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Low-Density Parity Check Codes over

Matthew C. Davey, David Mackay

Published Jan 1, 1998• 68 citations
OpenAlex

Abstract

Gallager's low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over for with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10 at dB.

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The class of low-density parity-check (LDPC) codes was introduced by Gallager in his thesis in 1961 [2]. Until the invention of turbo codes, they were largely neglected, but received a lot of interest lately, due to the possibility of designing LDPC codes within 0:0045 dB of the Shannon limit. We review in this project techniques used to analyze the performance of LDPC codes under a message-passing decoding algorithm, namely belief propagation (BP) [4][5]. This analysis gives us the tools to design good LDPC codes. We will flrst take the simpler example of transmission over the class of binary erasure channels (BEC). We will then see how it is possible to extend the results for the BEC to the much more general class of symmetric channels.

•Jan 1, 2005•0 citations
Gallager's low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over for with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10 at dB.
Gallager's (1962) low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over GF(q) for q>2, with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10/sup -5/ at E/sub b//N/sub 0/=0.2 dB.
Binary low density parity check (LDPC) codes have been shown to have near Shannon limit performance when decoded using a probabilistic decoding algorithm. The analogous codes defined over finite fields GF(q) of order q>2 show significantly improved performance. We present the results of Monte Carlo simulations of the decoding of infinite LDPC codes which can be used to obtain good constructions for finite codes. We also present empirical results for the Gaussian channel including a rate 1/4 code with bit error probability of 10/sup -4/ at E/sub b//N/sub 0/=-0.05 dB.
We construct new families of error-correcting codes based on Gallager's (1973) low-density parity-check codes. We improve on Gallager's results by introducing irregular parity-check matrices and a new rigorous analysis of hard-decision decoding of these codes. We also provide efficient methods for finding good irregular structures for such decoding algorithms. Our rigorous analysis based on martingales, our methodology for constructing good irregular codes, and the demonstration that irregular structure improves performance constitute key points of our contribution. We also consider irregular codes under belief propagation. We report the results of experiments testing the efficacy of irregular codes on both binary-symmetric and Gaussian channels. For example, using belief propagation, for rate 1/4 codes on 16000 bits over a binary-symmetric channel, previous low-density parity-check codes can correct up to approximately 16% errors, while our codes correct over 17%. In some cases our results come very close to reported results for turbo codes, suggesting that variations of irregular low density parity-check codes may be able to match or beat turbo code performance.
A generalization of Gallager's low-density parity-check codes is introduced, where as component codes single-error correcting Hamming codes are used instead of single-error detecting parity-check codes. Low-density (LD) parity-check codes were first introduced by Gallager in 1963. These codes are, in combination with iterative decoding, very promising for achieving low error probabilities at a reasonable cost. Results of computer simulations for long LD codes show, that a performance close to the Shannon capacity limit can be achieved. In this work, we consider a generalized version of low-density parity-check codes, where the decoding procedure can be based on the decoding of Hamming component codes.
The class of low-density parity-check (LDPC) codes was introduced by Gallager in his thesis in 1961 [2]. Until the invention of turbo codes, they were largely neglected, but received a lot of interest lately, due to the possibility of designing LDPC codes within 0:0045 dB of the Shannon limit. We review in this project techniques used to analyze the performance of LDPC codes under a message-passing decoding algorithm, namely belief propagation (BP) [4][5]. This analysis gives us the tools to design good LDPC codes. We will flrst take the simpler example of transmission over the class of binary erasure channels (BEC). We will then see how it is possible to extend the results for the BEC to the much more general class of symmetric channels.