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More MDS codes of non-Reed-Solomon type

Yansheng Wu, Ziling Heng, Chengju Li, Cunsheng Ding

TL;DR

The paper addresses the problem of identifying when the extended Roth–Lempel type code $\mathcal C_2$ is MDS, AMDS, or NMDS, and whether it is non-Reed-Solomon. It represents $\mathcal C_2$ as $\overline{RL({\boldsymbol a},\delta,k,n+2)}({\bf u})$ and derives exact MDS/NMDS/AMDS criteria via subset-sum and Vandermonde-determinant techniques, complemented by extended-code theory for duals and radii. Key contributions include explicit MDS conditions for $\mathcal C_2$, dual-AMDS/NMDS characterizations, an explicit parity-check matrix, covering-radius results for Roth–Lempel duals, and infinite families of (almost) optimally extendable dimension-3 codes, along with concrete examples. These results advance the understanding of non-Reed-Solomon MDS-type codes, with implications for code extension, dual-distance analysis, and combinatorial constructions.

Abstract

MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and establish some sufficient and necessary conditions for them being MDS. Notably, these codes differ from Reed-Solomon codes up to monomial equivalence. Additionally, we also explore the cases in which these codes are almost MDS or near MDS. Applying our main results, we determine the covering radii and deep holes of the dual codes associated with specific Roth-Lempel codes and discover an infinite family of (almost) optimally extendable codes with dimension three.

More MDS codes of non-Reed-Solomon type

TL;DR

The paper addresses the problem of identifying when the extended Roth–Lempel type code is MDS, AMDS, or NMDS, and whether it is non-Reed-Solomon. It represents as and derives exact MDS/NMDS/AMDS criteria via subset-sum and Vandermonde-determinant techniques, complemented by extended-code theory for duals and radii. Key contributions include explicit MDS conditions for , dual-AMDS/NMDS characterizations, an explicit parity-check matrix, covering-radius results for Roth–Lempel duals, and infinite families of (almost) optimally extendable dimension-3 codes, along with concrete examples. These results advance the understanding of non-Reed-Solomon MDS-type codes, with implications for code extension, dual-distance analysis, and combinatorial constructions.

Abstract

MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and establish some sufficient and necessary conditions for them being MDS. Notably, these codes differ from Reed-Solomon codes up to monomial equivalence. Additionally, we also explore the cases in which these codes are almost MDS or near MDS. Applying our main results, we determine the covering radii and deep holes of the dual codes associated with specific Roth-Lempel codes and discover an infinite family of (almost) optimally extendable codes with dimension three.
Paper Structure (15 sections, 13 theorems, 38 equations)

This paper contains 15 sections, 13 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two known constructions of MDS codes
  4. One class of special polynomials and generalized Vandermonde matrices
  5. A kind of extended codes of linear codes
  6. Extended codes of Roth-Lempel codes
  7. When is the code $\mathcal{C}_2$ MDS?
  8. When is the code $\mathcal{C}_2$ NMDS?
  9. When is the dual code ${\mathcal{C}}_2^{\bot}$ AMDS?
  10. Parity-check matrix of $\mathcal{C}_2$
  11. When is the code ${\mathcal{C}}_2^{}$ AMDS?
  12. Applications of the main results
  13. Covering radii of the dual of the Roth-Lempel codes
  14. (Almost) Optimal extendable linear codes
  15. Summary and concluding remarks

Key Result

Lemma 1

RL The Roth-Lempel code $RL({\boldsymbol{ \alpha}}, \delta, k,n+2)$ is MDS if and only if the set $\{\alpha_1, \ldots, \alpha_{n}\}$ is an $(n, k-1, \delta)$-set in ${\mathrm{GF}}(q)$, where a set $S\subseteq {\mathrm{GF}}(q)$ of size $n$ is called an $(n,k-1, \delta)$-set in ${\mathrm{GF}}(q)$ if

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 3
  • Lemma 5
  • Theorem 6
  • proof
  • ...and 19 more