Skew Cyclic Codes Over $$\mathbb {Z}_{4}+u\mathbb {Z}_{4}+u^{2}\mathbb {Z}_{4}$$ with Derivation and New $$\mathbb {Z}_{4}$$ Codes

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