LCD BCH codes are an important class of cyclic codes which have efficient encoding and decoding algorithms, but their parameters are difficult to determine. The objective of this paper is to study the LCD BCH codes of length $n=\frac{q^{m}+1}{\lambda}$, where $\lambda\mid (q+1)$ is an integer. Several types of LCD BCH codes with good parameters are presented, and many optimal linear codes are settled. Moreover, we present the first few largest coset leaders modulo $n=q^{m}+1, \frac{q^{m}+1}{2},\frac{3^{m}+1}{4}$, and partially solve two conjectures about BCH codes.

翻译：LCD BCH码是一类重要的循环码，具有高效的编码与译码算法，但其参数难以确定。本文旨在研究长度为 $n=\frac{q^{m}+1}{\lambda}$ 的LCD BCH码，其中 $\lambda\mid (q+1)$ 为整数。我们给出了若干类具有良好参数的LCD BCH码，并确定了多个最优线性码。此外，我们首次求出了模 $n=q^{m}+1, \frac{q^{m}+1}{2},\frac{3^{m}+1}{4}$ 的前几大陪集首，并部分解决了关于BCH码的两个猜想。