The dual codes of two families of BCH codes

Abstract

In this paper, we present an infinite family of MDS codes over $F_{2^s}$ and two infinite families of almost MDS codes over $F_{p^s}$ 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 $F_{3^s}$, resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost MDS codes and their dual codes hold infinite families of 3-designs over $F_{p^s}$ 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.