Skew Cyclic Codes Over ℤ4 + uℤ4 With Derivation Based on a New Gray Map and Automorphism - Eda Tekin (IEEE Access)
This paper investigates the algebraic structure and properties of skew cyclic codes over the finite chain ring $R = \mathbb {Z}_{4} + u\mathbb {Z}_{4}$ , where $u^{2} = 0$ . A central contribution of this work is the introduction and application of a novel Gray map, establishing a distance-preserving link between codes over R and linear codes over $\mathbb {Z}_{4}$ . We employ a specific, compatible pair consisting of a ring automorphism $\theta $ and a $\theta $ -derivation $\eta $ to define the appropriate skew polynomial ring structure $R[x; \theta , \eta]$ . Within this algebraic framework, we provide a comprehensive analysis of the fundamental structure of free $(\theta , \eta)$ -cyclic codes, detailing their generator polynomial structure and establishing their precise relationship with classical cyclic or quasi-cyclic codes. Furthermore, the structure of Euclidean dual codes for these free codes is examined for even lengths, and a construction for double $(\theta , \eta)$ -cyclic codes from free constituent codes is also presented.