BCH codes are an important class of linear codes and find extensive utilization in communication and disk storage systems.This paper mainly analyzes the negacyclic BCH code and cyclic BCH code of length $\frac{q^m-1}{2}$. For negacyclic BCH code, we give the dimensions of $C_{(n,-1,\left\lceil \frac{\delta+1}{2}\right\rceil,0)}$ for $\delta =a\frac{q^m-1}{q-1},aq^{m-1}-1$($1\leq a <\frac{q-1}{2}$) and $\delta =a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1},aq^{m-1}+(a+b)q^{m-2}-1$ $(2\mid m,1\leq a+b \leq q-1$,$\left\lceil \frac{q-a-2}{2}\right\rceil\geq 1)$. Furthermore, the dimensions of negacyclic BCH codes $C_{(n,-1,\delta,0)}$ with few nonzeros and $C_{(n,-1,\delta,b)}$ with $b\neq 0$ are settled. For cyclic BCH code, we give the weight distribution of extended code $\overline{C}_{(n,1,\delta,1)}$ and the parameters of dual code $C^{\perp}_{(n,1,\delta,1)}$, where $\delta_2\leq \delta \leq \delta_1$.

翻译：BCH 码是一类重要的线性码，在通信和磁盘存储系统中有着广泛的应用。本文主要分析了长度为 $\frac{q^m-1}{2}$ 的负循环 BCH 码和循环 BCH 码。对于负循环 BCH 码，我们给出了 $C_{(n,-1,\left\lceil \frac{\delta+1}{2}\right\rceil,0)}$ 在 $\delta =a\frac{q^m-1}{q-1},aq^{m-1}-1$（$1\leq a <\frac{q-1}{2}$）以及 $\delta =a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1},aq^{m-1}+(a+b)q^{m-2}-1$（$2\mid m,1\leq a+b \leq q-1$，$\left\lceil \frac{q-a-2}{2}\right\rceil\geq 1$）情况下的维数。此外，我们还确定了具有较少非零元素的负循环 BCH 码 $C_{(n,-1,\delta,0)}$ 以及 $b\neq 0$ 的负循环 BCH 码 $C_{(n,-1,\delta,b)}$ 的维数。对于循环 BCH 码，我们给出了扩展码 $\overline{C}_{(n,1,\delta,1)}$ 在 $\delta_2\leq \delta \leq \delta_1$ 时的重量分布，以及对偶码 $C^{\perp}_{(n,1,\delta,1)}$ 的参数。