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A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$

Hengfeng Liu, Chunming Tang, Zhengchun Zhou, Dongchun Han, Hao Chen

Abstract

A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial $t$-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for $t > 2$, all with length at most $q + 1$. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater than $q + 1$ are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the first \emph{generic construction} of $q$-ary NMDS codes supporting 2-designs with lengths \emph{exceeding $q + 1$}. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$

Abstract

A linear code with parameters is called maximum distance separable (MDS), and one with parameters is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial -designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for , all with length at most . Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater than are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the first \emph{generic construction} of -ary NMDS codes supporting 2-designs with lengths \emph{exceeding }. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

Paper Structure

This paper contains 8 sections, 29 theorems, 57 equations, 2 tables.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. The Assmus-Mattson Theorem and its generalized version
  4. NMDS codes and related $t$-designs
  5. Subset sums in finite abelian groups and derived $t$-designs
  6. NMDS elliptic curve codes supporting $2$-designs
  7. The weight distribution of the code
  8. Summary and Concluding Remarks

Key Result

Theorem 1

Let $\mathcal{C}$ be an $[n,k,d]$ linear code over $\mathbb{F}_{q}$. Let $t$ be an integer satisfying $1\leqslant t<\min\left\{d, d^{\perp}\right\}$. Assume that there are at most $d^{\perp}-t$ weights of $\mathcal{C}$ in $\{1,2,\ldots, n-t\}$. Then $\left(\mathcal{P}(\mathcal{C}),\mathcal{H}_{w}(\m

Theorems & Definitions (31)

  • Theorem 1: General version of Assmus-Mattson Theorem
  • Lemma 1
  • Theorem 2: Assmus-Mattson Theorem
  • Theorem 3: Generalized Assmus-Mattson theorem
  • Corollary 1
  • Theorem 4: Dodunekov1995
  • Theorem 5: Dodunekov1995
  • Theorem 6: Dodunekov1995
  • Corollary 2
  • Theorem 7: Faldum 1997
  • ...and 21 more