Polymer Expansions for Cycle LDPC Codes

TL;DR

This paper proves the exactness of the Bethe entropy expression for cycle LDPC codes on binary symmetric channels above the MAP threshold in the large block length limit, using polymer expansions and statistical physics methods.

Contribution

It introduces a polymer expansion approach to analyze finite size corrections for cycle LDPC codes, establishing the Bethe expression's exactness.

Findings

Bethe expression is exact above the MAP threshold in the large block limit

Finite size corrections are characterized by a polymer expansion

Methods from statistical physics are effectively applied to coding theory

Abstract

We prove that the Bethe expression for the conditional input-output entropy of cycle LDPC codes on binary symmetric channels above the MAP threshold is exact in the large block length limit. The analysis relies on methods from statistical physics. The finite size corrections to the Bethe expression are expressed through a polymer expansion which is controlled thanks to expander and counting arguments.