In this paper, we provide methods for constructing Hermitian dual-containing (HDC) matrix-product codes over \(\mathbb {F}_{q^2}\) from some non-singular matrices and a special sequence of HDC codes and determine parameters of obtained matrix-product codes when the input matrix and sequence of HDC codes satisfy some conditions. Then, using some nested HDC BCH codes with lengths \(n=\frac{q^4-1}{a} (a=1 ~\)or\(~ a=q\pm 1)\), we construct some HDC matrix-product codes with lengths \(N=\) 2n or 3n and derive nonbinary quantum codes with length N from these matrix-product codes via Hermitian construction. Four classes of quantum codes over \(\mathbb {F}_{q}\) (\(3\le q\le 5\)) are presented, whose parameters are better than those in the literature. Besides, some of our new quantum codes can exceed the quantum Gilbert-Varshamov (GV) bound.

在本文中，我们提供了从一些非奇异矩阵和特殊的HDC代码序列构造\（\ mathbb {F} _ {q ^ 2} \）上构造Hermitian对偶（HDC）矩阵乘积代码的方法。当输入矩阵和HDC码序列满足一定条件时，确定获得的矩阵乘积码的参数。然后，使用一些嵌套的长度为\（n = \ frac {q ^ 4-1} {a}（a = 1〜\）或\（〜a = q \ pm 1）\）的HDC BCH代码，我们构造了一些长度为\（N = \） 2 n或3 n的HDC矩阵乘积码，并通过Hermitian构造从这些矩阵乘积码中获得长度为N的非二进制量子码。四类量子码提出了\（\ mathbb {F} _ {q} \）（\（3 \ le q \ le 5 \）），其参数比文献中的参数要好。此外，我们的某些新量子代码可能会超出吉尔伯特-瓦尔沙莫夫（GV）量子范围。