A family of hemisystems on the parabolic quadrics

Abstract

We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d,q)$, for all ranks $d\ge 2$ and all odd prime powers $q$, that admit ${\Omega }_3(q)\cong {\mathrm{PSL}}_2(q)$. This yields the first known construction for $d\ge 4$.