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

Abstract

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