Short Cocircuits in Binary Matroids

Given a simple graph G having vertex set V, it is obvious that for any spanning tree T, there is an edge of T whose fundamental cutset has size at most |V|— 1. We extend this result to matroids.Call a cocircuit of a matroid M short if its size is at most the rank of M. Then we prove that for any simple binary matroid M having no Fano minor, and for any basis of M, there is a fundamental cocircuit which is short. This theorem gives a simple necessary condition for a matrix to be totally unimodular