On determining all codes in semi-simple group rings

Abstract

A group algebra code is an ideal in a group ring, FG. If char(F) χ order(G), then FG is semi-simple and every such code has an idempotent generator. The group ring and every group algebra code in it are direct sums of disjoint minimal ideals. Thus a list of all possible codes in FG may be produced by first determining idempotent generators of minimal codes which are direct summands of FG. When G is abelian, the task is straightforward and is facilitated by use of the character table for G. When G is non-abelian, however, two-sided ideals, which correspond to the group characters of G, in FG may decompose in multiple ways to one-sided ideals. The decomposition may be varied by altering the matrix representations afforded by the group representations. The number of ways the representation may be altered is limited by the structure of the representation space. With sufficient information about this space, all possible decompositions can be determined. A case study is presented in which a (125,20) two-sided ideal is found to contain 162 disjoint minimal left ideals, each with a distinct weight distribution.