# An infinite family of $m$-ovoids of $Q(4,q)$

## TL;DR

This paper constructs an infinite family of half-ovoids in the generalized quadrangle $Q(4,q)$ for certain prime powers, expanding the known examples and confirming their existence for all relevant $q$.

## Contribution

It introduces a new infinite family of $rac{q-1}{2}$-ovoids in $Q(4,q)$ for $q
ot r 4$ and $q > 5$, complementing previous constructions.

## Key findings

- Established existence of $rac{q-1}{2}$-ovoids for all odd prime powers $q$
- Constructed an explicit infinite family of such ovoids
- Extended known classifications of ovoids in $Q(4,q)$

## Abstract

In this paper, we construct an infinite family of $\frac{q-1}{2}$-ovoids of the generalized quadrangle $Q(4,q)$, for $q\equiv 1 (\text{mod}\ 4)$ and $q>5$. Together with the examples given by Bamberg et al. and constructions provided by Feng et al., this establishes the existence of $\frac{q-1}{2}$-ovoids in $Q(4,q)$ for each odd prime power $q$.