Application of constacyclic codes to entanglement-assisted quantum maximum distance separable codes

The entanglement-assisted stabilizer formalism overcomes the dual-containing constraint of standard stabilizer formalism for constructing quantum codes. This allows ones to construct entanglement-assisted quantum error-correcting codes (EAQECCs) from arbitrary linear codes by pre-shared entanglement between the sender and the receiver. However, it is not easy to determine the number c of pre-shared entanglement pairs required to construct an EAQECC from arbitrary linear codes. In this paper, let q be a prime power, we aim to construct new q-ary EAQECCs from constacyclic codes. Firstly, we define the decomposition of the defining set of constacyclic codes, which transforms the problem of determining the number c into determining a subset of the defining set of underlying constacyclic codes. Secondly, five families of non-Hermitian dual-containing constacyclic codes are discussed. Hence, many entanglement-assisted quantum maximum distance separable codes with c≤7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\le 7$$\end{document} are constructed from them, including ones with minimum distance d≥q+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge q+1$$\end{document}. Most of these codes are new, and some of them have better performance than ones obtained in the literature.