A family of $m$-ovoids of parabolic quadrics

TL;DR

This paper introduces the first infinite family of rac{(q-1)}{2}oids in parabolic quadrics of PG(4,q) for q quiv 3 mod 4, expanding known existence results.

Contribution

It constructs an infinite family of rac{(q-1)}{2}oids in Q(4,q) for q quiv 3 mod 4, and also provides new rac{q+1}{2}oids for q quiv 1 mod 4.

Findings

First infinite family of rac{(q-1)}{2}oids for q quiv 3 mod 4

Construction of rac{q+1}{2}oids for q quiv 1 mod 4

Extends known existence results for ovoids in parabolic quadrics.

Abstract

We construct a family of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$, the parabolic quadric of $\textup{PG}(4,q)$, for $q\equiv 3\pmod 4$. The existence of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$ was only known for $q=3, 7,$ or $11$. Our construction provides the first infinite family of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$.Along the way, we also give a construction of $\frac{q+1}{2}$-ovoids in $Q(4,q)$ for $q\equiv 1\pmod 4$.