The cocycle lattice of binary matroids, II Original Research Article

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) greater-or-equal, slanted k.