Innovations in Incidence Geometry

Small maximal partial ovoids of $H(3,q^2)$

Klaus Metsch

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Abstract

The trivial lower bound for the size of a maximal partial ovoid of H ( 3 , q 2 ) is q 2 + 1 . Ebert showed that this bound can be attained if and only if q is even. In the present paper it is shown that a maximal partial ovoid of H ( 3 , q 2 ) , q odd, has at least q 2 + 1 + 4 9 q points (previously, only q 2 + 3 was known). It is also shown that a maximal partial spread of H ( 3 , q 2 ) , q even, has size q 2 + 1 or size at least q 2 + 1 + 4 9 q .

Article information

Source
Innov. Incidence Geom., Volume 3, Number 1 (2006), 1-12.

Dates
Received: 14 March 2006
Accepted: 8 May 2006
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323233

Digital Object Identifier
doi:10.2140/iig.2006.3.1

Mathematical Reviews number (MathSciNet)
MR2267602

Zentralblatt MATH identifier
1108.51015

Subjects
Primary: 05B25: Finite geometries [See also 51D20, 51Exx] 51E12: Generalized quadrangles, generalized polygons 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx]

Keywords
ovoid polar space Hermitian variety

Citation

Metsch, Klaus. Small maximal partial ovoids of $H(3,q^2)$. Innov. Incidence Geom. 3 (2006), no. 1, 1--12. doi:10.2140/iig.2006.3.1. https://projecteuclid.org/euclid.iig/1551323233


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References

  • M. Cimráková, Search algorithms for substructures in generalized quadrangles, Ph.D. thesis, Ghent University, 2006.
  • A. Cossidente and G. Korchmáros, Transitive ovoids of the Hermitian surface of ${\rm PG}(3,q\sp 2)$, $q$ even, J. Combin. Theory Ser. A 101 (2003), 117–130.
  • A. Aguglia, A. Cossidente and G. L. Ebert, Complete spans on Hermitian varieties, Des. Codes Cryptogr. 29 (2003), 7–15.
  • A. Aguglia, G. L. Ebert and D. Luyckx, On partial ovoids of Hermitian surfaces, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 5, 641–650.
  • A. Klein and K. Metsch, New results on covers and partial spreads of polar spaces, Innov. Incidence Geom. 1 (2005), 19–34.
  • M. R. Brown, J. De Beule and L. Storme, Maximal partial spreads of $T_2(O)$ and $T_3(O)$, Europ. J. Combin. 24 (2003), 73–84.