St Timothy Kom - Department of Mathematics, Manipur University, Imphal, Manipur-۷۹۵۰۰۳, India.
O. Ratnabala Devi - Department of Mathematics, Manipur University, Imphal, Manipur-۷۹۵۰۰۳, India.
Th. Rojita Chanu - Department of Mathematics, D. M. College of Science, Imphal, Manipur-۷۹۵۰۰۱, India.

This paper studies \lambda-constacyclic codes and skew \lambda-constacyclic codes over the finite commutative non-chain ring R=\mathbb{Z}_۴+u\mathbb{Z}_۴+u^۲\mathbb{Z}_۴ with u^۳=۰ for \lambda= (۱+۲u+۲u^۲) and (۳+۲u+۲u^۲). We introduce distinct Gray maps and show that the Gray images of \lambda-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over \mathbb{Z}_۴. It is also shown that the Gray images of skew \lambda-constacyclic codes are quasi-cyclic codes of length ۲n and index ۲ over \mathbb{Z}_۴. Moreover, the structure of \lambda-constacyclic codes of odd length n over the ring R is determined and give some suitable examples.

constacyclic code, Cyclic code, Gray map, quasi-cyclic code, skew constacyclic code