Skew Cyclic Codes Over ℤ<sub>4</sub> + <italic>u</italic>ℤ<sub>4</sub> With Derivation Based on a New Gray Map and Automorphism
This paper investigates the algebraic structure and properties of skew cyclic codes over the finite chain ring <inline-formula> <tex-math notation="LaTeX">$R = \mathbb {Z}_{4} + u\mathbb {Z}_{4}$ </tex-math></inline-formula>, where <inline-formula> <tex-mat...
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| Format: | Article |
| Language: | English |
| Published: |
IEEE
2025-01-01
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| Series: | IEEE Access |
| Subjects: | |
| Online Access: | https://ieeexplore.ieee.org/document/11105441/ |
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| Summary: | This paper investigates the algebraic structure and properties of skew cyclic codes over the finite chain ring <inline-formula> <tex-math notation="LaTeX">$R = \mathbb {Z}_{4} + u\mathbb {Z}_{4}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$u^{2} = 0$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{4}$ </tex-math></inline-formula>. We employ a specific, compatible pair consisting of a ring automorphism <inline-formula> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> and a <inline-formula> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula>-derivation <inline-formula> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> to define the appropriate skew polynomial ring structure <inline-formula> <tex-math notation="LaTeX">$R[x; \theta , \eta]$ </tex-math></inline-formula>. Within this algebraic framework, we provide a comprehensive analysis of the fundamental structure of free <inline-formula> <tex-math notation="LaTeX">$(\theta , \eta)$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$(\theta , \eta)$ </tex-math></inline-formula>-cyclic codes from free constituent codes is also presented. |
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| ISSN: | 2169-3536 |