  Motivated by complexity questions in integer programming, this paper aims to
contribute to the understanding of combinatorial properties of integer matrices
of row rank $r$ and with bounded subdeterminants. In particular, we study the
column number question for integer matrices whose every $r \times r$ minor is
non-zero and bounded by a fixed constant $\Delta$ in absolute value.
Approaching the problem in two different ways, one that uses results from
coding theory, and the other from the geometry of numbers, we obtain linear and
asymptotically sublinear upper bounds on the maximal number of columns of such
matrices, respectively. We complement these results by lower bound
constructions, matching the linear upper bound for $r=2$, and a discussion of a
computational approach to determine the maximal number of columns for small
parameters $\Delta$ and $r$.
