A strong s-blocking set in a projective space is a set of points that intersects each codimension-s subspace in a spanning set of the subspace. We present an explicit construction of such sets in a \((k - 1)\) -dimensional projective space over \(\mathbb {F}_q\) of size \(O_s(q^s k)\) , which is optimal up to the constant factor depending on s. This also yields an optimal explicit construction of affine blocking sets in \(\mathbb {F}_q^k\) with respect to codimension- \((s+1)\) affine subspaces, and of s-minimal codes. Our approach is motivated by a recent construction of Alon, Bishnoi, Das, and Neri of strong 1-blocking sets, which uses expander graphs with a carefully chosen set of vectors as their vertex set. The main novelty of our work lies in constructing specific hypergraphs on top of these expander graphs, where tree-like configurations correspond to strong s-blocking sets. We also discuss some connections to size-Ramsey numbers of hypergraphs, which might be of independent interest.
