Two Classes of Quantum MDS Codes with Large Minimal Distance

In this paper, two classes of quantum MDS codes are constructed. The main tools are multiplicative structures on finite fields. Carefully choosing different cosets can make the corresponding generalized Reed-Solomon codes Hermitian self-orthogonal, which yields quantum MDS codes by utilizing Hermitian type CSS construction. In some cases, these codes have minimal distances larger than $q\mathrm{/}2$. Also, the parameters of these codes have never been reported before.