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

Abstract: Let $\mathrm{SLAut}(\mathbb{F}<em>{q}<sup>{n})$ denote the group of all semilinear isometries on $\mathbb{F}</em>{q}<sup>{n}$, where $q=p<sup>{e}$ 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 $$\mathrm{dim}<em>{\mathbb{F}</em>{q}}(\mathrm{Hull}<em>{\sigma}(\mathcal{C}))=\mathrm{dim}</em>{\mathbb{F}<em>{q}}(\mathrm{Hull}</em>{\sigma}(\mathcal{C}<sup>{\bot_{\sigma}}))$$. We prove that for any integer $h$ with $$\mathrm{max}{0,k_{1}-k_{2}}\leq h\leq \mathrm{dim}<em>{\mathbb{F}</em>{q}}(\mathcal{C}<em>{1}\cap\mathcal{C}</em>{2}<sup>{\bot_{\sigma}})$$, there exists a linear code $\mathcal{C}<em>{2,h}$ monomially equivalent to $\mathcal{C}</em>{2}$ such that $$\mathrm{dim}<em>{\mathbb{F}</em>{q}}(\mathcal{C}<em>{1}\cap\mathcal{C}</em>{2,h}<sup>{\bot_{\sigma}})=h$$, where $\mathcal{C}<em>{i}$ is an $[n,k</em>{i}]<em>{q}$ linear code for $i=1,2$. We show that given an $[n,k,d]</em>{q}$ linear code $\mathcal{C}$, there exists a monomially equivalent $[n,k,d]<em>{q}$ linear code $\mathcal{C}</em>{h}$, whose $\sigma$ dual code has minimum distance $d&#39;$, such that there exist an $[[n,k-h,d;n-k-h]]<em>{q}$ EAQECC and an $[[n,n-k-h,d&#39;;k-h]]</em>{q}$ EAQECC for every integer $h$ with $$0\leq h\leq \mathrm{dim}<em>{\mathbb{F}</em>{q}}(\mathrm{Hull}_{\sigma}(\mathcal{C}))$$. Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to $\sigma$ hulls.