An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $
Yuan Cao, Yonglin Cao, Yonglin Cao, Yonglin Cao, Hai Quang Dinh, Ramakrishna Bandi, Fang‐Wei Fu · Advances in Mathematics of Communications · 2020
In this paper, we give an explicit representation and enumeration for negacyclic codes of length \begin{document}$ 2^kn $\end{document} over the local non-principal ideal ring \begin{document}$ R = \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} \begin{document}$ (u^2 = 0) $\end{document} , where \begin{document}$ k, n $\end{document} are arbitrary positive integers and \begin{document}$ n $\end{document} is odd. In particular, we present all distinct negacyclic codes of length \begin{document}$ 2^k $\end{document} over \begin{document}$ R $\end{document} precisely. Moreover, we provide an exact mass formula for the number of negacyclic codes of length \begin{document}$ 2^kn $\end{document} over \begin{document}$ R $\end{document} and correct several mistakes in some literatures.