# 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.  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.  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.  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, (1991).   Google Scholar [5] D. Luyckx, On maximal partial spreads of $H(2n+1,q^2)$,, Discrete Math., 308 (2008), 375.  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.  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.   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.  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.  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.  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, (1991).   Google Scholar [5] D. Luyckx, On maximal partial spreads of $H(2n+1,q^2)$,, Discrete Math., 308 (2008), 375.  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.  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.   Google Scholar
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