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New families of non-Reed-Solomon MDS codes

Lingfei Jin, Liming Ma, Chaoping Xing, Haiyan Zhou

TL;DR

A general framework of constructing non-Reed-Solomon (non-RS) type 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 is introduced.

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.

New families of non-Reed-Solomon MDS codes

TL;DR

A general framework of constructing non-Reed-Solomon (non-RS) type 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 zeros in the evaluation set is introduced.

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 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 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.

Paper Structure

This paper contains 14 sections, 20 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Our approaches and results
  3. Organization
  4. Preliminaries
  5. Generalized Reed-Solomon codes
  6. Arithmetic progression sets
  7. Schur products of linear codes
  8. A general framework of non-Reed-Solomon MDS codes
  9. Existence results for non-Reed-Solomon MDS codes
  10. Explicit constructions of non-RS MDS codes
  11. Non-RS MDS codes via weight numerators
  12. Non-RS MDS codes via vector representations
  13. A generalization of explicit non-RS MDS codes
  14. Conclusions

Key Result

Lemma 2.1

Let $I$ be a set of $\ell$ integers. Then $|I+I|\geqslant 2\ell-1$. Moreover, if $I$ is a set of $\ell$ integers with $|I+I|=2\ell-1$, then $I$ is an arithmetic progression.

Theorems & Definitions (51)

  • Remark 1.2
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Definition 1: Schur product
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • ...and 41 more