Tight bounds for LDPC and LDGM codes under MAP decoding

TL;DR

This paper introduces a rigorous method based on spin glass theory to derive tight lower bounds on the entropy of LDPC and LDGM codes under MAP decoding, applicable to various graph ensembles and channels.

Contribution

It develops a new analytical approach using spin glass techniques to establish bounds on code performance under MAP decoding, extending to general degree distributions.

Findings

Method provides lower bounds on message entropy under MAP decoding.

Bounds are conjectured to be tight based on statistical mechanics heuristics.

Applicable to irregular ensembles over symmetric channels.

Abstract

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 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 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.