What is the minimal size of $S \subset \mathbb{F}_2^n \setminus \{0\}$ so that for any codimension $k$ subspace $W \subset \mathbb{F}_2^n$, there exists $s \in S$ such that $s \in W$? We can assume that $k$ is $O(1)$.
I thought about this for a while and cannot seem to get a good size. I know we have $\binom{n}{k}_2$ many subspaces, but choosing one vector per subspace is not correct as there is significant overlap between these subspaces. I'd be happy just showing that $|S|$ is $o(2^n)$, although I'm not even sure that is true.
Not a sharp result, but if $T$ is any $k+1$ dimensional subspace, then $S:=T- \{0\}$ will do, because $W$ must intersect $T$ non-trivially (the sum of the dimensions is bigger than the ambient space). $T$ has $2^{k+1}$ elements, so $|S|=2^{k+1}-1$, which is an upper bound for your question.
Edit: In the case $k=n-1$, the space $W$ is 1 dimensional, which in $\mathbb{F}_2^n$ is just a single vector (and the zero vector). Thus in this case, $S$ has to be the entire space (except zero), so $|S|=2^{k+1}-1$, suggesting that the above result may be optimal, at least asymptotically in $k$.