## 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.

### MSC:

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

### Keywords:

collineation group; cyclic $$q$$-clans
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