MacWilliams Identities and Matroid Polynomials

Abstract

We present generalisations of several MacWilliams type identities, including those by Kløve and Shiromoto, and of the theorems of Greene and Barg that describe how the Tutte polynomial of the vector matroid of a linear code determines the $r$th support weight enumerators of the code. One of our main tools is a generalisation of a decomposition theorem due to Brylawski.