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Several new infinite families of NMDS codes with arbitrary dimensions supporting $t$-designs

Yaozong Zhang, Dabin Zheng, Xiaoqiang Wang, Wei Lu

TL;DR

This paper advances NMDS codes by constructing two infinite families over $\mathbb{F}_q$ with lengths $q+1$ and $q+2$ and arbitrary dimension, plus two families over $\mathbb{F}_{2^m}$ that support $2$- and $3$-designs; it provides explicit weight distributions and shows how the minimum-weight structures yield $t$-designs via the Assmus–Mattson framework. The methods combine carefully chosen generator matrices with subset-sum counting to control dual distances and weight enumerators, leveraging Vandermonde-type determinants and the Li–Wan formulas for finite-field subset sums. The main contributions are the first infinite families with arbitrary dimensions supporting $t$-designs (for $t\ge 2$) and an affirmative resolution of Heng-Wang’s conjecture on flexible NMDS codes for $2$-designs, together with concrete examples and comparisons to prior fixed-dimension constructions. These results enrich the design-theoretic applications of NMDS codes and open avenue toward higher-design-strength families, with potential implications for combinatorial design, finite geometry, and cryptographic constructions.

Abstract

Near maximum distance separable (NMDS) codes, where both the code and its dual are almost maximum distance separable, play pivotal roles in combinatorial design theory and cryptographic applications. Despite progress in fixed dimensions (e.g., dimension 4 codes by Ding and Tang \cite{Ding2020}), constructing NMDS codes with arbitrary dimensions supporting $t$-designs ($t\geq 2$) has remained open. In this paper, we construct two infinite families of NMDS codes over $\mathbb{F}_q$ for any prime power $q$ with flexible dimensions and determine their weight distributions. Further, two additional families with arbitrary dimensions over $\mathbb{F}_{2^m}$ supporting $2$-designs and $3$-designs, and their weight distributions are obtained. Our results fully generalize prior fixed-dimension works~\cite{DingY2024,Heng2023,Heng20231,Xu2022}, and affirmatively settle the Heng-Wang conjecture \cite{Heng2023} on the existence of NMDS codes with flexible parameters supporting $2$-designs.

Several new infinite families of NMDS codes with arbitrary dimensions supporting $t$-designs

TL;DR

This paper advances NMDS codes by constructing two infinite families over with lengths and and arbitrary dimension, plus two families over that support - and -designs; it provides explicit weight distributions and shows how the minimum-weight structures yield -designs via the Assmus–Mattson framework. The methods combine carefully chosen generator matrices with subset-sum counting to control dual distances and weight enumerators, leveraging Vandermonde-type determinants and the Li–Wan formulas for finite-field subset sums. The main contributions are the first infinite families with arbitrary dimensions supporting -designs (for ) and an affirmative resolution of Heng-Wang’s conjecture on flexible NMDS codes for -designs, together with concrete examples and comparisons to prior fixed-dimension constructions. These results enrich the design-theoretic applications of NMDS codes and open avenue toward higher-design-strength families, with potential implications for combinatorial design, finite geometry, and cryptographic constructions.

Abstract

Near maximum distance separable (NMDS) codes, where both the code and its dual are almost maximum distance separable, play pivotal roles in combinatorial design theory and cryptographic applications. Despite progress in fixed dimensions (e.g., dimension 4 codes by Ding and Tang \cite{Ding2020}), constructing NMDS codes with arbitrary dimensions supporting -designs () has remained open. In this paper, we construct two infinite families of NMDS codes over for any prime power with flexible dimensions and determine their weight distributions. Further, two additional families with arbitrary dimensions over supporting -designs and -designs, and their weight distributions are obtained. Our results fully generalize prior fixed-dimension works~\cite{DingY2024,Heng2023,Heng20231,Xu2022}, and affirmatively settle the Heng-Wang conjecture \cite{Heng2023} on the existence of NMDS codes with flexible parameters supporting -designs.

Paper Structure

This paper contains 14 sections, 18 theorems, 79 equations, 2 tables.

Table of Contents

  1. Introduction
  2. NMDS codes supporting $t$-designs
  3. Our motivations and contributions
  4. Organization of this Paper
  5. Preliminaries
  6. NMDS codes
  7. The subset sum problem over finite fields
  8. New infinite families of NMDS codes with arbitrary dimensions
  9. The first infinite family of NMDS codes with parameters $[q+1,k,q+1-k]_q$
  10. The second infinite family of NMDS codes with parameters $[q+2,k,q+2-k]_q$
  11. New infinite families of NMDS codes with arbitrary dimensions supporting $t$-designs
  12. New NMDS codes with parameters $[2^m-1,k,2^m-1+k]_{2^m}$ supporting $2$-designs
  13. New NMDS codes with parameters $[2^m,k,2^m-k]_{2^m}$ supporting $3$-designs
  14. Summary and concluding remarks

Key Result

Lemma 1

DingT2022Dodunekov1995 Let $\mathcal{C}$ be an $[n,k,n-k]_q$ NMDS code. Then the weight distributions of $\mathcal{C}$ and $\mathcal{C}^\perp$ are given by for $t\in\{1,2,\ldots,n-k\}$, and for $t\in\{1,2,\ldots,k\}$.

Theorems & Definitions (34)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 24 more