New Optimal Asymmetric Quantum Codes Derived from Negacyclic Codes

The construction of quantum maximum-distance-separable (MDS) codes have been studied by many researchers for many years. Here, by using negacyclic codes, we construct two families of asymmetric quantum codes. The first family is the asymmetric quantum codes with parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[[q^{2}+1,q^{2}+1-2(t+k+1),(2k+2)/(2t+2)]]_{q^{2}}$\end{document}, where 0≤t≤k≤(q−1)/2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q \equiv1(\operatorname{mod} 4)$\end{document}, and k, t are positive integers. The second one is the asymmetric quantum codes with parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[[(q^{2}+1)/2,(q^{2}+1)/2-2(t+k),(2k+1)/(2t+1)]]_{q^{2}}$\end{document}, where 1≤t≤k≤(q−1)/2, and k, t are positive integers. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.