Non-Binary Quantum Codes from Cyclic Codes over Fp×(Fp+vFp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$\end{document}

In this paper, we study cyclic codes over the ring Fp×(Fp+vFp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$\end{document}, where p is an odd prime and v2 = v. We first investigate the properties of the ring Fp×(Fp+vFp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$\end{document} and the linear codes over this ring. We also define a distance-preserving Gray map from Fp×(Fp+vFp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p} \times (\mathbb {F}_{p}+v\mathbb {F}_{p})$\end{document} to Fp3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p}^{3}$\end{document}. We discuss cyclic codes and their dual codes over the ring. Also, we define a set of generators for these codes. As an implementation, we show that quantum error-correcting codes can be obtained from dual containing cyclic codes over the ring by using the Calderbank-Shor-Steane (CSS) construction. Furthermore, we give some illustrative examples. Finally, we tabulate the non-binary quantum error-correcting codes obtained from cyclic codes over the ring.