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Extensions of GQ(4,2), the description of hyperovals

Extensions of GQ(4,2), the description of hyperovals -- A subgraph of the point graph of the generalized quadrangle GQ(s, t) is called a hyperoval, if it is a regular graph without triangles of valence t + 1 with even number of vertices. In the triangular extensions of GQ(s, r) the role of -subgraphs can be played by hyperovals only. We give a classification of the hyperovals in GQ(4,2). For any even from 6 to 18 there exists a hyperoval with points. This research was supported by the Russian Foundation for Basic Research, grant 94-01-00802-a. INTRODUCTION A geometry G of rank 2 is the incidence system with the point set and the block set (without repeating blocks). Each block can be identified with the set of the incident to it points, and the incidence turns into the usual inclusion. The dual to G geometry GT is obtained by swapping the roles of points and blocks. Two different points from T are called collinear if they lie in one and the same block. The residue Ga of a geometry G at a point a is the geometry with the point set Ta of those points which are collinear with a and the block set 2. (2) = http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

Extensions of GQ(4,2), the description of hyperovals

Discrete Mathematics and Applications , Volume 7 (4) – Jan 1, 1997

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References (3)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1997.7.4.419
Publisher site
See Article on Publisher Site

Abstract

-- A subgraph of the point graph of the generalized quadrangle GQ(s, t) is called a hyperoval, if it is a regular graph without triangles of valence t + 1 with even number of vertices. In the triangular extensions of GQ(s, r) the role of -subgraphs can be played by hyperovals only. We give a classification of the hyperovals in GQ(4,2). For any even from 6 to 18 there exists a hyperoval with points. This research was supported by the Russian Foundation for Basic Research, grant 94-01-00802-a. INTRODUCTION A geometry G of rank 2 is the incidence system with the point set and the block set (without repeating blocks). Each block can be identified with the set of the incident to it points, and the incidence turns into the usual inclusion. The dual to G geometry GT is obtained by swapping the roles of points and blocks. Two different points from T are called collinear if they lie in one and the same block. The residue Ga of a geometry G at a point a is the geometry with the point set Ta of those points which are collinear with a and the block set 2. (2) =

Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1997

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