I am looking for a specific matroid. I found a source that claimed to discuss these matroids, but then, only discusses geometric lattice. Even more, in that paper, the geometric lattice that seems to be the right one was described as

... the lattice associated with the Steiner system $S(3,6,22)$.

It might be clear to some, how to translate between all these different constructs, but I have a hard time finding any source explaining to me (in short) how these concepts are linked.

I suppose, that the matroid of the geometric lattice $\mathcal L$ is defined on the set of atoms of $\mathcal L$, and independence of atoms $a_1,...,a_n\in\mathcal L$ means that the supremum $a_1\vee \cdots \vee a_n$ has rank $n$. But this is just a guess. Furthermore, the lattice that comes from $S(3,6,22)$ is said to be of rank 3, but there is not much more said about this.

Can someone tell me how to obtain the matroid from $S(3,6,22$)?

There are actually two papers I am talking about:

• Basis Transitive Matroids, by A. Delandtsheer, H. Li., which discusses matroids purely in terms of geometric lattices and dimensional linear spaces (DLS).
• Homogeneous Designs and Geometric Lattices, by W. M. Kantor (Theorem 2), which lists $S(3,6,22)$ as a Steiner system associated with a lattice.