$\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s $

Abstract

<p style='text-indent:20px;'>In this paper, we first study the skew cyclic codes of length <inline-formula><tex-math id="M3">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M4">\begin{document}$ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a prime number and <inline-formula><tex-math id="M6">\begin{document}$ u^3 = 0. $\end{document}</tex-math></inline-formula> Then we characterize the algebraic structure of <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}[u^2] $\end{document}</tex-math></inline-formula>-additive skew cyclic codes of length <inline-formula><tex-math id="M8">\begin{document}$ 2p^s. $\end{document}</tex-math></inline-formula> We will show that there are sixteen different types of these codes and classify them in terms of their generators.</p>