Quantum maximum-distance-separable (MDS for short) codes are an important
class of quantum codes. In this paper, by using Hermitian self-orthogonal
generalized Reed-Solomon (GRS for short) codes, we construct four new classes
of $q$-ary quantum MDS codes. The $q$-ary quantum MDS codes we construct have
larger minimum distances. And the minimum distance of these codes is greater
than $q/2+1$. Furthermore, it turns out that our quantum MDS codes generalize
the previous conclusions.Comment: 19 pages, 2 table