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IEEE Transactions on Information Theory

An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices

Eirik Rosnes, Øyvind Ytrehus

Published Aug 24, 2009• 63 citations
Publisher

Abstract

In this work, we introduce an efficient algorithm to find all stopping sets, of size less than some threshold, of a fixed low-density parity-check (LDPC) matrix. The solution is inspired by the algorithm proposed by Rosnes and Ytrehus in 2005 to find an exhaustive list of all small-size turbo stopping sets in a turbo code. The efficiency of the proposed algorithm is demonstrated by several numerical examples. For instance, we have applied the algorithm to the well-known (3, 5)-regular (155, 64) Tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer. Also, we have verified that the minimum stopping set size of the (4896, 2474) Ramanujan-Margulis code is indeed 24, and that the corresponding multiplicity is exactly 204. Furthermore, we have applied the algorithm to the IEEE 802.16e LDPC codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes. Finally, as an application, we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity-check matrix in order to remove all stopping sets in the corresponding Tanner graph of size less than the minimum distance. An extensive case study of the (155, 64) Tanner code illustrates the usefulness of the algorithm, and we present a 110 times 155 redundant parity-check matrix for this code with no stopping sets of size less than the minimum distance. Simulation results of iterative decoding on the binary erasure channel show performance improvements for low-to-medium erasure probabilities when this redundant parity-check matrix is used for decoding.

DOI
10.1109/tit.2009.2025573

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Addendum to “An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices”

Eirik Rosnes, Øyvind Ytrehus, Marcel Ambroze +1 more

In an earlier transactions paper, Rosnes and Ytrehus presented an efficient algorithm for determining all stopping sets of low-density parity-check (LDPC) codes, up to a specified weight, and also gave results for a number of well-known codes including the family of IEEE 802.16e LDPC codes, commonly referred to as the WiMax codes. It is the purpose of this short paper to review the algorithm for determining the initial part of the stopping set weight spectrum (which includes the codeword weight spectrum), and to provide some improvements to the algorithm. As a consequence, complete stopping set weight spectra up to weight 32 (for selected IEEE 802.16e LDPC codes) can be provided, while in previous work only stopping set weights up to 28 are reported. In the published standard for the IEEE 802.16e codes there are two methods of construction presented, depending upon the code rate and the code length. We compare the stopping sets of the resulting codes and provide complete stopping set weight spectra (up to five terms) for all IEEE 802.16e LDPC codes using both construction methods.

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The performance of low-density parity-check (LDPC) codes over the binary erasure channel for a low erasure probability is limited by small stopping sets. The frame error rate over the additive white Gaussian noise (AWGN) channel for a high signal-to-noise ratio is limited by small trapping sets and by the minimum distance of the LDPC code, which is equal to the codeword with the minimal Hamming weight. In this paper, we present a simple and fast algorithm based on the error impulse method to find small stopping sets in the Tanner graphs of LDPC codes. Since a codeword with small Hamming weight can be represented by a special stopping set, the algorithm can also be used to find the minimum distance of LDPC codes. Furthermore, we use the size of the small stopping sets and the minimum distance of the code to calculate asymptotic performance bounds for LDPC codes transmitted over the binary erasure and the AWGN channel, respectively and compare the bounds with simulations. 1

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In this work, we introduce an efficient algorithm to find all stopping sets, of size less than some threshold, of a fixed low-density parity-check (LDPC) matrix. The solution is inspired by the algorithm proposed by Rosnes and Ytrehus in 2005 to find an exhaustive list of all small-size turbo stopping sets in a turbo code. The efficiency of the proposed algorithm is demonstrated by several numerical examples. For instance, we have applied the algorithm to the well-known (3, 5)-regular (155, 64) Tanner code and found all stopping sets of size at most 18 in about 1 min on a standard desktop computer. Also, we have verified that the minimum stopping set size of the (4896, 2474) Ramanujan-Margulis code is indeed 24, and that the corresponding multiplicity is exactly 204. Furthermore, we have applied the algorithm to the IEEE 802.16e LDPC codes and determined the minimum stopping set size and the corresponding multiplicity exactly for these codes. Finally, as an application, we present a greedy algorithm to find a small number of redundant parity checks to add to the original parity-check matrix in order to remove all stopping sets in the corresponding Tanner graph of size less than the minimum distance. An extensive case study of the (155, 64) Tanner code illustrates the usefulness of the algorithm, and we present a 110 times 155 redundant parity-check matrix for this code with no stopping sets of size less than the minimum distance. Simulation results of iterative decoding on the binary erasure channel show performance improvements for low-to-medium erasure probabilities when this redundant parity-check matrix is used for decoding.
In this work, we introduce an efficient algorithm to find all stopping sets of size less than some threshold of a fixed low-density parity-check (LDPC) matrix. The solution is inspired by the algorithm proposed by Rosnes and Ytrehus in 2005 to find an exhaustive list of all small-size turbo stopping sets in a turbo code. The efficiency of the proposed algorithm is demonstrated by several numerical examples. For instance, we have applied the algorithm to the well-known (3, 5)-regular (155,64) Tanner code and found all stopping sets of size at most 18 in about one minute on a standard desktop computer. Also, we have verified that the minimum stopping set size of the (4896,2474) Ramanujan-Margulis code is indeed 24, and that the corresponding multiplicity is exactly 204.
On the binary erasure channel (BEC), the performance of low-density parity-check (LDPC) codes decoded by iterative algorithms is estimated by the small stopping sets in the codes. The size of smallest stopping set is called the stopping distance. Since it depends on the structure of parity-check matrix for the code, we can construct the parity-check matrix by adding dependent rows so that the stopping distance is equal to the minimum distance. The stopping redundancy is defined as the smallest number of rows in such a parity-check matrix. In this paper, we propose a probabilistic algorithm for computing the stopping redundancy of LDPC codes. Additionally, we show numerical results of computing the stopping redundancy of (155,64), (504,252) and (1008,504) LDPC codes.
In an earlier transactions paper, Rosnes and Ytrehus presented an efficient algorithm for determining all stopping sets of low-density parity-check (LDPC) codes, up to a specified weight, and also gave results for a number of well-known codes including the family of IEEE 802.16e LDPC codes, commonly referred to as the WiMax codes. It is the purpose of this short paper to review the algorithm for determining the initial part of the stopping set weight spectrum (which includes the codeword weight spectrum), and to provide some improvements to the algorithm. As a consequence, complete stopping set weight spectra up to weight 32 (for selected IEEE 802.16e LDPC codes) can be provided, while in previous work only stopping set weights up to 28 are reported. In the published standard for the IEEE 802.16e codes there are two methods of construction presented, depending upon the code rate and the code length. We compare the stopping sets of the resulting codes and provide complete stopping set weight spectra (up to five terms) for all IEEE 802.16e LDPC codes using both construction methods.
The performance of low-density parity-check (LDPC) codes over the binary erasure channel for a low erasure probability is limited by small stopping sets. The frame error rate over the additive white Gaussian noise (AWGN) channel for a high signal-to-noise ratio is limited by small trapping sets and by the minimum distance of the LDPC code, which is equal to the codeword with the minimal Hamming weight. In this paper, we present a simple and fast algorithm based on the error impulse method to find small stopping sets in the Tanner graphs of LDPC codes. Since a codeword with small Hamming weight can be represented by a special stopping set, the algorithm can also be used to find the minimum distance of LDPC codes. Furthermore, we use the size of the small stopping sets and the minimum distance of the code to calculate asymptotic performance bounds for LDPC codes transmitted over the binary erasure and the AWGN channel, respectively and compare the bounds with simulations. 1
It is known that redundant parity-check equations can improve the performance of an LDPC code by reducing the number of harmful substructures in the parity-check matrix. However, it is a difficult problem to design a parity-check matrix in such a way that it avoids substructures that are known to be harmful to iterative decoding while keeping the number of redundant parity-check equations moderate and ensuring other desirable properties. We explicitly give redundant parity-check matrices for cyclic regular LDPC codes of length n and minimum distance d ~ √n in which there are only n parity-check equations but no stopping sets of size d+1 or smaller except for those that correspond to the nonzero codewords of the smallest weight. We do this by showing that the well-known projective LDPC codes from the incidence matrices of projective planes PG(2, q) with q even have this property. This result may give insight into how the small number of redundant parity-check equations in the geometric LDPC codes may be contributing to the good performance reported in the literature. We also give a slightly improved upper bound on the size of a smallest generic erasure correcting set.