Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$

In this paper, we study the structure of duadic codes of an odd length \begin{document}$ n $\end{document} over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} , \begin{document}$ u^2 = 0 $\end{document} , (more generally over \begin{document}$ \mathbb{Z}_{q}+u\mathbb{Z}_{q} $\end{document} , \begin{document}$ u^2 = 0 $\end{document} , where \begin{document}$ q = p^r $\end{document} , \begin{document}$ p $\end{document} a prime and \begin{document}$ (n, p) = 1 $\end{document} ) using the discrete Fourier transform approach. We study these codes by considering them as a class of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented. Some conditions on the existence of self-dual augmented and extended duadic codes over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} are determined. We present a sufficient condition for abelian codes of the same length over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} to have the same minimum Hamming distance. A new Gray map over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} is defined, and it is shown that the Gray image of an abelian code over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} is an abelian code over \begin{document}$ \mathbb{Z}_4 $\end{document} . We have obtained five new linear codes of length \begin{document}$ 18 $\end{document} over \begin{document}$ \mathbb{Z}_4 $\end{document} from duadic codes of length \begin{document}$ 9 $\end{document} over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} through the Gray map and a new map from \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} to \begin{document}$ \mathbb{Z}_4^2 $\end{document} .