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

Let SLAut (Fₐ^n) denote the group of all semilinear isometries on Fₐ^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 hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be dual-containing and self-orthogonal. We prove that dim₅_ₐ (Hull_ (C) ) =dim₅_ₐ (Hull_ (C^_{}) ). We prove that for any integer h with max\0, k₁-k₂\ h dim₅_ₐ (C₁₂^_{}), there exists a linear code C₂, ₇ monomially equivalent to C₂ such that dim₅_ₐ (C₁₂, ₇^_{}) =h, where C₈ is an n, k₈ₐ linear code for i=1, 2. We show that given an n, k, dₐ linear code C, there exists a monomially equivalent n, k, dₐ linear code C₇, whose dual code has minimum distance d', such that there exist an [n, k-h, d;n-k-h]ₐ EAQECC and an [n, n-k-h, d';k-h]ₐ EAQECC for every integer h with 0 h dim₅_ₐ (Hull_ (C) ). Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to hulls.