Quantum synchronizable codes from finite rings

In this paper, we provide two methods of constructing quantum synchronizable codes from cyclic or constacyclic codes over finite rings. The first one is derived from the Calderbank-Shor-Steane (briefly, CSS) construction applied to dual-containing codes over finite chain rings. The second construction is derived from the CSS construction applied to Gray images of the constacyclic codes over semi-local rings Fp+vFp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{p}+v{\mathbb {F}}_{p}$$\end{document} with v2=v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^2=v$$\end{document}. By using two methods, concrete examples are presented to construct new quantum synchronizable codes.