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In this paper, an explicit representation and enumeration for negacycliccodes of length $2^kn$ over the local non-principal ideal ring$R=\mathbb{Z}_4+u\mathbb{Z}_4$ $(u^2=0)$ is provided, where $k, n$ are anypositive integers and $n$ is odd. As a corollary, all distinct negacyclic codesof length $2^k$ over $R$ are listed precisely. Moreover, a mass formula for thenumber of negacyclic codes of length $2^kn$ over $R$ is given and a mistake in[Cryptogr. Commun. (2017) 9: 241--272] is corrected.

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