Cycle Index, Weight Enumerator, and Tutte Polynomial

Abstract

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arises directly from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene's Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa. I propose a polynomial for IBIS groups which generalises both Tutte polynomial and cycle index.