New $${\mathbb {Z}}_4$$ codes from constacyclic codes over a non-chain ring

Abstract

Let $${\mathbb {Z}}_{4}$$ be the ring of integers modulo 4. This paper presents $$(1+2u+2v+2uv)$$ -constacyclic and skew $$(1+2u+2v+2uv)$$ -constacyclic codes over the ring $$ {\mathbb {Z}}_{4} +u{\mathbb {Z}}_{4}+v{\mathbb {Z}}_{4}+uv{\mathbb {Z}}_{4} $$ where $$u^2=u,v^{2}=v, uv=vu$$ . We define three Gray maps and show that the Gray images of $$(1+2u+2v+2uv)$$ -constacyclic and skew $$(1+2u+2v+2uv)$$ -constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over $${\mathbb {Z}}_4$$ . Also, we show that cyclic and $$(1+2u+2v+2uv)$$ -constacyclic codes of odd length are principally generated. As an application, several new quaternary linear codes from the Gray images of $$(1+2u+2v+2uv)$$ -constacyclic codes are obtained.