# Minimal number of nonzero points in $\mathbb{F}_2^n$ which cover all subspaces of codimension k

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 $$k$$.