A geometric proof of the upper bound on the size of partial spreads in $H(4n+1,$q2$)$

Frédéric Vanhove

doi:10.3934/amc.2011.5.157

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May 2011, 5(2): 157-160. doi: 10.3934/amc.2011.5.157

A geometric proof of the upper bound on the size of partial spreads in $H(4n+1,$q²$)$

Frédéric Vanhove ^1,

1.	Department of Mathematics, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium

Received March 2010 Revised August 2010 Published May 2011

Abstract
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We give a geometric proof of the upper bound of q²ⁿ⁺¹$+1$ on the size of partial spreads in the polar space $H(4n+1,$q²$)$. This bound is tight and has already been proved in an algebraic way. Our alternative proof also yields a characterization of the partial spreads of maximum size in $H(4n+1,$q²$)$.

Keywords: Hermitian varieties, partial spreads..

Mathematics Subject Classification: 05B25, 51E2.

Citation: Frédéric Vanhove. A geometric proof of the upper bound on the size of partial spreads in $H(4n+1,$q²$)$. Advances in Mathematics of Communications, 2011, 5 (2) : 157-160. doi: 10.3934/amc.2011.5.157

References:

[1]	A. Aguglia, A. Cossidente and G. L. Ebert, Complete spans on Hermitian varieties, Des. Codes Cryptogr., 29 (2003), 7-15. doi: 10.1023/A:1024179703511. Google Scholar
[2]	J. De Beule, A. Klein, K. Metsch and L. Storme, Partial ovoids and partial spreads in Hermitian polar spaces, Des. Codes Cryptogr., 47 (2008), 21-34. doi: 10.1007/s10623-007-9047-8. Google Scholar
[3]	J. De Beule and K. Metsch, The maximum size of a partial spread in $H(5,q^2)$ is $q^3+1$, J. Combin. Theory Ser. A, 114 (2007), 761-768. doi: 10.1016/j.jcta.2006.08.005. Google Scholar
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References:

[1]	A. Aguglia, A. Cossidente and G. L. Ebert, Complete spans on Hermitian varieties, Des. Codes Cryptogr., 29 (2003), 7-15. doi: 10.1023/A:1024179703511. Google Scholar
[2]	J. De Beule, A. Klein, K. Metsch and L. Storme, Partial ovoids and partial spreads in Hermitian polar spaces, Des. Codes Cryptogr., 47 (2008), 21-34. doi: 10.1007/s10623-007-9047-8. Google Scholar
[3]	J. De Beule and K. Metsch, The maximum size of a partial spread in $H(5,q^2)$ is $q^3+1$, J. Combin. Theory Ser. A, 114 (2007), 761-768. doi: 10.1016/j.jcta.2006.08.005. Google Scholar
[4]	J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'' The Clarendon Press, Oxford University Press, New York, 1991. Google Scholar
[5]	D. Luyckx, On maximal partial spreads of $H(2n+1,q^2)$, Discrete Math., 308 (2008), 375-379. doi: 10.1016/j.disc.2006.11.051. Google Scholar
[6]	J. A. Thas, Old and new results on spreads and ovoids of finite classical polar spaces, Ann. Discrete Math., 52 (1992), 529-544. doi: 10.1016/S0167-5060(08)70936-1. Google Scholar
[7]	F. Vanhove, The maximum size of a partial spread in $H(4n+1,q^2)$ is q²ⁿ⁺¹$+1$, Electron. J. Combin., 16 (2009), 1-6. Google Scholar

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A geometric proof of the upper bound on the size of partial spreads in $H(4n+1,$q²$)$

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