On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes

Let \begin{document}$ \mathbb{Z}_4 $\end{document} be the ring of integers modulo \begin{document}$ 4 $\end{document}. This paper studies mixed alphabets \begin{document}$ \mathbb{Z}_4\mathbb{Z}_4[u^3] $\end{document}-additive cyclic and \begin{document}$ \lambda $\end{document}-constacyclic codes for units \begin{document}$ \lambda = 1+2u^2,3+2u^2 $\end{document}. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain \begin{document}$ \mathbb{Z}_4 $\end{document}-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.