Extensions of the Critical Theorem

The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has previously been extended or generalised in a number of different ways. The main result of the present paper is a general form of the Critical Theorem that encompasses many of these results. Applications include generalisations of a theorem by Curtis Greene that describes how the weight enumerator of a linear code is determined by the Tutte polynomial of the associated vector matroid, as well as generalisations of the MacWilliams identity for linear codes.