Caps embedded in the Klein quadric (Q1971662)

Let \(\text{PG}(5,q)\) be the projective space of dimension \(5\) over the finite field with \(q\) elements. Let \(\mathcal K\) be the Klein quadric of order \(q\), i.e.\ the hyperbolic quadric contained in \(\text{PG}(5,q)\). A \(k\)-cap of \(\mathcal K\) is a set of \(k\) points of \(\mathcal K\) no three of which are collinear. If \(q\) is odd, then the maximal size of a cap contained in \(\mathcal K\) is \(q^3+q^2+q+1\), where \((q^3+q^2+q+1)\)-caps correspond via the Klein correspondence to linear complexes of \(\text{PG}(3,q)\). For even \(q\), \textit{G. L. Ebert, K. Metsch} and \textit{T. Szőnyi} [Geom. Dedicata 70, No. 2, 181-196 (1998; Zbl 0911.51013)] gave a construction of caps contained in \(\mathcal K\) having bigger size. Under more, they construct maximal caps of size \(q^3+q^2\pm \sqrt{2q}(q+1)-q+1\), where \(q\) is an odd power of 2. In the present paper, the author gives a new construction of such caps as a union of \(q+1\) Suzuki-Tits ovoids and one elliptic quadric (the only intersection points lying in the elliptic quadric). This construction has a more group-theoretical nature than the original construction.