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A family of linear codes that are either non-GRS MDS codes or NMDS codes

Yang Li, Zhonghua Sun, Shixin Zhu

TL;DR

The paper addresses constructing linear codes that are either non-GRS MDS codes or NMDS codes not derived from TGRS families. It introduces the $q$-ary class ${\mathcal{C}}_k({\mathcal{S}},{\bf v},\infty)$ and analyzes their parameters via a parity-check framework, subset-sum counts, and the Schur product to establish non-GRS properties and NMDS conditions. It then derives complete weight distributions using subset-sum results, proves that no self-dual codes occur in this class, and provides two explicit almost self-dual constructions. The work yields new infinite families of NMDS and non-GRS MDS codes with explicit weight enumerators, offering alternatives to TGRS-based constructions and potential cryptographic and storage applications.

Abstract

Both maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes (non-GRS MDS codes) and near MDS (NMDS) codes have nice applications in communication and storage systems. In this paper, we introduce and study a new family of linear codes involving their parameters, weight distributions, and self-orthogonal properties. We prove that such codes are either non-GRS MDS codes or NMDS codes, and hence, they can produce as many of the desired codes as possible. We also completely determine their weight distributions with the help of the solutions to some subset sum problems. A sufficient and necessary condition for such codes to be self-orthogonal is characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two new classes of almost self-dual codes.

A family of linear codes that are either non-GRS MDS codes or NMDS codes

TL;DR

The paper addresses constructing linear codes that are either non-GRS MDS codes or NMDS codes not derived from TGRS families. It introduces the -ary class and analyzes their parameters via a parity-check framework, subset-sum counts, and the Schur product to establish non-GRS properties and NMDS conditions. It then derives complete weight distributions using subset-sum results, proves that no self-dual codes occur in this class, and provides two explicit almost self-dual constructions. The work yields new infinite families of NMDS and non-GRS MDS codes with explicit weight enumerators, offering alternatives to TGRS-based constructions and potential cryptographic and storage applications.

Abstract

Both maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes (non-GRS MDS codes) and near MDS (NMDS) codes have nice applications in communication and storage systems. In this paper, we introduce and study a new family of linear codes involving their parameters, weight distributions, and self-orthogonal properties. We prove that such codes are either non-GRS MDS codes or NMDS codes, and hence, they can produce as many of the desired codes as possible. We also completely determine their weight distributions with the help of the solutions to some subset sum problems. A sufficient and necessary condition for such codes to be self-orthogonal is characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two new classes of almost self-dual codes.
Paper Structure (12 sections, 26 theorems, 31 equations, 6 figures, 2 tables)

This paper contains 12 sections, 26 theorems, 31 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notations
  4. MDS codes and NMDS codes
  5. The subset sum problem and the Schur product
  6. Some auxiliary results
  7. The class of linear codes ${\mathcal{C}}_k({\mathcal{S}},{\bf v},\infty)$
  8. Parameters of ${\mathcal{C}}_k({\mathcal{S}},{\bf v},\infty)$
  9. Non-GRS properties of ${\mathcal{C}}_k({\mathcal{S}},{\bf v},\infty)$
  10. Weight distributions of ${\mathcal{C}}_k({\mathcal{S}},{\bf v},\infty)$
  11. Self-orthogonal properties of ${\mathcal{C}}_k({\mathcal{S}},{\bf v},\infty)$
  12. Concluding remarks

Key Result

Lemma 2.1

( B2015) Let ${\mathcal{C}}$ be an $[n,k]_q$ linear code with a generator matrix $G$. Then ${\mathcal{C}}$ is MDS if and only if any $k$ columns of $G$ are linearly independent.

Theorems & Definitions (54)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • Definition 1
  • ...and 44 more