Finite-connectivity systems as error-correcting codes

TL;DR

This paper explores the performance of finite-connectivity parity check codes modeled as Ising spin systems, demonstrating their ability to approach Shannon's limit and analyzing decoding dynamics and improvements.

Contribution

It introduces a statistical physics approach to analyze finite-connectivity codes, extending understanding of decoding methods and performance near Shannon's bound.

Findings

Codes saturate Shannon's bound as K approaches infinity

Simulated annealing can effectively decode these codes

Analysis extends to noisy channels and decoding dynamics

Abstract

We investigate the performance of parity check codes using the mapping onto Ising spin systems proposed by Sourlas. We study codes where each parity check comprises products of K bits selected from the original digital message with exactly C checks per message bit. We show, using the replica method, that these codes saturate Shannon's coding bound for $K\to\infty$ when the code rate K/C is finite. We then examine the finite temperature case to asses the use of simulated annealing methods for decoding, study the performance of the finite K case and extend the analysis to accommodate different types of noisy channels. The connection between statistical physics and belief propagation decoders is discussed and the dynamics of the decoding itself is analyzed. Further insight into new approaches for improving the code performance is given.