Two Theorems in List Decoding

TL;DR

This paper proves new theorems in list decoding, showing that random errors can be corrected beyond worst-case limits and that concatenated codes can achieve list decoding capacity for erasures, using simple probabilistic methods.

Contribution

It introduces two theorems: one demonstrating improved error correction for random errors in any code, and another showing concatenated codes reach list decoding capacity for erasures.

Findings

Random errors can be corrected beyond worst-case limits.

Concatenated codes achieve list decoding capacity for erasures.

A simple decoding algorithm for Reed-Solomon codes from random errors.

Abstract

We prove the following results concerning the list decoding of error-correcting codes: (i) We show that for \textit{any} code with a relative distance of $\delta$ (over a large enough alphabet), the following result holds for \textit{random errors}: With high probability, for a $\rho\le \delta -\eps$ fraction of random errors (for any $\eps>0$), the received word will have only the transmitted codeword in a Hamming ball of radius $\rho$ around it. Thus, for random errors, one can correct twice the number of errors uniquely correctable from worst-case errors for any code. A variant of our result also gives a simple algorithm to decode Reed-Solomon codes from random errors that, to the best of our knowledge, runs faster than known algorithms for certain ranges of parameters. (ii) We show that concatenated codes can achieve the list decoding capacity for erasures. A similar result for worst-case errors was proven by Guruswami and Rudra (SODA 08), although their result does not directly imply our result. Our results show that a subset of the random ensemble of codes considered by Guruswami and Rudra also achieve the list decoding capacity for erasures. Our proofs employ simple counting and probabilistic arguments.