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.
Categories: math.RA cs.IT math.IT