New Set of Codes for the Maximum-Likelihood Decoding Problem

TL;DR

This paper introduces A-covered codes, a new class of codes that can be decoded efficiently using polynomial-time algorithms, including some binary BCH codes where maximum-likelihood decoding becomes feasible.

Contribution

The paper defines A-covered codes and demonstrates their applicability to binary BCH codes, enabling polynomial-time maximum-likelihood decoding for these codes.

Findings

Identification of A-covered codes with polynomial-time decoding

Examples of binary BCH codes that are A-covered

Maximum-likelihood decoding for some codes achieved in quasi-quadratic time

Abstract

The maximum-likelihood decoding problem is known to be NP-hard for general linear and Reed-Solomon codes. In this paper, we introduce the notion of A-covered codes, that is, codes that can be decoded through a polynomial time algorithm A whose decoding bound is beyond the covering radius. For these codes, we show that the maximum-likelihood decoding problem is reachable in polynomial time in the code parameters. Focusing on bi- nary BCH codes, we were able to find several examples of A-covered codes, including two codes for which the maximum-likelihood decoding problem can be solved in quasi-quadratic time.