Small 3-fold blocking sets in $\mathrm{PG}(2,p^n)$

TL;DR

This paper constructs specific 3-fold blocking sets in finite projective planes, advancing understanding of their minimal sizes and structures, especially for odd extension degrees where the problem remains open.

Contribution

It introduces a method to construct 3-fold blocking sets as unions of Rédé-type linear blocking sets on a common orbit, addressing cases where minimal sizes are unknown.

Findings

Constructed 3-fold blocking sets of conjectured size

Sets are unions of three Rédé-type linear blocking sets

Blocking sets lie on the same projective orbit

Abstract

A $t$-fold blocking set of the finite Desarguesian plane $\mathrm{PG}(2,p^n)$, $p$ prime, is a set of points meeting each line of the plane in at least $t$ points. The minimum size of such sets is of interest for numerous reasons; however, even the minimum size of nontrivial blocking sets (i.e. $1$-fold blocking sets not containing a line) in \(\mathrm{PG}(2,p^n)\) is an open question when $n\geq 5$ is odd. For $n>1$ the conjectured lower bound for this size is $(p^n+p^{n(s-1)/s}+1)$, where $p^{n/s}$ is the size of the largest proper subfield of $\mathbb{F}_{p^n}$. Since the union of $t$ pairwise disjoint nontrivial blocking sets is a $t$-fold blocking set, it is conjectured that when $p^{n/s}$ is large enough w.r.t. $t$, then the minimum size of a $t$-fold blocking set in $\mathrm{PG}(2,p^n)$ is $t(p^n+p^{n(s-1)/s}+1)$. If $n$ is even, then the decomposition of the plane into disjoint Baer subplanes gives a $t$-fold blocking set of this size. However, for odd $n$, the existence of such sets is an unsolved problem in most cases. In this paper, we construct $3$-fold blocking sets of conjectured size. These blocking sets are obtained as the disjoint union of three linear blocking sets of R\'edei type, and they lie on the same orbit of the projectivity $(x:y:z)\mapsto (z:x:y)$.