"Graded Codes"
Justo Peralta Lopez
University of Almeria, Spain.
jperalta@ualm.es
Abstract:
Since S.D.Berman showed in 1967 that cyclic codes and Reed Muller
codes can be studied as ideals in a group algebra $KG$,( $K$ been a
finite field and $G$ finite group cyclic and 2-group respectively),
several authors have considered these codes, since if you have more
algebraic structure then their study is more effective. Following
this way, we introduce the concept of graded code. A lineal code is
graded if it is a graded ideal in a graded $K$-algebra for some
suitable multiplicative group . We'll show that some important
properties about graded codes can be get from his homogeneous
components and generalize some results about codes as ideals in group
algebras. In particular,we'll consider a finite $K$-algebra $R$ graded
of type $G$, with $G$ a finite group,and the code $C=(M \otimes_{R_e}
R)$, with M a ideal in $R_e$, the homogeneous component of grade
1. We'll show some properties the graded code $C$, for $R$ graded and
strongly graded.