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IEEE International Symposium on Information Theory (ISIT)

Sharp Bounds for MAP Decoding of General Irregular LDPC Codes

Shrinivas Kudekar, Nicolas Macris

Published Jul 1, 2006• 14 citations
Publisher

Abstract

Consider communication over a binary input memoryless output symmetric channel with LDPC codes and MAP decoding. Recently Montanari proved that the replica solution is a lower bound to the conditional entropy for a class of LDPC ensembles. Here we extend this lower bound to any irregular LDPC ensemble for the BEC, BIAWGNC, BSC. Our work combines an analysis of the second derivative of the conditional entropy with respect to the noise and the interpolation method

DOI
10.1109/isit.2006.261969

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In the past decade, there has been tremendous amount of research on the analysis and design of the Low-Density Parity-Check (LDPC) codes with belief propagation decoding over different types of Binary-Input Output-Symmetric Memoryless (BIOSM) channels. However, with the exception of the Binary Erasure Channel (BEC), analytical results on LDPC codes over such channels are limited and most results are based on numerical methods and optimization. In particular, systematic design of provably capacity achieving sequences of LDPC code ensembles over general class of BIOSM channels has remained a fundamental open problem. Such sequences have been designed only for the binary erasure channel (BEC). In this paper, we first prove some novel analytical properties of the LDPC code ensembles over BIOSM channels. In particular, we prove a result that generalizes the previously known stability condition over symmetric channels. This suggests that a modified version of the flatness condition, a property which has been shown to be critical for capacity achieving sequences over the BEC, can be used to devise capacity achieving sequences for general class of BIOSM channels. Based on this assumption, we propose a method which could result in the systematic design of such sequences over BIOSM channels. Numerical evidence is promising and provides a consistent convergence behavior to capacity over the considered BIOSM channels as the average check node degree increases.

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Tight Bounds for LDPC and LDGM Codes Under MAP Decoding

Andrea Montanari A

A new method for analyzing low-density parity-check (LDPC) codes and low-density generator-matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows one to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The result holds for standard irregular ensembles when used over binary-input output-symmetric (BIOS) channels. The method is first developed for Tanner-graph ensembles with Poisson left-degree distribution. It is then generalized to "multi-Poisson" graphs, and, by a completion procedure, to arbitrary degree distribution

arXiv:cs/0407060•IEEE Transactions on Information Theory•Aug 24, 2005•105 citations
Consider communication over a binary input memoryless output symmetric channel with LDPC codes and MAP decoding. Recently Montanari proved that the replica solution is a lower bound to the conditional entropy for a class of LDPC ensembles. Here we extend this lower bound to any irregular LDPC ensemble for the BEC, BIAWGNC, BSC. Our work combines an analysis of the second derivative of the conditional entropy with respect to the noise and the interpolation method
Consider communication over a binary-input memoryless output-symmetric channel with low-density parity-check (LDPC) codes and maximumaposteriori(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 ldquointerpolation method,rdquo in particular, we show how channel symmetry allows to control the fluctuations of the ldquooverlap parametersrdquo.
An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly \emph{punctured} output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded \emph{complexity}. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described.
An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly punctured output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded complexity. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described.
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.
In the past decade, there has been tremendous amount of research on the analysis and design of the Low-Density Parity-Check (LDPC) codes with belief propagation decoding over different types of Binary-Input Output-Symmetric Memoryless (BIOSM) channels. However, with the exception of the Binary Erasure Channel (BEC), analytical results on LDPC codes over such channels are limited and most results are based on numerical methods and optimization. In particular, systematic design of provably capacity achieving sequences of LDPC code ensembles over general class of BIOSM channels has remained a fundamental open problem. Such sequences have been designed only for the binary erasure channel (BEC). In this paper, we first prove some novel analytical properties of the LDPC code ensembles over BIOSM channels. In particular, we prove a result that generalizes the previously known stability condition over symmetric channels. This suggests that a modified version of the flatness condition, a property which has been shown to be critical for capacity achieving sequences over the BEC, can be used to devise capacity achieving sequences for general class of BIOSM channels. Based on this assumption, we propose a method which could result in the systematic design of such sequences over BIOSM channels. Numerical evidence is promising and provides a consistent convergence behavior to capacity over the considered BIOSM channels as the average check node degree increases.
A new method for analyzing low-density parity-check (LDPC) codes and low-density generator-matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows one to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The result holds for standard irregular ensembles when used over binary-input output-symmetric (BIOS) channels. The method is first developed for Tanner-graph ensembles with Poisson left-degree distribution. It is then generalized to "multi-Poisson" graphs, and, by a completion procedure, to arbitrary degree distribution