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IEEE International Symposium on Information Theory (ISIT)

Evaluating zero error noise thresholds by the replica method for Gallager code ensembles

Yoshiyuki Kabashima, Naoya Sazuka A, K. Nakamura, David Saad

Published Jun 25, 2003• 4 citations
Publisher

Abstract

The replica method, developed in statistical physics, is employed in conjunction with Gallager's methodology to accurately evaluate zero error noise thresholds for Gallager code ensembles. Our approach generally provides more optimistic evaluations than those reported in the information theory literature for sparse matrices; the difference vanishes as the parity check matrix becomes dense.

DOI
10.1109/isit.2002.1023527

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The replica method, developed in statistical physics, is employed in conjunction with Gallager's methodology to accurately evaluate zero error noise thresholds for Gallager code ensembles. Our approach generally provides more optimistic evaluations than those reported in the information theory literature for sparse matrices; the difference vanishes as the parity check matrix becomes dense.
Statistical physics is employed to evaluate the performance of error-correcting codes in the case of finite message length for an ensemble of Gallager's error correcting codes. We follow Gallager's approach of upper bounding the average decoding error rate, but invoke the replica method to reproduce the tightest general bound to date, and to improve on the most accurate zero-error noise level threshold reported in the literature. The relation between the methods used and those presented in the information theory literature are explored.
A variation of Gallager error-correcting codes is investigated using statistical mechanics. In codes of this type, a given message is encoded into a codeword that comprises Boolean sums of message bits selected by two randomly constructed sparse matrices. The similarity of these codes to Ising spin systems with random interaction makes it possible to assess their typical performance by analytical methods developed in the study of disordered systems. The typical case solutions obtained via the replica method are consistent with those obtained in simulations using belief propagation decoding. We discuss the practical implications of the results obtained and suggest a computationally efficient construction for one of the more practical configurations.
We review recent theoretical progress on the statistical mechanics of error correcting codes, focusing on low-density parity-check (LDPC) codes in general, and on Gallager and MacKay–Neal codes in particular. By exploiting the relation between LDPC codes and Ising spin systems with multi-spin interactions, one can carry out a statistical mechanics based analysis that determines the practical and theoretical limitations of various code constructions, corresponding to dynamical and thermodynamical transitions, respectively, as well as the behaviour of error-exponents averaged over the corresponding code ensemble as a function of channel noise. We also contrast the results obtained using methods of statistical mechanics with those derived in the information theory literature, and show how these methods can be generalized to include other channel types and related communication problems.