$(1+2u)$-constacyclic codes over ${\mathbb{Z}}_4+u{\mathbb{Z}}_4$

Abstract

Let $R={\mathbb{Z}}_4+u{\mathbb{Z}}_4,$ where ${\mathbb{Z}}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to ${\mathbb{Z}}_4^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over ${\mathbb{Z}}_4$. Further, a number of linear codes over ${\mathbb{Z}}_4$ as the images of $(1+2u)$-constacyclic codes over $R$ are obtained.