a study on constacyclic codes over the ring z4 + uz4 + u²z4

This paper studies λ-constacyclic codes and skew λ-constacyclic codes over the finite commutative non-chain ring r = z4 + uz4 + u²z4 with u³ = 0 for λ = (1 + 2u + 2u²) and (3 + 2u + 2u²). we introduce distinct gray maps and show that the gray images of λ-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over z4. it is also shown that the gray images of skew λ-constacyclic codes are quasi-cyclic codes of length 2n and index 2 over z4. moreover, the structure of λ-constacyclic codes of odd length n over the ring r is determined and give some suitable examples.