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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.

An explicit representation and enumeration for negacyclic codes of length 2^kn over \mathbb{Z}_4+u\mathbb{Z}_4