  An important family of quantum codes is the quantum
maximum-distance-separable (MDS) codes. In this paper, we construct some new
classes of quantum MDS codes by generalized Reed-Solomon (GRS) codes and
Hermitian construction. In addition, the length $n$ of most of the quantum MDS
codes we constructed satisfies $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$, which is
different from previously known code lengths. At the same time, the quantum MDS
codes we construct have large minimum distances that are greater than $q/2+1$.
