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.

翻译：在射影空间中，一个强$s$-阻塞集是指一个点集，它与每个余维数为$s$的子空间相交于该子空间的一个生成集。我们给出了在$\mathbb{F}_q$上的$(k-1)$维射影空间中此类集合的一个显式构造，其规模为$O_s(q^s k)$，该结果在依赖于$s$的常数因子范围内是最优的。这也导出了在$\mathbb{F}_q^k$中关于余维数为$(s+1)$的仿射子空间的最优显式仿射阻塞集构造，以及$s$-极小码的构造。我们的方法受到Alon、Bishnoi、Das和Neri近期提出的强$1$-阻塞集构造的启发，该构造使用了以一组精心选择的向量作为顶点集的扩展图。我们工作的主要创新点在于在这些扩展图之上构建特定的超图，其中树状结构对应于强$s$-阻塞集。我们还讨论了与超图的大小拉姆齐数的一些联系，这可能具有独立的研究意义。