New families of non-Reed-Solomon MDS codes

TL;DR

This paper introduces a general framework for constructing new families of non-Reed-Solomon MDS codes with flexible lengths, expanding the known options and addressing a significant open problem in coding theory.

Contribution

The paper presents a novel, general construction method for non-Reed-Solomon MDS codes, including explicit constructions and combinatorial approaches, broadening the scope of available codes.

Findings

New families of non-RS MDS codes with flexible lengths

Construction method based on evaluation polynomials and points

Most new codes are not covered by existing results

Abstract

MDS codes have garnered significant attention due to their wide applications in practice. To date, most known MDS codes are equivalent to Reed-Solomon codes. The construction of non-Reed-Solomon (non-RS) type MDS codes has emerged as an intriguing and important problem in both coding theory and finite geometry. Although some constructions of non-RS type MDS codes have been presented in the literature, the parameters of these MDS codes remain subject to strict constraints. In this paper, we introduce a general framework of constructing $[n,k]$ MDS codes using the idea of selecting a suitable set of evaluation polynomials and a set of evaluation points such that all nonzero polynomials have at most $k-1$ zeros in the evaluation set. Moreover, these MDS codes can be proved to be non-Reed-Solomon by computing their Schur squares. Furthermore, several explicit constructions of non-RS MDS codes are given by converting to combinatorial problems. As a result, new families of non-RS MDS codes with much more flexible lengths can be obtained and most of them are not covered by the known results.