LDPC Codes Achieve List Decoding Capacity

TL;DR

This paper demonstrates that Gallager's LDPC codes can achieve list-decoding capacity with high probability, establishing a new class of graph-based codes with optimal list-decoding performance and potential for linear-time decoding.

Contribution

It proves that LDPC codes from Gallager's ensemble achieve list-decoding capacity, a first for graph-based codes, and introduces a general framework linking local properties to random linear codes.

Findings

LDPC codes achieve list-decoding capacity with high probability.

Sharp thresholds are established for local properties in random linear codes.

A characterization of the threshold rate for local properties is provided.

Abstract

We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieves list-decoding capacity with high probability. These are the first graph-based codes shown to have this property. This result opens up a potential avenue towards truly linear-time list-decodable codes that achieve list-decoding capacity. Our result on list decoding follows from a much more general result: any $\textit{local}$ property satisfied with high probability by a random linear code is also satisfied with high probability by a random LDPC code from Gallager's distribution. Local properties are properties characterized by the exclusion of small sets of codewords, and include list-decodability, list-recoverability and average-radius list-decodability. In order to prove our results on LDPC codes, we establish sharp thresholds for when local properties are satisfied by a random linear code. More precisely, we show that for any local property $\mathcal{P}$, there is some $R^*$ so that random linear codes of rate slightly less than $R^*$ satisfy $\mathcal{P}$ with high probability, while random linear codes of rate slightly more than $R^*$, with high probability, do not. We also give a characterization of the threshold rate $R^*$.