The $σ$ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes

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.