Construction of quantum MDS codes from Hermitian self-orthogonal generalized Reed-Solomon codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In recent years, Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes have been widely used to construct quantum MDS codes. In this paper, we give some sufficient conditions under which a certain system of equations over \({\mathbb {F}}_{q^2}\) ${\mathbb{F}}_{q^2}$ has a solution over \({\mathbb {F}}_q^*\) ${\mathbb{F}}_q^{\ast }$ , which effectively unify similar known techniques for constructing Hermitian self-orthogonal codes. Moreover, we construct five new classes of q-ary quantum MDS codes with flexible parameters from Hermitian self-orthogonal GRS codes. Compared to the previous literature, the quantum MDS codes we construct have different lengths, or the same length but larger distances. In particular, some of the quantum MDS codes we construct have distances that can be taken to the maximum distance of the quantum MDS codes from GRS codes.