On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes.

We study mixed alphabet cyclic and constacyclic codes over the two alphabets Z 4 , the ring of integers modulo 4, and its quadratic extension Z 4 [u] = Z 4 + uZ 4 , u 2 = 0. Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we nd cyclic, quasi-cyclic codes over Z 4 as the Gray images of both λ-constacyclic and skew λ-constacyclic codes over Z 4 [u]. Moreover, it is proved that the Gray images of Z 4 Z 4 [u]-additive constacyclic and skew Z 4 Z 4 [u]-additive constacyclic codes are generalized quasi-cyclic codes over Z 4. Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.