In this paper, we give an explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over the local non-principal ideal ring $ R = \mathbb{Z}_4+u\mathbb{Z}_4 $ $ (u^2 = 0) $, where $ k, n $ are arbitrary positive integers and $ n $ is odd. In particular, we present all distinct negacyclic codes of length $ 2^k $ over $ R $ precisely. Moreover, we provide an exact mass formula for the number of negacyclic codes of length $ 2^kn $ over $ R $ and correct several mistakes in some literatures.
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