On the polycyclic codes over $ \mathbb{F}_q+u\mathbb{F}_q $

Abstract

<p style='text-indent:20px;'>In this article, we mainly study the polycyclic codes over <inline-formula><tex-math id="M2">\begin{document}$ S $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ S = \mathbb{F}_q+u\mathbb{F}_q $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ u^2 = u $\end{document}</tex-math></inline-formula>. First, the annihilator self-dual codes, annihilator self-orthogonal codes and annihilator <inline-formula><tex-math id="M5">\begin{document}$ {{{\rm{LCD}}}} $\end{document}</tex-math></inline-formula> codes over <inline-formula><tex-math id="M6">\begin{document}$ S $\end{document}</tex-math></inline-formula> are also introduced and studied. Next, we define a Gray map from <inline-formula><tex-math id="M7">\begin{document}$ S^n $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{F}^{2n}_q $\end{document}</tex-math></inline-formula> and investigate the structure properties of polycyclic codes over <inline-formula><tex-math id="M9">\begin{document}$ S $\end{document}</tex-math></inline-formula> using the decomposition method. The Hamming distances of the Gray images are also determined by their decompositions. Finally, we obtain some good codes based on the results.</p>