Bases

A matroid on a finite set $E$ can be specified by its bases: a nonempty family of subsets satisfying the exchange axiom (if $B_1, B_2$ are bases and $x \in B_1 \setminus B_2$, then $(B_1 \setminus \{x\}) \cup \{y\}$ is a basis for some $y \in B_2 \setminus B_1$). All bases have the same size, the rank. For the matroid of the columns of a matrix, the bases are the sets of columns forming a basis of the column space.

This database stores each matroid through its bases, via the revlex encoding.