Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several commensurate classical codes with a defining matrix. We give an explicit formula for calculating the dimension of the $\sigma$ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be $\sigma$ dual-containing and $\sigma$ self-orthogonal. We prove that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_{\sigma}(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_{\sigma}(\mathcal{C}^{\bot_{\sigma}}))$. We prove that for any integer $h$ with $\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_{\sigma}})$, there exists a linear code $\mathcal{C}_{2,h}$ monomially equivalent to $\mathcal{C}_{2}$ such that $\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_{\sigma}})=h$, where $\mathcal{C}_{i}$ is an $[n,k_{i}]_{q}$ linear code for $i=1,2$. We show that given an $[n,k,d]_{q}$ linear code $\mathcal{C}$, there exists a monomially equivalent $[n,k,d]_{q}$ linear code $\mathcal{C}_{h}$, whose $\sigma$ dual code has minimum distance $d'$, such that there exist an $[[n,k-h,d;n-k-h]]_{q}$ EAQECC and an $[[n,n-k-h,d';k-h]]_{q}$ EAQECC for every integer $h$ with $0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_{\sigma}(\mathcal{C}))$. Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to $\sigma$ hulls.

翻译：设$\mathrm{SLAut}(\mathbb{F}_{q}^{n})$表示$\mathbb{F}_{q}^{n}$上所有半线性等距构成的群，其中$q=p^{e}$为素数幂。矩阵积码是一类通过组合若干同维经典码与定义矩阵生成的长经典码。本文给出计算矩阵积码σ壳维数的显式公式，进而给出矩阵积码为σ对偶包含和σ自正交的充要条件。我们证明$\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_{\sigma}(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_{\sigma}(\mathcal{C}^{\bot_{\sigma}}))$。对于任意整数$h$满足$\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_{\sigma}})$，存在与$\mathcal{C}_{2}$单项等价的线性码$\mathcal{C}_{2,h}$使得$\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_{\sigma}})=h$，其中$\mathcal{C}_{i}$为$[n,k_{i}]_{q}$线性码（$i=1,2$）。进一步证明：给定$[n,k,d]_{q}$线性码$\mathcal{C}$，存在与$\mathcal{C}$单项等价的$[n,k,d]_{q}$线性码$\mathcal{C}_{h}$，其σ对偶码具有最小距离$d'$，使得对任意满足$0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_{\sigma}(\mathcal{C}))$的整数$h$，存在$[[n,k-h,d;n-k-h]]_{q}$型EAQECC和$[[n,n-k-h,d';k-h]]_{q}$型EAQECC。基于此结果，我们提出从与σ壳相关的矩阵积码中构造参数灵活的EAQECC的通用方法。