  Take a prime power $q$, an integer $n\geq 2$, and a coordinate subspace
$S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$
an $n$-partite $n$-uniform clutter $\mathcal{C}$, where every part has size $q$
and there is a bijection between the vectors in $S$ and the members of
$\mathcal{C}$.
  In this paper, we determine when the clutter $\mathcal{C}$ is ideal, a
property developed in connection to Packing and Covering problems in the areas
of Integer Programming and Combinatorial Optimization. Interestingly, the
characterization differs depending on whether $q$ is $2,4$, a higher power of
$2$, or otherwise. Each characterization uses crucially that idealness is a
minor-closed property: first the list of excluded minors is identified, and
only then is the global structure determined. A key insight is that idealness
of $\mathcal{C}$ depends solely on the underlying matroid of $S$.
  Our theorems also extend from idealness to the stronger max-flow min-cut
property. As a consequence, we prove the Replication and $\tau=2$ Conjectures
for this class of clutters.
