Several new infinite families of NMDS codes with arbitrary dimensions supporting $t$-designs
Year: 2025
Type: preprint
Abstract: Near maximum distance separable (NMDS) codes, where both the code and its dual are almost maximum distance separable, play pivotal roles in combinatorial design theory and cryptographic applications. Despite progress in fixed dimensions (e.g., dimension 4 codes by Ding and Tang \cite{Ding2020}), constructing NMDS codes with arbitrary dimensions supporting $t$-designs ($t\geq 2$) has remained open. In this paper, we construct two infinite families of NMDS codes over $\mathbb{F}_q$ for any prime power $q$ with flexible dimensions and determine their weight distributions. Further, two additional families with arbitrary dimensions over $\mathbb{F}_{2^m}$ supporting $2$-designs and $3$-designs, and their weight distributions are obtained. Our results fully generalize prior fixed-dimension works~\cite{DingY2024,Heng2023,Heng20231,Xu2022}, and affirmatively settle the Heng-Wang conjecture \cite{Heng2023} on the existence of NMDS codes with flexible parameters supporting $2$-designs.
Source: arXiv (Cornell University)
Authors: Zhang, Yaozong, Zheng, Dabin, Wang, Xiaoqiang, Lu, Wei
Institution: Hubei University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G7246510406, G8996357861
OpenAlex ID: W4417244030