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

TL;DR

This paper introduces a novel, generic method to construct $q$-ary near-MDS codes supporting 2-designs with lengths beyond the traditional limit of $q+1$, using advanced mathematical connections.

Contribution

It provides the first generic construction of such codes with lengths exceeding $q+1$, expanding the possibilities for NMDS codes supporting 2-designs.

Findings

Constructed an infinite family of $q$-ary NMDS codes supporting 2-designs with length > $q+1$

Established new connections between elliptic curve codes, finite abelian groups, and combinatorial designs

Derived weight distributions for the constructed codes

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.