Transforming Skew Cyclic Codes into Generalized Quasi-Cyclic Codes via a New Gray Map over Z_4+uZ_4

We introduce a new Gray map Q(a+ub)=(a,a+3b), along with an automorphism θ(a+ub)=a+u(3b). Using these, we construct quasi-cyclic codes as left submodules of the skew polynomial ring R[x;θ] over R=Z_4+uZ_4. Although the original codes are skew cyclic, we show that their Gray images under Q are invariant under a modified cyclic shift operator, and hence are generalized quasi-cyclic codes over Z_4. Our analysis of the Lee weight transformation, supported by examples, demonstrates that this approach yields codes with predictable structure and favorable properties.