On the Minimum Distances of Some Families of BCH Codes

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 $d=\delta$, where $d$ is the minimum distance of a $q$-ary BCH code of length $n$, designed distance $\delta$, and offset $b$, denoted by $\mathbf{C}_{(q, n, \delta, b)}$. For primitive BCH codes, we obtain infinite families of BCH codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ satisfying $d=\delta$, where $\delta \in \{5,6,7,8\}$. Moreover, we construct several infinite families of $q$-ary BCH codes with $d=\delta$, where $2 \le \delta \le q-1$. For $\delta=q^t+1$, we prove that the BCH code $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ has $d=\delta$ 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 $\mathbf{C}_{(q,\frac{q^p-1}{\lambda},p+1,1)}$ with $d=\delta=p+1$, where $p$ is an odd prime, $q=p^e$ with $p \nmid e$ and $\lambda \mid q-1$.