Applications of correlation inequalities to low density graphical codes

TL;DR

This paper explores how correlation inequalities from spin glass theory can be applied to analyze Low Density Parity Check (LDPC) codes, enabling rigorous comparison of decoding methods.

Contribution

It introduces a novel connection between correlation inequalities in spin glasses and the analysis of LDPC codes, providing new theoretical insights.

Findings

Correlation inequalities help compare MAP and belief propagation decoders.

Theoretical analysis of LDPC codes is enhanced through spin glass methods.

Rigorous bounds are established for decoding performance.

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.