Innovations in Incidence Geometry

Bounds on partial ovoids and spreads in classical generalized quadrangles

Stefaan De Winter and Koen Thas

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Abstract

We present an improvement on a recent bound for small maximal partial ovoids of W ( q 3 ) . We also classify maximal partial ovoids of size ( q 2 1 ) of Q ( 4 , q ) which allow a certain large automorphism group, and discuss examples for small q .

Article information

Source
Innov. Incidence Geom., Volume 11, Number 1 (2010), 19-33.

Dates
Received: 27 March 2007
Accepted: 22 June 2009
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323082

Digital Object Identifier
doi:10.2140/iig.2010.11.19

Mathematical Reviews number (MathSciNet)
MR2795055

Zentralblatt MATH identifier
1266.51005

Subjects
Primary: 51E12: Generalized quadrangles, generalized polygons 51E23: Spreads and packing problems

Keywords
partial ovoid partial spread generalized quadrangle

Citation

De Winter, Stefaan; Thas, Koen. Bounds on partial ovoids and spreads in classical generalized quadrangles. Innov. Incidence Geom. 11 (2010), no. 1, 19--33. doi:10.2140/iig.2010.11.19. https://projecteuclid.org/euclid.iig/1551323082


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References

  • A. Aguglia, A. Cossidente and G. Ebert, Complete spans on Hermitian varieties, Designs Codes Cryptogr. 21 (2003), 7–15.
  • A. Aguglia, G. Ebert and D. Luyckx, On partial ovoids of Hermitian surfaces, Bull. Belg. Math. Soc. | Simon Stevin 12 (2005), 641–650.
  • S. Ball, On ovoids of $O(5,q)$, Adv. Geom. 4 (2004), 1–7.
  • M. Cimrakova, S. De Winter, V. Fack and L. Storme, On the smallest maximal partial ovoids and spreads of the generalized quadrangles $\hW(q)$ and $\cQ(4,q)$, European J. Combin. 28 (2007), 1934–1942.
  • M. Cimrakova and V. Fack, Searching for maximal partial ovoids and spreads in generalized quadrangles, Bull. Belg. Math. Soc. | Simon Stevin 12 (2005), 697–705.
  • J. De Beule and A. Gács, Complete arcs on the parabolic quadric ${\rm Q}(4,q)$, Finite Fields Appl. 14 (2008), 14–21.
  • G. Ebert and J. Hirschfeld, Complete systems of lines on a Hermitian surface over a finite field, Des. Codes Crytogr. 17 (1999), 253–268.
  • D. Glynn, A lower bound for maximal partial spreads in $\PG(3,q)$, Ars Combin. 13 (1982), 39–40.
  • J. Hirschfeld and G. Korchmáros, Caps on Hermitian varieties and maximal curves, Adv. Geom. (2003), S206–S214.
  • B. Huppert, Endliche Gruppen I, Grundlehren Math. Wiss., Springer Verlag, Berlin, Heidelberg, New York, 1967.
  • K. Metsch, Small maximal partial ovoids of $H(3,q\sp 2)$, Innov. Incidence Geom. 3 (2006), 1–12.
  • K. Metsch and L. Storme, Maximal partial ovoids and maximal partial spreads in Hermitian generalized quadrangles, J. Combin. Des. 16 (2008), 101–116.
  • S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, Res. Notes Math. 110, Pitman Advanced Publishing Program, Boston/London/Melbourne, 1984.
  • T. Penttila, Complete $(q^2-1)$-arcs in $\cQ(4,q)$, $q=5,7,11$, Private communication, September 2003.
  • G. Tallini, Blocking sets with respect to planes of $\PG(3,q)$ and maximal spreads of a non-singular quadric of $\cQ(4,q)$, Mitt. Math. Sem. Giessen 201 (1991), 141–147.
  • J. A. Thas, Ovoidal translation planes, Arch. Math. 23 (1972), 110–112.
  • ––––, Semipartial geometries and spreads of classical polar spaces, J. Combin. Theory Ser. A 35 (1983), 58–66.
  • ––––, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1984), 135–144.
  • ––––, Flocks, maximal exterior sets, and inversive planes, in Finite Geometries and Combinatorial Designs, Contemp. Math. 111 (1990), pp. 187–218.
  • J. A. Thas and S. E. Payne, Spreads and ovoids of finite generalized quadrangles, Geom. Dedicata 52 (1994), 227–253.
  • K. Thas, Nonexistence of complete $(st-t/s)$-arcs in generalized quadrangles of order $(s,t)$, I, J. Combin. Theory Ser. A 97 (2002), 394–402.
  • ––––, Symmetry in Finite Generalized Quadrangles, Front. Math. 1, Birkhäuser-Verlag, Basel|Boston|Berlin, 2004.
  • J. Tits, Sur la trialité et certains groupes qui s'en déduisent, Ins. Hautes Etudes Sci. Publ. Math. 2 (1959), 13–60.