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.

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MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field F q is completely solved for q is even. In this paper, for finite field with odd characteristic, we construct some new classes of MDS self-dual codes over F q by (extended) generalized Reed-Solomon codes.