Several classes of BCH codes of length $n=\frac{q^{m}-1}{2}$

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 $\delta_2\leq \delta \leq \delta_1$.