Conical embeddings of Steiner systems (Q1107815)

The classical Steiner system \(S(3,2^ a+1;2^{ab}+1)\) is also known as inversive space. A construction of \(S(3,2^ a+1;2^{ab}+1)\) uses an ovoid in \(PG(3,2^ a)\) the blocks being determined by intersections of the ovoid with planes. While this provides an embedding into space, in this paper the authors give a natural embedding into a plane \(PG(2,2^{ab})\) of such Steiner systems. More precisely, they use a conic of \(PG(2,2^{ab})\) as the point set of \(S(3,2^ a+1;2^{ab}+1);\) the blocks will be determined by the intersection with certain subplanes of order \(2^ a\). Consequently, every such Steiner system is naturally embedded into PG(2,K) where K is the algebraic closure of GF(2).