Some remarks on Steiner systems (Q1404328)

The author presents a new and more explicit construction of known Steiner systems SS\(_2\)(H\((q)\)) derived from the finite classical generalized hexagons H\((q)\) of order \(q\). The points of SS\(_2\)(H\((q)\)) are the points of H\((q)\) and the lines are the lines of the quadric \(Q\) in PG(\(6,q\)) defined by the equation \(x_{0}x_{4}+x_{1}x_{5}+x_{2}x_{6}=x_{3}^{2}\). The blocks of SS\(_2\)(H\((q)\)) are the lines of \(Q\) together with the sets \(C(a,b)\), which are defined for points \(a,b\) of distance 6 from each other as the sets of those points that have distance 3 from any line at distance 3 from both \(a\) and \(b\). In fact, the sets \(C(a,b)\) consist of exactly \(q+1\) points and form a conic on \(Q\). The author proves that for \(q\) even the Steiner system SS\(_2\)(H\((q)\)) is isomorphic to SS\(_2\)(PG(\(5,q\))). For \(q\) odd the automorphism group of SS\(_2\)(H\((q)\)) is isomorphic to the type preserving automorphism group of H\((q)\) itself. Consequently, for \(q\) odd the two Steiner systems SS\(_2\)(H\((q)\)) and SS\(_2\)(PG(\(5,q\))) are not isomorphic. Moreover, he shows that for every prime power \(q\) the generalized hexagon H\(^{*}(q)\) can be embedded in an \(S(2,q+1,(q^{6}-1)/(q-1))\) Steiner system whose blocks are lines and traces of H\(^{*}(q)\) such that two intersecting lines of H\(^{*}(q)\) are contained in a unique subsystem isomorphic to a Desarguesian projective plane. The paper closes with a rather free construction of Steiner systems out of existing ones.