Critical Noise Levels for LDPC decoding

TL;DR

This paper introduces a novel method for determining the critical noise level in LDPC decoding using the magnetization enumerator, clarifying decoding scheme relationships and explaining performance differences.

Contribution

It presents a new approach based on the magnetization enumerator for analyzing LDPC decoding, offering clearer insights and more optimistic results than traditional information theory methods.

Findings

The magnetization enumerator method aligns well with recent physics-based results.

It explains performance differences between MN and Gallager codes.

Provides a more optimistic estimate of critical noise levels.

Abstract

We determine the critical noise level for decoding low density parity check error correcting codes based on the magnetization enumerator ($\cM$), rather than on the weight enumerator ($\cW$) employed in the information theory literature. The interpretation of our method is appealingly simple, and the relation between the different decoding schemes such as typical pairs decoding, MAP, and finite temperature decoding (MPM) becomes clear. In addition, our analysis provides an explanation for the difference in performance between MN and Gallager codes. Our results are more optimistic than those derived via the methods of information theory and are in excellent agreement with recent results from another statistical physics approach.