Constructing minimal blocking sets using field reduction

TL;DR

This paper introduces a new method for constructing minimal blocking sets in projective spaces over finite fields using field reduction and blocking cones, extending known constructions and relating to the linearity conjecture.

Contribution

It presents a novel construction for minimal blocking sets via field reduction and blocking cones, generalizing the MPS-construction and characterizing linearity of small minimal blocking sets.

Findings

The new construction yields larger blocking sets than previous planar-based methods.

Every minimal blocking set with respect to hyperplanes can be obtained through field reduction from smaller spaces.

Small minimal blocking sets from the cone construction are linear, linking to the linearity conjecture.

Abstract

We present a construction for minimal blocking sets with respect to $(k-1)$-spaces in $\mathrm{PG}(n-1,q^t)$, the $(n-1)$-dimensional projective space over the finite field $\mathbb{F}_{q^t}$ of order $q^t$. The construction relies on the use of blocking cones in the {\em field reduced} representation of $\mathrm{PG}(n-1,q^t)$, extending the well-known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino and Storme ({\em the MPS-construction}); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in \cite{pol}. Furthermore we show that every minimal blocking set with respect to the hyperplanes in $\mathrm{PG}PG(n-1,q^t)$ can be obtained by applying field reduction to a minimal blocking set with respect to $(nt-t-1)$-spaces in $\mathrm{PG}(nt-1,q)$. We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS-construction, it is of R\'edei-type whereas a small minimal blocking set arises from our cone construction if and only if it is linear.