Cyclic codes over rings of matrices

Abstract

<p style='text-indent:20px;'>In this paper, we consider the ring of matrices <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> of order <inline-formula><tex-math id="M2">\begin{document}$ 2 $\end{document}</tex-math></inline-formula> over the ring <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_2 [u] / \langle u^k \rangle $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> is an indeterminate with <inline-formula><tex-math id="M5">\begin{document}$ u^k = 0 $\end{document}</tex-math></inline-formula>, i.e. <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A} = M_2 ( \mathbb{F}_2 [u] / \langle u^k \rangle) $\end{document}</tex-math></inline-formula>. We derive the structure theorem for cyclic codes of odd length <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula> over the ring <inline-formula><tex-math id="M8">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> with the help of isometry map from <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M10">\begin{document}$ \mathbb{F}_4 [u, v] / \langle u^k, v^2, u v - v u \rangle $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M11">\begin{document}$ v $\end{document}</tex-math></inline-formula> is an indeterminate satisfying <inline-formula><tex-math id="M12">\begin{document}$ v^2 = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ u v = v u $\end{document}</tex-math></inline-formula>. We define a map <inline-formula><tex-math id="M14">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> which takes the linear codes of odd length <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M16">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> to linear codes of even length <inline-formula><tex-math id="M17">\begin{document}$ 2 k n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M18">\begin{document}$ \mathbb{F}_4 $\end{document}</tex-math></inline-formula>. We also define a weight on the ring <inline-formula><tex-math id="M19">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula> which is an extension of the weight defined over the ring <inline-formula><tex-math id="M20">\begin{document}$ M_2 ( \mathbb{F}_2) $\end{document}</tex-math></inline-formula>. An example is also given as applications to construct the linear codes of odd length <inline-formula><tex-math id="M21">\begin{document}$ n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M22">\begin{document}$ \mathcal{A} $\end{document}</tex-math></inline-formula>.</p>