Construction of new entanglement-assisted quantum MDS codes via cyclic codes

Entanglement-assisted quantum error-correcting (EAQEC) codes can be transformed from classical linear codes through entanglement-assisted formalism by loosing the dual-containing condition and using pre-shared entanglement. It has become a challenging task to construct optimal EAQEC codes and determine the required number of pre-shared entanglement pairs. In this work, we explore the structure of q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^2$$\end{document}-ary cyclic codes through analyzing two classes of cyclotomic cosets independently. By computing the number of maximally entangled states, we construct three classes of q-ary entanglement-assisted quantum maximum distance separable (EAQMDS) codes. This construction produces new EAQMDS codes with minimum distance more than q+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q+1$$\end{document}.