BCH codes form an important subclass of cyclic codes, and are widely used in compact discs, digital audio tapes and other data storage systems to improve data reliability. As far as we know, there are few results on $q$-ary BCH codes of length $n=\frac{q^{m}+1}{q+1}$. This is because it is harder to deal with BCH codes of such length. In this paper, we study $q$-ary BCH codes with lengths $n=\frac{q^{m}+1}{q+1}$ and $n=q^m+1$. These two classes of BCH codes are always LCD codes. For $n=\frac{q^{m}+1}{q+1}$, the dimensions of narrow-sense BCH codes of length $n$ with designed distance $\delta=\ell q^{\frac{m-1}{2}}+1$ are determined, where $q>2$ and $2\leq \ell \leq q-1$. Moreover, the largest coset leader is given for $m=3$ and the first two largest coset leaders are given for $q=2$. The parameters of BCH codes related to the first few largest coset leaders are investigated. Some binary BCH codes of length $n=\frac{2^m+1}{3}$ have optimal parameters. For ternary narrow-sense BCH codes of length $n=3^m+1$, a lower bound on the minimum distance of their dual codes is developed, which is good in some cases.

翻译：BCH码是循环码的重要子类，广泛应用于光盘、数字磁带及其他数据存储系统以提高数据可靠性。据我们所知，关于长度为$n=\frac{q^{m}+1}{q+1}$的$q$元BCH码的研究成果较少，这是因为此类长度的BCH码处理难度较大。本文研究了长度为$n=\frac{q^{m}+1}{q+1}$和$n=q^m+1$的两类$q$元BCH码，这两类码始终是LCD码。对于$n=\frac{q^{m}+1}{q+1}$的情况，确定了设计距离为$\delta=\ell q^{\frac{m-1}{2}}+1$的窄意义BCH码的维数，其中$q>2$且$2\leq \ell \leq q-1$。此外，给出了$m=3$时最大的陪集首项，以及$q=2$时前两个最大的陪集首项。研究了与前几个最大陪集首项相关的BCH码参数，其中长度为$n=\frac{2^m+1}{3}$的部分二元BCH码具有最优参数。对于长度为$n=3^m+1$的三元窄意义BCH码，建立了其对偶码最小距离的下界，该下界在某些情况下具有良好效果。