give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity-check matrices. Further, we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator.

DOI
10.1109/tit.2007.915923

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The performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying Tanner graph, or, equivalently, the parity-check matrix representing the code. In this correspondence, we introduce the notion of dead-end sets to explicitly demonstrate this dependency. The choice of the parity-check matrix entails a tradeoff between performance and complexity. We give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity-check matrices. Further, we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator.
Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let CCC be a binary [n,k][n,k][n,k] linear code with parity-check matrix HHH, where the rows of HHH may be dependent. A stopping set SSS of CCC with parity-check matrix HHH is a subset of column indices of HHH such that the restriction of HHH to SSS does not contain a row of weight one. The stopping set distribution {Ti(H)}i=0n\{T_i(H)\}_{i=0}^n{Ti​(H)}i=0n​ enumerates the number of stopping sets with size iii of CCC with parity-check matrix HHH. Note that stopping sets and stopping set distribution are related to the parity-check matrix HHH of CCC. Let H∗H^{*}H∗ be the parity-check matrix of CCC which is formed by all the non-zero codewords of its dual code C⊥C^{\perp}C⊥. A parity-check matrix HHH is called BEC-optimal if Ti(H)=Ti(H∗),i=0,1,...,nT_i(H)=T_i(H^*), i=0,1,..., nTi​(H)=Ti​(H∗),i=0,1,...,n and HHH has the smallest number of rows. On the BEC, iterative decoder of CCC with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.
It is now well known that the performance of a linear code CCC under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for CCC. Several recent papers refer to this parameter as the \emph{stopping distance} sss of CCC. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for CCC depends on the corresponding choice of a parity-check matrix. It is easy to see that s≤ds \le ds≤d, where ddd is the minimum Hamming distance of CCC, and we show that it is always possible to choose a parity-check matrix for CCC (with sufficiently many dependent rows) such that s=ds = ds=d. We thus introduce a new parameter, termed the \emph{stopping redundancy} of CCC, defined as the minimum number of rows in a parity-check matrix HHH for CCC such that the corresponding stopping distance s(H)s(H)s(H) attains its largest possible value, namely s(H)=ds(H) = ds(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 35 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.
It is now well known that the performance of a linear code Copf under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for Copf. Several recent papers refer to this parameter as the stopping distance s of Copf. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for Copf depends on the corresponding choice of a parity-check matrix. It is easy to see that s les d, where d is the minimum Hamming distance of Copf, and we show that it is always possible to choose a parity-check matrix for Copf (with sufficiently many dependent rows) such that s = d. We thus introduce a new parameter, termed the stopping redundancy of Copf, defined as the minimum number of rows in a parity-check matrix H for Copf such that the corresponding stopping distance s(H) attains its largest possible value, namely s(H) = d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes
It is now well known that the performance of a linear code Copf under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for Copf. Several recent papers refer to this parameter as the stopping distance s of Copf. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for Copf depends on the corresponding choice of a parity-check matrix. It is easy to see that s les d, where d is the minimum Hamming distance of Copf, and we show that it is always possible to choose a parity-check matrix for Copf (with sufficiently many dependent rows) such that s=d. We thus introduce a new parameter, the stopping redundancy of Copf, defined as the minimum number of rows in a parity- check matrix H for Copf such that the corresponding stopping distance s(H) attains its largest possible value, namely, s(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes
We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of up to a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies for data transmission over the binary erasure channel that combine iterative message passing and permutation decoding in order to avoid errors confined to stopping sets. We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond to stopping sets within a judiciously chosen parity-check matrix.