Double blocking sets of size 3q-1 in PG(2,q)

The main purpose of this paper is to find double blocking sets in PG(2,q) of size less than 3q, in particular when q is prime. To this end, we study double blocking sets in 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 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.