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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 all members of $Y$ can not be exchanged for $a$, it follows that all elements of $Y$ are in the block containing $X \setminus a$, which means $Y$ is the subset of the block by uniqueness. Contradiction.

Corollary. $B$ is the basis of a matroid with rank $4$.

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 $4$.

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 $X,Y\in B$, $a\in X$. $X\text{\\}a$ is a 3-element set, so it is contained in exactly one block of $S(3,6,22)$. If all members 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 $Y$ is the subset of the block by uniqueness. Contradiction.

Corollary. $B$ is the basis of a matroid with rank $4$.

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 all members of $Y$ can not be exchanged for $a$, it follows that all elements of $Y$ are in the block containing $X \setminus a$, which means $Y$ is the subset of the block by uniqueness. Contradiction.

Corollary. $B$ is the basis of a matroid with rank $4$.

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 $4$.

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 $X,Y\in B$, $a\in X$. $X\text{\\}a$ is a 3-element set, so it is contained in exactly one block of $S(3,6,22)$. If all members 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 $Y$ is the subset of the block by uniqueness. Contradiction.

Corollary. $B$ is the basis of a matroid with rank $4$.