A family of m-ovoids of parabolic quadrics

We construct a family of (q−1)2-ovoids of Q(4,q)Q(4,q), the parabolic quadric of PG(4,q)PG(4,q), for q≡3(mod4). The existence of (q−1)2-ovoids of Q(4,q)Q(4,q) was only known for q=3,7q=3,7, or 11. Our construction provides the first infinite family of (q−1)2-ovoids of Q(4,q)Q(4,q). Along the way, we also give a construction of (q+1)2-ovoids in Q(4,q)Q(4,q) for q≡1(mod4).