Explicit constructions of optimal blocking sets and minimal codes

TL;DR

This paper provides explicit, near-optimal constructions of strong blocking sets and minimal codes in projective and affine spaces, advancing combinatorial design and coding theory.

Contribution

It introduces a novel hypergraph-based method to explicitly construct strong s-blocking sets and minimal codes, improving upon previous probabilistic approaches.

Findings

Constructed strong s-blocking sets of size O_s(q^s k) in projective spaces

Achieved optimal explicit affine blocking sets with respect to certain subspaces

Connected constructions to size-Ramsey numbers of hypergraphs

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 $(k - 1)$-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.