In this paper, we study a class of skew-cyclic codes over the ring \(R=\mathbb {Z}_{4}+u\mathbb {Z}_{4}+u^{2}\mathbb {Z}_{4}\) , where \(u^{3}=0\) with an automorphism \(\theta \) and a derivation \(\delta _{\theta }\) and we call such codes: \(\delta _{\theta }\) -cyclic codes. Some structural properties of the skew polynomial ring \(R[x,\theta ,\delta _{\theta }]\) are discussed and these codes are considered as left \(R[x,\theta ,\delta _{\theta }]\) -submodules. Generator and parity-check matrices of a free \(\delta _{\theta }\) -cyclic code of even length over R are determined. A Gray map on R is used to obtain the \(\mathbb {Z}_{4}\) -images. Furthermore, these codes are generalized to double skew-cyclic codes. As an application of \(\delta _{\theta }\) -cyclic codes, we have obtained new quaternary linear codes from the Gray images of \(\delta _{\theta }\) -cyclic codes over R and added them to Aydin’s codetable.
