BCH codes constitute an important class of cyclic codes, many of which are optimal and have wide applications in communication systems. However, determining their parameters remains a challenging problem. In this paper, we investigate BCH codes and LCD cyclic codes over finite fields $\mathbb{F}_q$ of length $n=λ(q^m+1)$, where $m$ is a positive integer and $λ\mid q-1$. We begin by analyzing the cyclotomic cosets modulo $n$, establishing the sufficient and necessary conditions for $γ$ is a coset leader for any $0\le γ<n$ and determining the two largest coset leaders. Based on these, we determine the dimensions of several families of BCH codes, and improve the lower bound on their minimal distances. Notably, some of the codes we constructed are optimal. Additionally, when $m$ is odd, we establish necessary and sufficient conditions for a BCH code of length $n$ to be dually-BCH. Furthermore, we enumerate all LCD cyclic codes of length $n$. Finally, several open problems are proposed for further study.

翻译：BCH 码是一类重要的循环码，其中许多码是最优的，并在通信系统中具有广泛应用。然而，确定其参数仍然是一个具有挑战性的问题。本文研究了长度为 $n=λ(q^m+1)$ 的有限域 $\mathbb{F}_q$ 上的 BCH 码和 LCD 循环码，其中 $m$ 为正整数且 $λ\mid q-1$。我们首先分析了模 $n$ 的分圆陪集，建立了对于任意 $0\le γ<n$，$γ$ 为陪集首的充分必要条件，并确定了两个最大的陪集首。基于此，我们确定了几类 BCH 码的维数，并改进了其最小距离的下界。值得注意的是，我们构造的部分码是最优的。此外，当 $m$ 为奇数时，我们建立了长度为 $n$ 的 BCH 码成为对偶 BCH 码的充分必要条件。同时，我们枚举了所有长度为 $n$ 的 LCD 循环码。最后，提出了几个有待进一步研究的开放性问题。