The cyclic \(q\)-clans with \(q=2^e\). (English) Zbl 1038.51009

This paper deals with GQs arising from \(q\)–clans, \(q=2^e\). Using a computer, S. E. Payne, T. Penttila and G. F. Royle found several GQs of order \((q^2,q)\). These GQs were called cyclic because they admit a collineation group acting cyclically on the \(q+1\) lines through the point \((\infty)\). Later on, W. E. Cherowitzo, C. M. O’Keefe and T. Penttila discovered a new infinite family of GQs that appeared to include the examples by Payne, Penttila and Royle. In particular, they provided a unified construction of the previously known families as well the new family. However, it was not so clear that the unified construction always gave a cyclic GQs. In this paper the authors provide a proof of this fact, clarifying the relationship between the collineation group of a GQs and the “magic action”, and between cyclic GQs and the corresponding flocks of the quadratic cone.


51E21 Blocking sets, ovals, \(k\)-arcs
05B25 Combinatorial aspects of finite geometries
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