Hermitian dual-containing constacyclic codes over Fq2+v1Fq2+⋯+vrFq2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q^{2}}+{v_{1}}\mathbb {F}_{q^{2}}+\cdots +{v_{r}}\mathbb {F}_{q^{2}}$$\end{document} and new quantum codes

In this paper, we study Hermitian dual-containing constacyclic codes over the finite ring Rr=Fq2+v1Fq2+⋯+vrFq2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{r}=\mathbb {F}_{q^{2}}+{v_{1}}\mathbb {F}_{q^{2}}+\cdots +{v_{r}}\mathbb {F}_{q^{2}}$$\end{document}, where q is a prime power and vi2=vi,vivj=vjvi=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${v_{i}}^{2}={v_{i}},v_{i}v_{j}=v_{j}v_{i}=0$$\end{document} for 1 ≤ i,j ≤ r,i≠j. A necessary and sufficient condition is provided to determine whether a constacyclic code C over Rr is Hermitian dual-containing. Moreover, we propose a generalized Gray map ΦM to preserve the property of Hermitian dual-containing. Compared with some existing Gray maps, ΦM increases the possibility of making the minimum distance of ΦM(C) larger. As an application, some new quantum codes over Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q}$$\end{document} are constructed from constacyclic codes over Rr.