$2$-Modular Matrices

A rank-$r$ integer matrix $A$ is $\mathrm{\Delta }$-modular if the determinant of each $r\times r$ submatrix has absolute value at most $\mathrm{\Delta }$. 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 rows of a rank-$r$ unimodular matrix is $\left(\genfrac{}{}{0px}{}{r+1}{2}\right)$. We prove that, for each sufficiently large integer $r$, the maximum number of nonzero, pairwise non-parallel rows of a rank-$r$ $2$-modular matrix is $\left(\genfrac{}{}{0px}{}{r+2}{2}\right)-2$.