A MacWilliams type identity for matroids

In this paper, we consider the coboundary polynomial for a matroid as a generalization of the weight enumerator of a linear code. By describing properties of this polynomial and of a more general polynomial, we investigate the matroid analogue of the MacWilliams identity. From coding-theoretical approaches, upper bounds are given on the size of circuits and cocircuits of a matroid, which generalizes bounds on minimum Hamming weights of linear codes due to I. Duursma.