On one-lee weight and two-lee weight $ \mathbb{Z}_2\mathbb{Z}_4[u] $ additive codes and their constructions

Abstract

This paper mainly study \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}-additive codes. A Gray map from \begin{document}$ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $\end{document} to \begin{document}$ \mathbb{Z}_{4}^{\alpha+2\beta} $\end{document} is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}-additive code and its dual is proved. Some properties of one-weight \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}-additive codes and two-weight projective \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}-additive codes are discussed. As main results, some construction methods for one-weight and two-weight \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}-additive codes are studied, meanwhile several examples are presented to illustrate the methods.