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IEEE International Symposium on Information Theory (ISIT)

Upper Bounds on the Error Exponents of LDPC Code Ensembles

David Burshtein, Ohad Barak

Published Jul 1, 2006• 5 citations
Publisher

Abstract

We consider the ensemble of regular LDPC codes and use recent concentration results on the distance spectrum to derive upper bounds on the error exponent of a randomly chosen code from the ensemble. These bounds hold with some confidence level that approaches one as the connectivity of the graph increases. We show that the bounds can be used to obtain the true error exponent over some range of channel parameter values, with the above confidence level

DOI
10.1109/isit.2006.261699

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We consider the ensemble of regular LDPC codes and use recent concentration results on the distance spectrum to derive upper bounds on the error exponent of a randomly chosen code from the ensemble. These bounds hold with some confidence level that approaches one as the connectivity of the graph increases. We show that the bounds can be used to obtain the true error exponent over some range of channel parameter values, with the above confidence level
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We consider the ensemble of regular LDPC codes and obtain an expression for the second moment of the distance spectrum. We show how this expression can be used to derive a lower bound on the probability that the growth rate of a randomly chosen code from the ensemble is equal to the growth rate of the average distance spectrum, when the block length is sufficiently large. In particular, when the connectivity of the code is sufficiently large, the distance spectrum of a code in the ensemble is concentrated. We then derive a lower bound on the probability (confidence level) that the minimum distance and error rate, respectively, of a randomly chosen code from the ensemble are upper and lower bounded by some values (which depend on the confidence level)
We derive bounds on the error probability of ML decoded LDPC (low density parity check) codes, for any binary-input symmetric-output channel. For appropriately chosen ensembles of LDPC codes, reliable communication is possible up to channel capacity. The lower and upper bounds coincide asymptotically, indicating a polynomially decreasing ensemble averaged error probability. For ensembles with suitably chosen parameters, the error probability of almost all codes is exponentially decreasing. Furthermore, the error exponent can be set arbitrarily close to the standard random coding exponent.