Lets define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.

I have seen a lot work related to $(n-1)$-blockings set. Covering finite fields with cosets of subspaces. Here is proved that minimal $(n-1)$-blocking has $n(q-1)+1$ points.

I cant find any result about $(n-2)$-blocking sets. I have managed to get following bounds about $(n-2)$-blocking set in $AG(n,2)$. It has at least $2n-1$ and no more than $3n^{\log_{2} 3}+1$.

I am very interested in the solution of the problem. Does anyone know information about it?