The cocycle lattice of binary matroids, II

Abstract We continue our previous study of the lattice (grid) generated by the incidence vectors of cocycles of a binary matroid and its dual lattice. We have already proved that every denominator in the dual lattice is a power of 2 and have characterized those binary matroids M for which the largest exponent k ( M ) is 1. In this paper, we characterize the matroids with k ( M ) = 2 and, for each constant k , give a polynomial time algorithm to decide whether k ( M ) ⩾ k .