I am looking for a specific [matroid](https://en.wikipedia.org/wiki/Matroid). I found a source that claimed to discuss these matroids, but then, only discusses [geometric lattice](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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 you tell me how to obtain the matroid from $S(3,6,22$)?