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

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.