A geometric proof of the upper bound on the size of partial spreads in H(4n+1,<I>q</I><SUP><I>2</I></SUP>)

Previous Article
On $q$-analogs of Steiner systems and covering designs
AMC Home
This Issue
Next Article
On the structure of non-full-rank perfect $q$-ary codes

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
Related Papers

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.
[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.

show all references

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.

[1]	V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
[2]	Lawrence Ein, Wenbo Niu, Jinhyung Park. On blowup of secant varieties of curves. Electronic Research Archive, 2021, 29 (6) : 3649-3654. doi: 10.3934/era.2021055
[3]	Ian Blake, V. Kumar Murty, Hamid Usefi. A note on diagonal and Hermitian hypersurfaces. Advances in Mathematics of Communications, 2016, 10 (4) : 753-764. doi: 10.3934/amc.2016039
[4]	Anton Izosimov. Pentagrams, inscribed polygons, and Prym varieties. Electronic Research Announcements, 2016, 23: 25-40. doi: 10.3934/era.2016.23.004
[5]	G. Mashevitzky, B. Plotkin and E. Plotkin. Automorphisms of categories of free algebras of varieties. Electronic Research Announcements, 2002, 8: 1-10.
[6]	A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.
[7]	Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109
[8]	Aravind Asok, James Parson. Equivariant sheaves on some spherical varieties. Electronic Research Announcements, 2011, 18: 119-130. doi: 10.3934/era.2011.18.119
[9]	Ghislain Fourier, Gabriele Nebe. Degenerate flag varieties in network coding. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021027
[10]	Jesús Carrillo-Pacheco, Felipe Zaldivar. On codes over FFN$(1,q)$-projective varieties. Advances in Mathematics of Communications, 2016, 10 (2) : 209-220. doi: 10.3934/amc.2016001
[11]	Sylvain E. Cappell, Anatoly Libgober, Laurentiu Maxim and Julius L. Shaneson. Hodge genera and characteristic classes of complex algebraic varieties. Electronic Research Announcements, 2008, 15: 1-7. doi: 10.3934/era.2008.15.1
[12]	Fabrizio Catanese, Luca Cesarano. Canonical maps of general hypersurfaces in Abelian varieties. Electronic Research Archive, 2021, 29 (6) : 4315-4325. doi: 10.3934/era.2021087
[13]	Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2037-2053. doi: 10.3934/dcdsb.2020365
[14]	Seung-Yeal Ha, Myeongju Kang, Hansol Park. Collective behaviors of the Lohe Hermitian sphere model with inertia. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2613-2641. doi: 10.3934/cpaa.2021046
[15]	Alexander Gorodnik, Frédéric Paulin. Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. Journal of Modern Dynamics, 2014, 8 (1) : 25-59. doi: 10.3934/jmd.2014.8.25
[16]	Eric Férard. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 497-509. doi: 10.3934/amc.2014.8.497
[17]	Yang Cao, Wei- Wei Tan, Mei-Qun Jiang. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 811-821. doi: 10.3934/naco.2012.2.811
[18]	Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054
[19]	Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991
[20]	Vincent Astier, Thomas Unger. Signatures, sums of hermitian squares and positive cones on algebras with involution. Electronic Research Announcements, 2018, 25: 16-26. doi: 10.3934/era.2018.25.003

2020 Impact Factor: 0.935

Tools

Metrics

Other articles
by authors

on AIMS
Frédéric Vanhove
on Google Scholar
Frédéric Vanhove

[Back to Top]