Some results on <inline-formula><tex-math id="M1111">$ \mathbb{Z}_p\mathbb{Z}_p[v] $</tex-math></inline-formula>-additive cyclic codes

Abstract

<p style='text-indent:20px;'><inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}</tex-math></inline-formula>-Additive cyclic codes of length <inline-formula><tex-math id="M2">\begin{document}$ (\alpha,\beta) $\end{document}</tex-math></inline-formula> can be viewed as <inline-formula><tex-math id="M3">\begin{document}$ R[x] $\end{document}</tex-math></inline-formula>-submodules of <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ R = \mathbb{Z}_p+v\mathbb{Z}_p $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ v^2 = v $\end{document}</tex-math></inline-formula>. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as <inline-formula><tex-math id="M7">\begin{document}$ R[x] $\end{document}</tex-math></inline-formula>-submodules of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $\end{document}</tex-math></inline-formula>. We also determine the generator polynomials of the dual codes of <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}</tex-math></inline-formula>-additive cyclic codes. Some optimal <inline-formula><tex-math id="M10">\begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}</tex-math></inline-formula>-linear codes and MDSS codes are obtained from <inline-formula><tex-math id="M11">\begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}</tex-math></inline-formula>-additive cyclic codes. Moreover, we also get some quantum codes from <inline-formula><tex-math id="M12">\begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}</tex-math></inline-formula>-additive cyclic codes.