A Class of Skew-Cyclic Codes over (Z_(2^m ) [u])/〈u^2-r〉 with Derivation

Abstract

Let R_r=Z_(2^m )+uZ_(2^m ) be a finite ring, where u^2=r for r∈Z_(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_(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_(θ_r )-cyclic codes. We investigate the structures of a skew polynomial ring R_r [x,θ_r,δ_(θ_r ) ]. A δ_(θ_r )-cyclic code is showed to be a left R_r [x,θ_r,δ_(θ_r ) ]-submodule of (R_r [x,θ_r,δ_(θ_r ) ])/〈x^n-1〉 . We give the generator matrix of a δ_(θ_r )-cyclic code of length n over R_r. Also, we present the generator matrix of the dual of a free δ_(θ_r )-cyclic code of even length n over R_r.