2-modular matrices

Y An integer matrix A is Delta-modular if the determinant of each rank(A) x rank(A) submatrix has absolute value at most Delta. The class of 1-modular, or unimodular, matrices is of fundamental significance in both integer programming theory and matroid theory. A 1957 result of Heller shows that the maximum number of nonzero, pairwise non-parallel columns of a rank-r unimodular matrix is (r+1 2). We prove that, for each sufficiently large integer r, the maximum number of nonzero, pairwise non-parallel columns of a rank-r 2-modular matrix is (r+2 2)-2.