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

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{Z}_4 $\end{document}</tex-math></inline-formula> be the ring of integers modulo <inline-formula><tex-math id="M3">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>. This paper studies mixed alphabets <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z}_4\mathbb{Z}_4[u^3] $\end{document}</tex-math></inline-formula>-additive cyclic and <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic codes for units <inline-formula><tex-math id="M6">\begin{document}$ \lambda = 1+2u^2,3+2u^2 $\end{document}</tex-math></inline-formula>. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z}_4 $\end{document}</tex-math></inline-formula>-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.</p>