COMPUTING THE DIMENSION OF IDEALS IN GROUP ALGEBRAS, WITH AN APPLICATION TO CODING THEORY

Abstract

Abstract We study the problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F, by relating the dimension to the rank of an appropriatematrix, originating from a regular right/left representation of G. In particular, the dimension of a principal ideal is equal to the rank of the matrix representing a generator. From this observation, we establish a bound and an efficient algorithm for the computation of the dimension of an ideal in a group ring. Since group codes are ideals in finite group rings, our algorithm allows to efficiently compute their dimension.