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A hemisystem of a nonclassical generalised quadrangle

(2009) DESIGNS CODES AND CRYPTOGRAPHY. 51(2). p.157-165
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Abstract
The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points H such that every line l meets H in half of the points of l. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q (2)) were those of the elliptic quadric Q(-)(5, q) , q odd. We show in this paper that there exists a hemisystem of the Fisher-Thas-Walker-Kantor generalised quadrangle of order (5, 5(2)), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3 . A (7)-hemisystem of Q-(5, 5), first constructed by Cossidente and Penttila.
Keywords
Strongly regular graph, Partial quadrangle, Hemisystem, Association scheme, QUADRATIC CONE, FLOCKS

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Citation

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Chicago
Bamberg, John, Frank De Clerck, and Nicola Durante. 2009. “A Hemisystem of a Nonclassical Generalised Quadrangle.” Designs Codes and Cryptography 51 (2): 157–165.
APA
Bamberg, J., De Clerck, F., & Durante, N. (2009). A hemisystem of a nonclassical generalised quadrangle. DESIGNS CODES AND CRYPTOGRAPHY, 51(2), 157–165.
Vancouver
1.
Bamberg J, De Clerck F, Durante N. A hemisystem of a nonclassical generalised quadrangle. DESIGNS CODES AND CRYPTOGRAPHY. 2009;51(2):157–65.
MLA
Bamberg, John, Frank De Clerck, and Nicola Durante. “A Hemisystem of a Nonclassical Generalised Quadrangle.” DESIGNS CODES AND CRYPTOGRAPHY 51.2 (2009): 157–165. Print.
@article{513745,
  abstract     = {The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points H such that every line l meets H in half of the points of l. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q (2)) were those of the elliptic quadric Q(-)(5, q) , q odd. We show in this paper that there exists a hemisystem of the Fisher-Thas-Walker-Kantor generalised quadrangle of order (5, 5(2)), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3 . A (7)-hemisystem of Q-(5, 5), first constructed by Cossidente and Penttila.},
  author       = {Bamberg, John and De Clerck, Frank and Durante, Nicola},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  keyword      = {Strongly regular graph,Partial quadrangle,Hemisystem,Association scheme,QUADRATIC CONE,FLOCKS},
  language     = {eng},
  number       = {2},
  pages        = {157--165},
  title        = {A hemisystem of a nonclassical generalised quadrangle},
  url          = {http://dx.doi.org/10.1007/s10623-008-9251-1},
  volume       = {51},
  year         = {2009},
}

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