Constructions of new entanglement-assisted quantum MDS and almost MDS codes

In this work, we construct new entanglement-assisted quantum MDS codes with length n=q2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=q^2+1$$\end{document}. We also construct new entanglement-assisted quantum almost MDS codes with length n=q4-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=q^4-1$$\end{document}. Most of entanglement-assisted quantum MDS and almost MDS codes presented here are new in the sense that their parameters are different from all the previously known ones.