Polynomial aspects of codes, matroids and permutation groups

Index 74 Preface The three subjects of the title (codes, matroids, and permutation groups) have many interconnections. In particular, in each case, there is a polynomial which captures a lot of information about the structure: we have the weight enumerator of a code, the Tutte polynomial (or rank polynomial) of a matroid, and the cycle index of a permutation group. With any code is associated a matroid in a natural way. A celebrated theorem of Curtis Greene asserts that the weight enumerator of the code is a specialisation of the Tutte polynomial of the matroid. It is less well known that with any code is associated a permutation group, and the weight enumerator of the code is the same (up to normalisation) as the cycle index of the permutation group. There is a class of permutation groups, the so-called IBIS groups, which are closely associated with matroids. More precisely, the IBIS groups are those for which the irredundant bases (in the sense of computational group theory) are the bases of a matroid. The permutation group associated with a code is an IBIS group, and the matroid associated to the group differs only inessentially from the matroid obtained directly from the code. For some IBIS groups, the cycle index can be extracted from the Tutte polynomial of the matroid but not vice versa; for others, the Tutte polynomial can be obtained from the cycle index but not vice versa. This leads us to wonder whether there is a more general polynomial for IBIS groups which " includes " both the Tutte polynomial and the cycle index. Such a polynomial (the Tutte cycle index) is given in the last chapter of these notes (an expanded version of [5]). Whether or not there is a more general concept extending both matroids and arbitrary permutation groups, and giving rise to a polynomial extending both the Tutte polynomial and the cycle index, I do not know; I cannot even speculate what such a concept might be.