On \(m\)-ovoids of \({\mathcal W}_3 (q)\) (Q2469473)

The authors are interested in \(m\)-ovoids of the symplectic generalized quadrangle \({\mathcal W}_3(q)\), which has order \((q, q)\). It is known that \({\mathcal W}(q)\), \(q\) odd, has no ovoids, which makes it interesting in this case to look for \(m\)-ovoids, with \(m> 1\). Since \({\mathcal W}_3(q)\) is isomorphic to the dual of \({\mathcal Q}(4, q)\), a \({1\over 2}(q+ 1)\)-ovoid of \({\mathcal W}_3(q)\) for odd \(q\) corresponds to a hemisystem of \({\mathcal Q}(4, q)\). They construct several infinite families of \({1\over 2}(q+ 1)\)-ovoids of \({\mathcal W}_3(q)\) based on the existence of certain line-spreads of \(\text{PG}(3, q)\), \(q\) odd. Some computational results for small \(q\) are also provided. In addition, the authors show that for \(q\) even, \(m\)-ovoid of \({\mathcal W}_3(q)\) admitting the semidirect product of a cyclic group of order \(q^2+ 1\) by a cyclic group of order 4 exists for all integers \(m\), \(1\leq m\leq q\).