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

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 $q$.