A class of constacyclic codes over the ring Z_4[u,v]/< u^2,v^{2},uv-vu> and their Gray images

Abstract

In this paper, we study (1+2u+2v)-constacyclic and skew (1+2u+2v)-constacyclic codes over the ring $ \mathbb{Z}_{4} +u\mathbb{Z}_{4} +v\mathbb{Z}_{4} +uv\mathbb{Z}_{4} $ where $u^2=v^{2}=0, uv=vu$. We define some new Gray maps and show that the Gray images of $(1+2u+2v)$-constacyclic and skew $(1+2u+2v)$-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over $\mathbb{Z}_4$. Further, we determine the structure of $(1+2u+2v)$-constacyclic codes of odd length $n$.