On the equivalence of NMDS codes

TL;DR

The paper addresses the monomial equivalence problem for NMDS codes by exploiting a geometric framework that ties NMDS codes to point-sets in projective spaces. It constructs an infinite family of $[q+3,3,q]$ NMDS codes for even $q$ from hyperovals in $ ext{PG}(2,q)$ and fully classifies their monomial equivalence via hyperoval stabilizers; it also provides dimension-4 constructions from arcs in $ ext{PG}(3,q)$ with added points, together with orbit-based equivalence classifications and weight enumerators. The results unify and extend prior constructions (e.g., from Wang-Li and DingXu) by leveraging stabilizer actions to count equivalence classes and by detailing the weight distributions across cases $q$ even/odd and modular constraints. Collectively, the work advances explicit NMDS code families with known invariants and clarifies when distinct-looking constructions are monomially equivalent, offering a geometric perspective for NMDS code design and analysis.

Abstract

An $[n,k,d]$ linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) if $d=n-k+1$ or $d=n-k$, respectively. If a code and its dual code are both AMDS, then the code is said to be near maximum distance separable (NMDS). For $k=3$ and $k=4$, there are many constructions of NMDS codes by adding some suitable projective points to arcs in $\mathrm{PG}(k-1,q)$. In this paper, we consider the monomial equivalence problem for some NMDS codes with the same weight distributions and present new constructions of NMDS codes.

Theorems & Definitions (51)

• Lemma 1.1
• Lemma 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• Lemma 2.7
• Theorem 3.1