A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation

In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over \begin{document}$R = \mathbb{Z}_4+u\mathbb{Z}_4;u^2 = 1$ \end{document} , with an automorphism \begin{document}$θ$ \end{document} and a derivation \begin{document}$δ_θ$ \end{document} . We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as \begin{document}$δ_θ$ \end{document} -cyclic codes. Some properties of skew polynomial ring \begin{document}$R[x, θ, {δ_θ}]$ \end{document} are presented. A \begin{document}$δ_θ$ \end{document} -cyclic code is proved to be a left \begin{document}$R[x, θ, {δ_θ}]$ \end{document} -submodule of \begin{document}$\frac{R[x, θ, {δ_θ}]}{\langle x^n-1 \rangle}$ \end{document} . The form of a parity-check matrix of a free \begin{document}$δ_θ$ \end{document} -cyclic codes of even length \begin{document}$n$ \end{document} is presented. These codes are further generalized to double \begin{document}$δ_θ$ \end{document} -cyclic codes over \begin{document}$R$ \end{document} . We have obtained some new good codes over \begin{document}$\mathbb{Z}_4$ \end{document} via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of \begin{document}$\mathbb{Z}_4$ \end{document} -codes [ 2 ].