The dual codes of two families of BCH codes

TL;DR

This paper constructs new families of MDS and almost MDS codes from BCH codes, proves their design properties, and explores their subfield subcodes, including resolving a recent conjecture.

Contribution

It introduces infinite families of MDS and near MDS codes from BCH codes, including a resolution of a 2022 conjecture, and studies their combinatorial design properties.

Findings

Constructed infinite families of MDS codes over _{2^s}

Established almost MDS codes over _{p^s} for any prime p

Proved these codes and their duals hold infinite families of 3-designs

Abstract

In this paper, we present an infinite family of MDS codes over $\mathbb{F}_{2^s}$ and two infinite families of almost MDS codes over $\mathbb{F}_{p^s}$ for any prime $p$, by investigating the parameters of the dual codes of two families of BCH codes. Notably, these almost MDS codes include two infinite families of near MDS codes over $\mathbb{F}_{3^s}$, resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost AMDS codes and their dual codes hold infinite families of $3$-designs over \(\mathbb{F}_{p^s}\) for any prime $p$. Additionally, we study the subfield subcodes of these families of MDS and near MDS codes, and provide several binary, ternary, and quaternary codes with best known parameters.