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

In this article, we mainly study the polycyclic codes over \begin{document}$ S $\end{document} , where \begin{document}$ S = \mathbb{F}_q+u\mathbb{F}_q $\end{document} with \begin{document}$ u^2 = u $\end{document} . First, the annihilator self-dual codes, annihilator self-orthogonal codes and annihilator \begin{document}$ {{{\rm{LCD}}}} $\end{document} codes over \begin{document}$ S $\end{document} are also introduced and studied. Next, we define a Gray map from \begin{document}$ S^n $\end{document} to \begin{document}$ \mathbb{F}^{2n}_q $\end{document} and investigate the structure properties of polycyclic codes over \begin{document}$ S $\end{document} 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.