Weight Enumerators of Normalized Codes II. The Hermitian Case

Abstract

Let F be a finite field whose order is a square. A linear code C in $F^n $ is self-dual if C coincides with its vector space dual with respect to the natural “Hermitian” form on $F^n $. If C contains the all-ones vector, then C is said to be normalized. The complete weight enumerator of a normalized self-dual code over F is invariant under the action of a linear group G which is explicitly determined. The character of this linear group is then used to calculate the Molien series.Other conditions may be imposed on C which lead to its weight enumerator being invariant under the action of a larger linear group containing G. However, there are only finitely many finite linear groups containing G with the property that the only scalar matrices appearing are those already contained in G. In fact, if the characteristic of F is odd and if $G^0 $ is the unimodular subgroup of G, then the finite unimodular subgroups containing $G^0 $ are contained in a unique maximal such linear group.