- Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi
- Volume:16 Issue:2
- A Class of Skew-Cyclic Codes over (Z_(2^m ) [u])/〈u^2-r〉 with Derivation
A Class of Skew-Cyclic Codes over (Z_(2^m ) [u])/〈u^2-r〉 with Derivation
Authors : Hayrullah ÖZİMAMOĞLU
Pages : 327-344
Doi:10.18185/erzifbed.1120896
View : 150 | Download : 68
Publication Date : 2023-08-31
Article Type : Research Paper
Abstract :Let R_r=Z_insert ignore into journalissuearticles values(2^m );+uZ_insert ignore into journalissuearticles values(2^m ); be a finite ring, where u^2=r for r∈Z_insert ignore into journalissuearticles values(2^m );, m is a positive integer, and m≥2. In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over R_r with an automorphism θ_r and a derivation δ_insert ignore into journalissuearticles values(θ_r );. We generalize the skew-cyclic codes over Z_4+uZ_4; u^2=1 to the skew-cyclic codes over R_r, and call such codes as δ_insert ignore into journalissuearticles values(θ_r );-cyclic codes. We investigate the structures of a skew polynomial ring R_r [x,θ_r,δ_insert ignore into journalissuearticles values(θ_r ); ]. A δ_insert ignore into journalissuearticles values(θ_r );-cyclic code is showed to be a left R_r [x,θ_r,δ_insert ignore into journalissuearticles values(θ_r ); ]-submodule of insert ignore into journalissuearticles values(R_r [x,θ_r,δ_insert ignore into journalissuearticles values(θ_r ); ]);/〈x^n-1〉 . We give the generator matrix of a δ_insert ignore into journalissuearticles values(θ_r );-cyclic code of length n over R_r. Also, we present the generator matrix of the dual of a free δ_insert ignore into journalissuearticles values(θ_r );-cyclic code of even length n over R_r.Keywords : cyclic codes, skew polynomial rings, skew cyclic codes
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