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

 We report on constructions of minimal double blocking sets of size $3q-1$ in $\PG(2,q)$ for $q=13$, $16$, $19$, $25$, $31$, $37$ and $43$. These are particularly interesting when $q$ is prime, because in that case, no examples of double blocking sets of size less than $3q$ has been known except for $q=13$. All of our examples admit two $(q-1)$-secants and have been found using a computer. Furthermore, we show that a double blocking set in $\PG(2,q)$ of size $3q-1$ cannot have three $(q-1)$-secants. These results partially prove and disprove some conjectures of R. Hill from 1984. Joint work with Bence Csajbok.