Weight Enumerators of Normalized Codes

Abstract

A linear code C over a finite field is normalized if it contains the all ones vector. If a normalized code C is also self-dual then its complete weight enumerator is invariant under the action of a linear group G, which is explicitly determined. The character of this representation is then used to calculate the Molien series for G. Further restrictions on C may lead to larger finite linear groups containing G. It is determined here that if the field is not $GF(2)$ or $GF(4)$ then there are only finitely many linear groups containing G with the property that the only scalar matrices appearing are those already contained in G. In fact, if the characteristic is odd and $\tilde G$ is the unimodular subgroup of G, then the finite unimodular subgroups containing $\tilde G$ are contained in a unique, such maximal linear group. The classification of the finite simple groups is used for the proof of this last result.