Generalized quadrangles with an ovoid that is translation with respect to opposite flags (Q1777326)

The authors deal with finite generalized quadrangles with an order which is a pair \((s,t)\) of integers. They are particularly interested in the classical finite generalized quadrangle \(Q(4,q)\) and the ovoid \({\mathcal O}\) of the generalized quadrangle \(\Gamma\), which is a set of points such that on each line of \(\Gamma\) there is exactly one point of \({\mathcal O}\). The ovoid acquired as the set of absolute points of the polarity of \(Q(4,q)\) is called Suzuki-Tits ovoid. The aim of the present article is to show that if a finite generalized quadrangle \(\Gamma\) has an ovoid \({\mathcal O}\) that is translation with respect to two opposite flags but not with respect to two non-opposite flags, then \(\Gamma\) is \(Q(4,q)\) and \({\mathcal O}\) is a Suzuki-Tits ovoid. As the order \((s,t)\) of \(\Gamma\) is such that \(t\) is even or \(s= t\), the authors improve on the result given previously by \textit{A. Offer} and \textit{H. Van Maldeghem} [Des. Codes Cryptography 32, 351--367 (2004; Zbl 1064.51004), Theorem 5.3].