The column number for 3-modular matrices

Abstract

An integer-valued matrix $\mathbf{A}$ is $\hat{I}”$-modular if each $\text{rank}(\mathbf{A})\times \text{rank}(\mathbf{A})$ submatrix has determinant at most $\hat{I}”$ in absolute value. The column number problem is to determine the maximum number of pairwise non-parallel columns of a rank-$r$, $\hat{I}”$-modular matrix. Exact values for the column number are only known for $r\le 2$ or $\hat{I}”\le 2$. We prove that if $r$ is sufficiently large, then the maximum number of pairwise non-parallel columns of a rank-$r$, $3$-modular matrix is $\left(\genfrac{}{}{0ex}{}{r+1}{2}\right)+2(r-1)$. This settles a conjecture by Lee, Paat, Stallknecht, and Xu on the column number in the case $\hat{I}”=3$. We complement this main result by showing that there are at least three $3$-modular matrices with pairwise non-isomorphic vector matroids that attain this upper bound. More generally, we show that if $r>\hat{I}”$, then the number of $\hat{I}”$-modular matrices with $\left(\genfrac{}{}{0ex}{}{r+1}{2}\right)+(\hat{I}”-1)(r-1)$ pairwise non-parallel columns and pairwise non-isomorphic vector matroids is at least exponential in $\sqrt{\hat{I}}”$; previously only one matrix was known due to Lee et al.