Let $B$ be the subsets of $\{1,2,...,22\}$ of size $4$ which are not contained in any block of $S(3,6,22)$. $B$ satisfies the basis exchange property.
Proof:
Let $X,Y\in B$, $a\in X$. $X \setminus a$ is a 3-element set, so it is contained in exactly one block of $S(3,6,22)$.
If $∀y\in Y :(X \setminus a) \cup y \notin B$, it follows that all elements of $Y$ are in some block containing $X \setminus a$, and there's only one such block, call it $A$. It follows that $Y\subset A$, contradiction.
Corollary. $B$ is the basis of a matroid with rank $4$.