Decay of correlations: An application to low-density parity check codes

Kudekar, Shrinivas; Macris, Nicolas

doi:10.1109/turbocoding.2008.4658665

Cited by 3 publications

(5 citation statements)

References 11 publications

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“…Indeed one can eliminate the degree one nodes and convert the problem to a new graphical model containing a mixture of hard parity check constraints and soft LDGM type weigths. If the density of soft weights is high enough the analysis of the present paper can be extended (see [26] for a summary). Combining theses ideas with duality one may also treat special ensembles of LDGM codes for low noise.…”

Section: Resultsmentioning

confidence: 98%

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Decay of Correlations for Sparse Graph Error Correcting Codes

Kudekar

Macris

2011

SIAM J. Discrete Math.

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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 sub-optimal belief propagation decoder, are investigated. We consider the correlation (or covariance) between two codebits, 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 fixed general low-density generator-matrix codes and high enough noise parameter, and also for fixed general lowdensity parity-check codes and low enough noise parameter. This has many consequences. Appropriate performance curves -called 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.

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Section: Resultsmentioning

confidence: 98%

“…Indeed one can then compute the n → +∞ limit term by term in (26) and then resum the resulting series (which is again absolutely convergent, uniformly with respect to d) to obtain (22).…”

Section: Ldgm Codes: High Noisementioning

confidence: 99%

Decay of Correlations for Sparse Graph Error Correcting Codes

Kudekar

Macris

2011

SIAM J. Discrete Math.

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“…The proof of the replica formulas is, in general, an open problem 1 . In the context of communication they have been proven for a class of low density parity check codes (LDPC) codes on the BEC [11], [12] (see also [13] for recent work going beyond the BEC) and for low density generator codes (LDGM) on a class of channels [14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Sharp Bounds for Optimal Decoding of Low-Density Parity-Check Codes

Kudekar¹,

Macris²

2009

IEEE Trans. Inform. Theory

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Consider communication over a binary-input memoryless outputsymmetric channel with low density parity check (LDPC) codes and maximum a posteriori (MAP) decoding. The replica method of spin glass theory allows to conjecture an analytic formula for the average input-output conditional entropy per bit in the infinite block length limit. Montanari proved a lower bound for this entropy, in the case of LDPC ensembles with convex check degree polynomial, which matches the replica formula. Here we extend this lower bound to any irregular LDPC ensemble. The new feature of our work is an analysis of the second derivative of the conditional input-output entropy with respect to noise. A close relation arises between this second derivative and correlation or mutual information of codebits. This allows us to extend the realm of the "interpolation method", in particular we show how channel symmetry allows to control the fluctuations of the "overlap parameters".

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“…We are interested in the behavior of the correlation between two code bits as a function of their graph distance. In [1] we treated this problem, for the regime of high noise, for a special code ensemble containing a sufficiently large fraction of degree one variable nodes. In the present contribution we attack the problem in the low noise regime.…”

Section: Introductionmentioning

confidence: 99%

“…For example it is easy to see that the standard Dobrushin type methods [2] fail due to the presence of hard constraints. In the high noise regime we were able to convert the problem (at least in the special case of [1]) to a spin glass containing a mixture of soft and hard constaints for which appropriate cluster expansions can be applied. These expansions have been applied to the simpler case of low density generator matrix codes (LDGM) in [3] for the high noise regime, which boils down to a high temperature spin glass.…”

Section: Introductionmentioning

confidence: 99%

Decay of correlations in low density parity check codes: Low noise regime

Kudekar

Macris

2009

2009 IEEE International Symposium on Information Theory

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Abstract-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.

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Decay of correlations: An application to low-density parity check codes

Cited by 3 publications

References 11 publications

Decay of Correlations for Sparse Graph Error Correcting Codes

Decay of Correlations for Sparse Graph Error Correcting Codes

Sharp Bounds for Optimal Decoding of Low-Density Parity-Check Codes

Decay of correlations in low density parity check codes: Low noise regime

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