Abstract 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) (k - 1) -dimensional projective space over Fq F q of size Oₛ (qˢ k) 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 Fqᵏ F q k with respect to codimension- (s+1) (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.
Bishnoi et al. (Thu,) studied this question.