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arXiv:0911.4522 · IEEE Transactions on Information Theory

On the Number of Errors Correctable with Codes on Graphs

Alexander Barg, Arya Mazumdar

Published Jan 21, 2011• 12 citations
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Abstract

We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum distance of codes in these ensembles grows linearly with the code length. We show that these codes can correct a linearly growing number of errors under simple iterative decoding algorithms. In particular, we show that this property extends to codes constructed by parallel concatenation of Hamming codes and other codes with small minimum distance. Previously known results that proved this property for graph codes relied on graph expansion and required the choice of local codes with large distance relative to their length.

DOI
10.1109/tit.2010.2094812

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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.