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IEEE Transactions on Information Theory

Griffith–Kelly–Sherman Correlation Inequalities: A Useful Tool in the Theory of Error Correcting Codes

Nicolas Macris

Published Jan 23, 2007• 49 citations
Publisher

Abstract

It is shown that a correlation inequality of statistical mechanics can be applied to linear low-density parity-check codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular low-density parity-check (LDPC) codes, can be computed exactly by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. Then, considering communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code we prove that, under a natural assumption, part of the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation algorithm. The correlation inequality yields a sharp lower bound on the GEXIT curve. We also make an extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem

DOI
10.1109/tit.2006.889002

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•Jan 1, 2006•0 citations
It is shown that a correlation inequality of statistical mechanics can be applied to linear low-density parity-check codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular low-density parity-check (LDPC) codes, can be computed exactly by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. Then, considering communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code we prove that, under a natural assumption, part of the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation algorithm. The correlation inequality yields a sharp lower bound on the GEXIT curve. We also make an extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem
It is shown that a correlation inequality of statistical mechanics can be applied to low-density parity-check codes. Thanks to this tool we prove that the growth rate of regular LDPC codes, can be exactly calculated by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. We also consider communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code and prove that (at least part of) the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation decoder. In both problems, the correlation inequality yields sharp lower bounds. We also use a non trivial extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem
The subject of this paper is transmission over a general class of binary-input memoryless symmetric channels using error correcting codes based on sparse graphs, namely, low-density generator-matrix and low-density parity-check codes. The optimal (or ideal) decoder based on the posterior measure over the code-bits and its relationship to the suboptimal belief propagation decoder are investigated. We consider the correlation (or covariance) between two code-bits, averaged over the noise realizations, as a function of the graph distance for the optimal decoder. Our main result is that this correlation decays exponentially fast for given low-density generator-matrix codes and a high enough noise parameter and also for given low-density parity-check codes and a low enough noise parameter. This has many consequences. Appropriate performance curves—called generalized extrinsic information transfer (GEXIT) functions—of the belief propagation and optimal decoders match in high/low noise regimes. This means that in high/low noise regimes the performance curves of the optimal decoder can be computed by density evolution. Another interpretation is that the replica predictions of spin-glass theory are exact. Our methods are rather general and use cluster expansions first developed in the context of mathematical statistical mechanics.
Recently the decay of correlations between bits of low density generator matrix (LDGM) codes have been investigated by using high temperature expansions from statistical physics. In this work we apply these ideas to a special class of low density parity check codes (LDPC) on the binary input Gaussian white noise channel (BIAWGNC). We give a rigorous derivation of the maximum a posteriori (MAP) Generalized EXIT curve (the derivative with respect to the noise parameter of the input-output conditional entropy) for high values of the noise. Our result agrees with the formal expressions obtainable from replica calculations, and is the first result that fully justifies the replica formulas beyond the binary erasure channel (BEC). It also shows that the MAP and BP-GEXIT curves are equal in the high noise regime. The ensemble of LDPC codes considered here is constructed by adding randomly a sufficient fraction p of degree one variable nodes to a standard irregular LDPC(Lambda, P) Tanner graphs.
Consider transmission over a binary additive white gaussian noise channel using a fixed low-density parity check code. We consider the posterior measure over the code bits and the corresponding correlation between two codebits, averaged over the noise realizations. We show that for low enough noise variance this average correlation decays exponentially fast with the graph distance between the code bits. One consequence of this result is that for low enough noise variance the GEXIT functions (further averaged over a standard code ensemble) of the belief propagation and optimal decoders are the same.
We consider communication over binary-input memoryless symmetric channels with low-density parity-check (LDPC) codes. The relationship between maximum a posteriori and belief propagation decoding is investigated using a set of correlation inequalities that first appeared in statistical mechanics of Gaussian spin glasses. We prove bounds on generalized extrinsic information transfer (EXIT) functions, that are believed to be tight, and discuss their relationship with the ones obtained by the interpolation method.
Abstract.: This contribution is based on the contents of a talk delivered at the Next-SigmaPhi conference held in Crete in August 2005. It is adressed to an audience of physicists with diverse horizons and does not assume any background in communications theory. Capacity approaching error correcting codes for channel communication known as Low Density Parity Check (LDPC) codes have attracted considerable attention from coding theorists in the last decade. Surprisingly strong connections with the theory of diluted spin glasses have been discovered. In this work we elucidate one new connection, namely that a class of correlation inequalities valid for Gaussian spin glasses can be applied to the theoretical analysis of LDPC codes. This allows for a rigorous comparison between the so called (optimal) maximum a posteriori and the computationaly efficient belief propagation decoders. The main ideas of the proofs are explained and we refer to recent works for the more lengthy technical details