Flocks, ovoids of \(Q(4,q)\) and designs (Q1360954)

If \(K\) is a quadratic cone in \(PG(3,q)\), a flock of \(K\) is a set of disjoint conics on \(K\) which cover the points other than the vertex. Associated with the flock is a spread which defines a translation plane. Associated with a generalized quadrangle \(Q(4,q)\) in \(PG(4,q)\) is an ovoid which has exactly one point in common with each line of \(Q(4,q)\). By the Klein quadric there corresponds a line spread \(S(O)\) of \(PG(3,q)\) whose lines are totally isotropic with respect to a symplectic polarity defined by \(Q(4,q)\) via the Klein correspondence. The author shows that an ovoid \(O\) of \(Q(4,q)\) is a Thas ovoid associated with a semifield flock (the translation plane is a semi-field plane) iff \(O\) represents, on the Klein quadric, a symplectic spread of \(PG(3,q)\) which defines a semi-field plane.