Graded Codes

Abstract.

 Since S.D. Berman showed in 1967 that cyclic codes and Reed Muller codes can be studied as ideals in a group algebra KG (over a finite field K and G is considered, in each case, a finite cyclic group and a 2-group respectively), several authors have investigated these codes. It has been observed, that the presence of additional algebraic structure make their study more effective. Following these ideas, we consider codes which are (graded) left ideals of a graded ring R or more generally, codes as (graded) left R-modules. In this paper, a linear code is graded if it is a graded R-module for some multiplicative group G and a G-graded K-algebra R. We will show that some important properties of graded codes can be obtained from their homogeneous components and then we generalize some results about codes as ideals in group algebras. In particular, we study cocyclic codes as ideals in a twisted group algebra, and more generally, in a crossed product.