An important family of quantum codes is the quantum maximum-distance-separable (MDS) codes. In this paper, we construct some new classes of quantum MDS codes by generalized Reed-Solomon (GRS) codes and Hermitian construction. In addition, the length $n$ of most of the quantum MDS codes we constructed satisfies $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$, which is different from previously known code lengths. At the same time, the quantum MDS codes we construct have large minimum distances that are greater than $q/2+1$.

翻译：量子最大距离可分（MDS）码是一类重要的量子码。本文通过广义里德-所罗门（GRS）码与厄米特构造法，构建了若干新类别的量子MDS码。此外，所构造的大部分量子MDS码的长度 $n$ 满足 $n\equiv 0,1($mod$\,\frac{q\pm1}{2})$，这不同于以往已知的码长。同时，我们构造的量子MDS码具有大于 $q/2+1$ 的大最小距离。