We can define a partial order $\leq$ on loopless matroids, such that $M_1\leq M_2$ if $M_1$ and $M_2$ are on the same groundset and $B_1\subseteq B_2$, where $B_1$ and $B_2$ are the set of bases of $M_1$ and $M_2$, respectively.
A matroid is minimal, if it is minimal under this partial order.
What can we say about number of non-isomorphic minimal matroids of size $n$ and rank $r$?
In particular, I'm interested in $n=2r$ case.
