# American Institute of Mathematical Sciences

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$)$

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

Received  March 2010 Revised  August 2010 Published  May 2011

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$)$.
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. [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. [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. [4] J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'' The Clarendon Press, Oxford University Press, New York, 1991. [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. [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. [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.

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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. [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. [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. [4] J. W. P. Hirschfeld and J. A. Thas, "General Galois Geometries,'' The Clarendon Press, Oxford University Press, New York, 1991. [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. [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. [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.
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