Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $

Abstract

<p style='text-indent:20px;'>For any odd prime <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>, the structures and duals of <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic codes of length <inline-formula><tex-math id="M5">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M6">\begin{document}$ \mathcal R = \mathbb F_{p^m}+u\mathbb F_{p^m} $\end{document}</tex-math></inline-formula> are completely determined for all unit <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> of the form <inline-formula><tex-math id="M8">\begin{document}$ \lambda = \xi^l\in \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M9">\begin{document}$ l $\end{document}</tex-math></inline-formula> is even. In addition, the algebraic structures of all cyclic and negacyclic codes of length <inline-formula><tex-math id="M10">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M11">\begin{document}$ \mathcal R $\end{document}</tex-math></inline-formula> are established in term of their generator polynomials. Dual codes of all cyclic and negacyclic codes of length <inline-formula><tex-math id="M12">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M13">\begin{document}$ \mathcal R $\end{document}</tex-math></inline-formula> are also investigated. Furthermore, we give the number of codewords in each of those cyclic and negacyclic codes. We also obtain the number of cyclic and negacyclic codes of length <inline-formula><tex-math id="M14">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M15">\begin{document}$ \mathcal R $\end{document}</tex-math></inline-formula>.</p>

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