Quantum MDS Codes with length $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$

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 $q/2+1$.