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

This is the authors' abstract. We don't add key points for this paper.

Let SLAut(Fqⁿ) denote the group of all semilinear isometries on Fqⁿ, where q=pᵉ 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 σ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be σ dual-containing and σ self-orthogonal. We prove that dim_Fq(Hullσ(C))=dim_Fq(Hullσ(C^σ)). We prove that for any integer h with max\0,k₁-k₂\≤ h≤ dim_Fq(C₁₂^σ), there exists a linear code C2,h monomially equivalent to C₂ such that dim_Fq(C₁2,h^σ)=h, where Cᵢ is an [n,kᵢ]q linear code for $i=1,2$. We show that given an [n,k,d]q linear code C, there exists a monomially equivalent [n,k,d]q linear code Cₕ, whose σ 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≤ h≤ dim_Fq(Hullσ(C)). Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to σ hulls.