BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes with $d=δ$, where $d$ is the minimum distance of a $q$-ary BCH code of length $n$, designed distance $δ$, and offset $b$, denoted by $\mathbf{C}_{(q, n, δ, b)}$. For primitive BCH codes, we obtain infinite families of BCH codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ satisfying $d=δ$, where $δ\in \{5,6,7,8\}$. Moreover, we construct several infinite families of $q$-ary BCH codes with $d=δ$, where $2 \le δ\le q-1$. For $δ=q^t+1$, we prove that the BCH code $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ has $d=δ$ for all $m$ satisfying $m \equiv 0 \pmod{pt}$, where $p$ denotes the characteristic of $\mathbb{F}_q$. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ is always equal to its Bose distance $d_B$. Our result confirms this conjecture for the case $m \equiv 0 \pmod{pt}$. For non-primitive BCH codes, we construct a family of BCH codes $\mathbf{C}_{(q,\frac{q^p-1}λ,p+1,1)}$ with $d=δ=p+1$, where $p$ is an odd prime, $q=p^e$ with $p \nmid e$ and $λ\mid q-1$.

翻译：BCH码是一类重要的循环码，在通信和数据存储系统中有着广泛应用。尽管BCH界给出了BCH码最小距离的下界，但确定BCH码的真实最小距离是一个极具挑战性的问题。本文解决了多个无限族窄义BCH码的最小距离问题。通过显式构造最小重量码字的位置多项式，我们获得了许多满足$d=δ$的原始和非原始BCH码族，其中$d$是长度为$n$、设计距离为$δ$、偏移量为$b$的$q$元BCH码$\mathbf{C}_{(q, n, δ, b)}$的最小距离。对于原始BCH码，我们获得了$\mathbb{F}_3$和$\mathbb{F}_4$上满足$d=δ$的无限族BCH码，其中$δ\in \{5,6,7,8\}$。此外，我们构造了多个满足$d=δ$的$q$元BCH码无限族，其中$2 \le δ\le q-1$。对于$δ=q^t+1$，我们证明了对于所有满足$m \equiv 0 \pmod{pt}$的$m$（其中$p$表示$\mathbb{F}_q$的特征），BCH码$\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$具有$d=δ$。在Ding等人发表于IEEE Trans. Inf. Theory 61(5): 2351-2356的论文中，猜想$\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$的最小距离始终等于其Bose距离$d_B$。我们的结果证实了该猜想在$m \equiv 0 \pmod{pt}$情况下的正确性。对于非原始BCH码，我们构造了满足$d=δ=p+1$的BCH码族$\mathbf{C}_{(q,\frac{q^p-1}λ,p+1,1)}$，其中$p$为奇素数，$q=p^e$且$p \nmid e$，$λ\mid q-1$。