Skew Cyclic Codes Over Z₄+uZ₄+u²Z₄ with Derivation and New Z₄ Codes

In this paper, we study a class of skew-cyclic codes over the ring R=Z₄+uZ₄+u²Z₄ R = Z 4 + u Z 4 + u 2 Z 4 , where u³=0 u 3 = 0 with an automorphism θ θ and a derivation δ θ δ θ and we call such codes: δ θ δ θ -cyclic codes. Some structural properties of the skew polynomial ring R[x,θ ,δ θ] R [ x , θ , δ θ ] are discussed and these codes are considered as left R[x,θ ,δ θ] R [ x , θ , δ θ ] -submodules. Generator and parity-check matrices of a free δ θ δ θ -cyclic code of even length over R are determined. A Gray map on R is used to obtain the Z₄ Z 4 -images. Furthermore, these codes are generalized to double skew-cyclic codes. As an application of δ θ δ θ -cyclic codes, we have obtained new quaternary linear codes from the Gray images of δ θ δ θ -cyclic codes over R and added them to Aydin’s codetable.