New Entanglement-Assisted Quantum MDS Codes with Maximal Entanglement

The construction of maximum distance separable (MDS) linear complementary dual (LCD) codes and entanglement-assisted quantum MDS (EAQMDS) codes have been of a great interest. In this paper, for arbitrary prime power q, we construct two new families of MDS Hermitian LCD codes of length n=q2+1λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n=\frac {{{q^{2}}+1}}{\lambda }$\end{document} and n=q2−1r,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n=\frac {q^{2}-1}{r},$\end{document} where rq + 1. By applying the obtained MDS Hermitian LCD codes to the EAQMDS codes, we derive new maximal entanglement EAQMDS codes of parameters q2+1λ,q2+1λ−l+1,l;l−1q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\left [ \kern -0.30em\left [ {\frac {{{q^{2}}+1}}{\lambda },\frac {{{q^{2}}+1}}{\lambda }-l+1,l;l-1}\right ] \kern -0.30em\right ]_{q}}$\end{document} where 2≤l≤q2+1+2λ2λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\leq l\leq \left \lfloor \frac {{{q^{2}}+1+2\lambda }}{2\lambda }\right \rfloor $\end{document} and q2−1r,q2−1r−γ,γ+1;γq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\left [ \kern -0.30em\left [ {\frac {{{q^{2}-}1}}{r},\frac {{{q^{2}-}1}}{r} -\gamma ,\gamma +1;\gamma }\right ] \kern -0.30em\right ]_{q}}$\end{document} where 1≤γ≤q2−12r.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq \gamma \leq \frac {{{q^{2}}-1}}{2r}.$\end{document}