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.

翻译：本文通过研究两类BCH码对偶码的参数，构造了$\mathbb{F}_{2^s}$上的一个无限族MDS码，以及对于任意素数$p$在$\mathbb{F}_{p^s}$上的两个无限族几乎MDS码。特别地，这些几乎MDS码包含了$\mathbb{F}_{3^s}$上的两个无限族近MDS码，从而解决了Geng等人于2022年提出的一个猜想。此外，我们证明了对于任意素数$p$，这些几乎MDS码及其对偶码在\(\mathbb{F}_{p^s}\)上均包含无限族$3$-设计。另外，我们还研究了这些MDS码与近MDS码族的子域子码，并给出了若干具有已知最优参数的二进制、三进制及四进制码。