On the Number of Errors Correctable with Codes on Graphs

TL;DR

This paper demonstrates that ensembles of graph-based codes, including LDPC and hypergraph codes, can correct a linearly growing number of errors using simple iterative decoding, extending previous results to codes with small minimum distance.

Contribution

It shows that codes constructed from random graphs and fixed local codes can correct many errors with simple decoding, even when local codes have small minimum distance.

Findings

Codes on graphs can correct errors linearly proportional to code length.

This property extends to codes formed by concatenating Hamming and similar codes.

Previous results required local codes with large minimum distance.

Abstract

We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum distance of codes in these ensembles grows linearly with the code length. We show that these codes can correct a linearly growing number of errors under simple iterative decoding algorithms. In particular, we show that this property extends to codes constructed by parallel concatenation of Hamming codes and other codes with small minimum distance. Previously known results that proved this property for graph codes relied on graph expansion and required the choice of local codes with large distance relative to their length.