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: 

Let $X,Y\in B$, $a\in X$. $X\text{\\}a$ is a 3-element set, so it is contained in a unique block $A$ of $S(3,6,22)$. If any member of $Y$ can not be exchanged for $a$, it follows that all elements of $Y$ are in the block containing $X\text{\\}a$, which means that $Y\subset A$. Contradiction, as $Y$ is not contained in any block.

So there's a matroid with $B$ as a basis, and it has rank $4$.