In this paper, the sufficient and necessary condition for the minimum distance of the BCH codes over $\mathbb{F}_q$ with length $q+1$ and designed distance 3 to be 3 and 4 are provided. Let $d$ be the minimum distance of the BCH code $\mathcal{C}_{(q,q+1,3,h)}$. We prove that (1) for any $q$, $d=3$ if and only if $\gcd(2h+1,q+1)>1$; (2) for $q$ odd, $d=4$ if and only if $\gcd(2h+1,q+1)=1$. By combining these conditions with the dimensions of these codes, the parameters of this BCH code are determined completely when $q$ is odd. Moreover, several infinite families of MDS and almost MDS (AMDS) codes are shown. Furthermore, the sufficient conditions for these AMDS codes to be distance-optimal and dimension-optimal locally repairable codes are presented. Based on these conditions, several examples are also given.

翻译：本文给出了有限域$\mathbb{F}_q$上长为$q+1$、设计距离为3的BCH码最小距离为3和4的充要条件。设$d$为BCH码$\mathcal{C}_{(q,q+1,3,h)}$的最小距离，我们证明了：(1) 对任意$q$，$d=3$当且仅当$\gcd(2h+1,q+1)>1$；(2) 对奇数$q$，$d=4$当且仅当$\gcd(2h+1,q+1)=1$。结合这些条件与码的维数，当$q$为奇数时完全确定了该BCH码的参数。此外，本文给出了多个MDS码与几乎MDS（AMDS）码的无穷族，并进一步给出了这些AMDS码成为距离最优与维数最优局部可修复码的充分条件，同时基于这些条件给出了若干实例。