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 combinatorialt-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 fort> 2, all with length at mostq+ 1. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater thanq+ 1 are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the firstgeneric constructionofq-ary NMDS codes supporting 2-designs with lengthsexceedingq+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.

The defect of an$[n,k,d]$linear code is defined as$s({\mathcal{ C}})=n-k+1-d$. Codes with$s({\mathcal{ C}})=0$are called maximum distance separable (MDS), while codes with$s({\mathcal{ C}})=s({\mathcal{ C}}^{\perp})=1$are called near maximum distance separable (NMDS). NMDS codes correspond to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. There have been many constructions of NMDS codes, but most of them are focus on fixed$q$or$k$, except for constructions from elliptic curves. Roth and Lempel (IEEE Trans. Inf. Theory 1989) constructed a type of linear codes (referred as Roth-Lempel codes), and presented the necessary and sufficient conditions of Roth-Lempel code to be MDS. Especially, they pointed out that the resultant MDS codes is not linearly equivalent to Reed-Solomn codes. In this paper, the NMDS properties of Roth-Lempel codes will be analyzed. We also obtain the necessary and sufficient condition of Roth-Lempel codes to be NMDS, and further completely determine the weight distributions of Roth-Lempel codes with length$q+2$and dimension$3\leq k\leq q$. Besides, by analyzing the upper bound for the code lengths of elliptic curve MDS codes, we illustrate the linearly inequivalence of Roth-Lempel NMDS codes and elliptic curve NMDS codes when their corresponding code lengths exceed$4(q+2\sqrt {q}+1)/5+1$.

Determining the maximal length of MDS codes with certain dimension has been an interesting research topic in coding theory. The objective of this paper is to derive an upper bound for the maximal length of MDS elliptic codes over$\mathbb {F}_{q}$with dimension$3\leq k\leq \frac {q+1-2\sqrt {q}}{10}$. For such a range of dimension$k$, our result improves an earlier bound of Munuera and gives an affirmative solution to the conjecture of Li, Wan, and Zhang. Most notably, the proposed upper bound is tight for odd$k$in the sense that it can be achieved by some well-designed MDS elliptic codes.

In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $M(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the reciprocity $|M(G,|H|)|=|M(H,|G|)|$. Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. Among others, we partially answer a question proposed by Panyushev in a generalized version.