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IEEE Transactions on Information Theory

On the Error Correction of Regular LDPC Codes Using the Flipping Algorithm

David Burshtein

Published Jan 29, 2008• 43 citations
Publisher

Abstract

The iterative bit flipping algorithm is applied to the standard regular low-density parity-check (LDPC) code ensemble. In the past, it was shown, for a typical code in the ensemble with left degree at least five and block length sufficiently large, that this algorithm can correct a linear (in the block length) number of worst case errors. In this paper, this result is extended to the case where the left degree is at least four. For the case where the left degree is larger than four, an improvement, compared to existing results, of several orders of magnitude is obtained on the fraction of worst case errors that can be corrected. It is also shown how the results can be further improved when random errors produced by the channel (as opposed to worst case errors) are considered.

DOI
10.1109/tit.2007.913261

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The iterative bit flipping algorithm is applied to the standard regular low-density parity-check (LDPC) code ensemble. In the past, it was shown, for a typical code in the ensemble with left degree at least five and block length sufficiently large, that this algorithm can correct a linear (in the block length) number of worst case errors. In this paper, this result is extended to the case where the left degree is at least four. For the case where the left degree is larger than four, an improvement, compared to existing results, of several orders of magnitude is obtained on the fraction of worst case errors that can be corrected. It is also shown how the results can be further improved when random errors produced by the channel (as opposed to worst case errors) are considered.
We apply the iterative bit flipping algorithm to the standard regular low-density parity-check (LDPC) code ensemble. In the past it was shown, for a typical code in the ensemble with left degree at least five and block length sufficiently large, that this algorithm can correct all error patterns with some linear (in the block length) number of errors. We extend this result to the case where the left degree is at least four. For the case where the left degree is larger than four, we obtain an improvement of several orders of magnitude compared to the existing results on the fraction of worst case errors that can be corrected. We also show how our results can be improved when we consider random errors (as opposed to worst case errors) produced by the channel.
Consider an ensemble of regular generalized LDPC (GLDPC) codes and assume that the same component code is associated with each parity check node. To decode a GLDPC code from the ensemble, we use the bit flipping bounded distance decoding algorithm, which is an extension of the bit flipping algorithm for LDPC codes. Previous work has shown conditions, under which, for a typical code in the ensemble with blocklength sufficiently large, a positive constant fraction of worst case errors can be corrected. In this work we first show that these requirements can be relaxed for ensembles with small left degrees. While previous work on GLDPC codes has considered expander graph arguments, our analysis formulates a necessary condition that the Tanner graph needs to satisfy for a failure event and then shows that the probability of this event vanishes for a sufficiently large blocklength. We then extend the analysis to random error correction and derive a lower bound on the fraction of random errors that can be corrected asymptotically. We discuss the extension of our results to non-binary GLDPC codes and present numerical examples.
The discovery in 1993 of the turbo codes Berrou C., A. and P. Glavieux Thitimajshima at the International Conférence of Communication Florida has revolutionized communication. This innovation made it possible to rediscovery the error correcting codes invented by Robert Gallager in 1963 and rediscovered by Mackay in 1995, currently called Low Density Parity Check (LDPC). This code is one of the error correction codes introduced by the transmission channel that have the fastest and most effective level of protection and correction of the information transmitted during the communication. This code has been almost generalized in communication standards such as satellites. The objective of our article is to minimize the number of iterations and in some cases completely remove the iteration produced by the LDPC code bit flipping algorithm. The result of this algorithm, if we take for example the LDPC code (n, k), the number of possible syndromes which we can find is 2k-1 and for each syndrome different from zero, we can do almost four iterations to correct the errors, which gives 4x(2k-1) iterations in totality. On the other hand, the proposed algorithm removes the iteration for k syndromes and reduces iterations for the rest of the cases from four to two iterations. This study of the new design gave more performance results and the results of the simulation below according to a hardware description language used in digital electronics using quartus software tools show the importance of this algorithm.
In this paper we analyze the effect of hardware unreliability to performance of bit-flipping decoders of low-density parity-check (LDPC) codes. We apply expander arguments to show that the simple parallel bit flipping decoder, built partially from faulty gates, can correct a linear fraction of worst case channel errors, when gate failures are correlated and dependent on the switching activity of logic gates. In addition, we provide a lower bound on the guaranteed error correction of LDPC codes with left degree of at least eight.
The relation between the girth and the guaranteed error correction capability of¿-left-regular low-density parity-check (LDPC) codes when decoded using the bit flipping (serial and parallel) algorithms is investigated. A lower bound on the size of variable node sets which expand by a factor of at least3 ¿/4is found based on the Moore bound. This bound, combined with the well known expander based arguments, leads to a lower bound on the guaranteed error correction capability. The decoding failures of the bit flipping algorithms are characterized using the notions of trapping sets and fixed sets. The relation between fixed sets and a class of graphs known as cage graphs is studied. Upper bounds on the guaranteed error correction capability are then established based on the order of cage graphs. The results are extended to left-regular and right-uniform generalized LDPC codes. It is shown that this class of generalized LDPC codes can correct a linear number of worst case errors (in the code length) under the parallel bit flipping algorithm when the underlying Tanner graph is a good expander. A lower bound on the size of variable node sets which have the required expansion is established.