Efficient Covering Using Reed--Solomon Codes

TL;DR

This paper introduces a novel algorithm for efficiently finding Reed-Solomon codewords near any point in the Hamming space, advancing the understanding of their covering properties and decoding capabilities.

Contribution

It presents the first algorithm for the covering problem in Reed-Solomon codes, utilizing existing decoding methods to analyze their covering radius.

Findings

The algorithm effectively finds codewords within the covering radius.

Numerical results show most of the space is covered by spheres around codewords.

The average covering radius approaches the Guruswami-Sudan decoding radius.

Abstract

We propose an efficient algorithm to find a Reed-Solomon (RS) codeword at a distance within the covering radius of the code from any point in its ambient Hamming space. To the best of the authors' knowledge, this is the first attempt of its kind to solve the covering problem for RS codes. The proposed algorithm leverages off-the-shelf decoding methods for RS codes, including the Berlekamp-Welch algorithm for unique decoding and the Guruswami-Sudan algorithm for list decoding. We also present theoretical and numerical results on the capabilities of the proposed algorithm and, in particular, the average covering radius resulting from it. Our numerical results suggest that the overlapping Hamming spheres of radius close to the Guruswami-Sudan decoding radius centered at the codewords cover most of the ambient Hamming space.