Typical Performance of Gallager-type Error-Correcting Codes

TL;DR

This paper analyzes the performance of Gallager-type error-correcting codes using statistical physics methods, showing many codes reach Shannon capacity and exploring decoding via the TAP approach.

Contribution

It applies statistical physics techniques to evaluate Gallager codes and demonstrates capacity saturation for many code variants.

Findings

Many codes saturate Shannon capacity

Decoding performance is analyzed using the TAP approach

Some codes have higher practical relevance

Abstract

The performance of Gallager's error-correcting code is investigated via methods of statistical physics. In this approach, the transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse matrices; the number of non-zero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity is saturated for many of the codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the TAP approach which is identical to the commonly used belief-propagation-based decoding.