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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 $\Delta$. Wecomplement this and obtain an upper bound of the form $O(\Delta)$ for fixed$m$, and with the implied constant depending polynomially on $m$.

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