The matrix-product (MP) code $\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot A$ with a non-singular by column (NSC) matrix $A$ plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix $A$ satisfies the condition that $AA^{\dag}$ is $(D,\tau)$-monomial. We give an explicit formula for calculating the dimension of the Hermitian hull of a MP code. We provide the necessary and sufficient conditions that a MP code is Hermitian dual-containing (HDC), almost Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD, respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code with these properties, respectively. We give alternative necessary and sufficient conditions for a MP code to be AHDC and AHSO, respectively, and show several cases where a MP code is not AHDC or AHSO. We provide the construction methods of HDC and AHDC MP codes, including those with optimal minimum distance lower bounds.

翻译：矩阵乘积（MP）码 $\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot A$ 结合非奇异按列（NSC）矩阵 $A$ 在构建优质量子纠错码中发挥着重要作用。本文研究了当定义矩阵 $A$ 满足 $AA^{\dag}$ 为 $(D,\tau)$-单项矩阵时MP码的性质。我们给出了MP码Hermitian核维数的显式计算公式，并分别刻画了MP码为Hermitian对偶包含（HDC）、几乎Hermitian对偶包含（AHDC）、Hermitian自正交（HSO）、几乎Hermitian自正交（AHSO）以及Hermitian LCD的充要条件。从理论上确定了各组成码之间所有可能的关联方式，使得MP码分别具有这些性质。我们还提出了AHDC和AHSO性质的替代充要条件，并展示了若干MP码不满足AHDC或AHSO的实例。最终给出了HDC与AHDC MP码的构造方法，包括具有最优最小距离下界的码字构造。