Double blocking sets of size $3q-1$ in $\mathrm{PG}(2,q)$

TL;DR

This paper investigates double blocking sets of size less than 3q in projective planes, providing structural insights, impossibility results, and explicit constructions for certain q, thereby resolving longstanding conjectures.

Contribution

It introduces new structural properties of double blocking sets of size 3q-1, proves certain configurations are impossible, and constructs explicit examples for multiple q, solving two conjectures from 1984.

Findings

Double blocking sets of size 3q-1 admit at least two (q-1)-secants.

Such sets cannot have three (q-1)-secants.

Constructed minimal double blocking sets for specific q values.

Abstract

The main purpose of this paper is to find double blocking sets in $\mathrm{PG}(2,q)$ of size less than $3q$, in particular when $q$ is prime. To this end, we study double blocking sets in $\mathrm{PG}(2,q)$ of size $3q-1$ admitting at least two $(q-1)$-secants. We derive some structural properties of these and show that they cannot have three $(q-1)$-secants. This yields that one cannot remove six points from a triangle, a double blocking set of size $3q$, and add five new points so that the resulting set is also a double blocking set. Furthermore, we give constructions of minimal double blocking sets of size $3q-1$ in $\mathrm{PG}(2,q)$ for $q=13$, $16$, $19$, $25$, $27$, $31$, $37$ and $43$. If $q>13$ is a prime, these are the first examples of double blocking sets of size less than $3q$. These results resolve two conjectures of Raymond Hill from 1984.