  It is well known that linear rank-metric codes give rise to $q$-polymatroids.
Analogously to classical matroid theory one may ask whether a given
$q$-polymatroid is representable by a rank-metric code. We provide a partial
answer by presenting examples of $q$-matroids that are not representable by
${\mathbb F}_{q^m}$-linear rank-metric codes. We then go on and introduce
deletion and contraction for $q$-polymatroids and show that they are mutually
dual and that they correspond to puncturing and shortening of rank-metric
codes. Finally, we introduce a closure operator along with the notion of flats
and show that the generalized rank weights of a rank-metric code are fully
determined by the flats of the associated $q$-polymatroid.
