MDS codes with Euclidean and Hermitian hulls of flexible dimensions and their applications to EAQECCs

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of maximum distance separable (MDS) codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. As a consequence, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. As applications, for the former, we further construct four new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new families of MDS EAQECCs of length n>q+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>q+1$$\end{document}. Meanwhile, many of the distance parameters of our MDS EAQECCs are greater than ⌈q2⌉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lceil \frac{q}{2} \rceil $$\end{document} or q; for the latter, we show some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes. In addition, two new general methods for constructing extended GRS codes with (k-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k-1)$$\end{document}-dimensional Hermitian hull and Hermitian self-orthogonal extended GRS codes are also provided.