On the Maximal Number of Columns of a $\Delta$-modular Integer Matrix: Bounds and Computations

We study the maximal number of pairwise distinct columns in a $\Delta$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $\Delta$. We complement this and obtain an upper bound of the form $O(\Delta)$ for fixed $m$, and with the implied constant depending polynomially on $m$.