BCH codes are an important class of cyclic codes, and have wide applications in communication and storage systems. In this paper, we study the negacyclic BCH code and cyclic BCH code of length $n=\frac{q^m-1}{2}$.For negacyclic BCH code, we give the dimensions of $\mathcal C_{(n,-1,\delta,0)}$ for $\widetilde{\delta} =a\frac{q^m-1}{q-1},aq^{m-1}-1$($1\leq a <\frac{q-1}{2}$) and $\widetilde{\delta} =a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1},aq^{m-1}+(a+b)q^{m-2}-1$ $(2\mid m,1\leq a+b \leq q-1$,$\left\lceil \frac{q-a-2}{2}\right\rceil\geq 1)$. The dimensions of negacyclic BCH codes $\mathcal C_{(n,-1,\delta,0)}$ with few nonzeros and $\mathcal C_{(n,-1,\delta,b)}$ with $b\neq 1$ are settled.For cyclic BCH code, we give the weight distributions of extended codes $\overline{\mathcal C}_{(n,1,\delta,1)}$ for $\delta=\delta_1,\delta_2$ and the parameters of dual code $\mathcal C^{\perp}_{(n,1,\delta,1)}$ for $\delta_2\leq \delta \leq \delta_1$.

翻译：BCH码是一类重要的循环码，在通信与存储系统中具有广泛应用。本文研究了长度为$n=\frac{q^m-1}{2}$的负循环BCH码与循环BCH码。对于负循环BCH码，我们给出了$\mathcal C_{(n,-1,\delta,0)}$在$\widetilde{\delta}=a\frac{q^m-1}{q-1},aq^{m-1}-1$（$1\leq a <\frac{q-1}{2}$）以及$\widetilde{\delta}=a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1},aq^{m-1}+(a+b)q^{m-2}-1$（$2\mid m,1\leq a+b \leq q-1$,$\left\lceil \frac{q-a-2}{2}\right\rceil\geq 1$）情形下的维数。同时确定了具有少量非零位置的负循环BCH码$\mathcal C_{(n,-1,\delta,0)}$以及$b\neq 1$时$\mathcal C_{(n,-1,\delta,b)}$的维数。对于循环BCH码，我们给出了$\delta=\delta_1,\delta_2$时扩展码$\overline{\mathcal C}_{(n,1,\delta,1)}$的重量分布，以及$\delta_2\leq \delta \leq \delta_1$时对偶码$\mathcal C^{\perp}_{(n,1,\delta,1)}$的参数。