  In this paper, we present three new classes of $q$-ary quantum MDS codes
utilizing generalized Reed-Solomon codes satisfying Hermitian self-orthogonal
property. Among our constructions, the minimum distance of some $q$-ary quantum
MDS codes can be bigger than $\frac{q}{2}+1$. Comparing to previous known
constructions, the lengths of codes in our constructions are more flexible.
