The dual codes of two families of BCH codes

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.