2025/06/20 by Liu, Hengfeng, Tang, Chunming, Zhou, Zhengchun +2 · 2 citations
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2506.16793
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 generic construction of q-ary NMDS codes supporting 2-designs with lengths 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.