Gallager error correcting codes for binary asymmetric channels

TL;DR

This paper analyzes Gallager error-correcting codes for binary asymmetric channels using statistical mechanics, identifying critical noise levels, decoding regimes, and the relation between algorithm convergence and solution complexity.

Contribution

It introduces a novel analysis of Gallager codes on asymmetric channels, linking decoding performance with statistical mechanics concepts.

Findings

Critical noise levels depend on input bias and temperature.

Decoding regimes are characterized by entropy and codeword space structure.

Convergence of message passing relates to solution complexity and endogeny.

Abstract

We derive critical noise levels for Gallager codes on asymmetric channels as a function of the input bias and the temperature. Using a statistical mechanics approach we study the space of codewords and the entropy in the various decoding regimes. We further discuss the relation of the convergence of the message passing algorithm with the endogeny property and complexity, characterizing solutions of recursive equations of distributions for cavity fields.