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Primitive Ovoids in O+8(q)

Athula Gunawardena
2000 Journal of combinatorial theory. Series A  
Let O be a primitive ovoid in O + 8 (q) space. Then O is either the Cooperstein ovoid in O + 8(5), or a 2-transitive ovoid (and so appears in Kleidman's list [12] ).  ...  All the primitive ovoids in O + 8 (q) are shown in 7 . 7 G 0 =PSU 3 (3) and q=3.8. G 0 =PSL 4 (2)$A 8 and q=3.9.  ... 
doi:10.1006/jcta.1999.3004 fatcat:luzazmk5j5gmhl6y2ri6tt4pee

Generalizing flocks of Q+(3,q)

Laura Bader, Antonio Cossidente, Guglielmo Lunardon
2001 Advances in Geometry  
We define flocks of Segre varieties S n; n as a generalization of flocks of Q þ ð3; qÞ, studying the connections with translation planes.  ...  the polarity defined by Q þ ð7; qÞ. The 3dimensional space hp 0 U p 00 i intersects Q þ ð7; qÞ in some Q À ð3; qÞ, as O is an ovoid, hence hp 0 U p 00 i ?  ...  The polar plane of each plane of the flock intersects Q þ ð5; qÞ in an irreducible conic containing the points a and b, the union of these conics is an ovoid O, and the Klein correspondence f maps O to  ... 
doi:10.1515/advg.2001.019 fatcat:szzvbnowpbdlzezv3cjkzwixve

On m-ovoids of W3(q)

A. Cossidente, C. Culbert, G.L. Ebert, G. Marino
2008 Finite Fields and Their Applications  
For q even, we show that W 3 (q) has m-ovoids for all integers m, 1 m q. Stabilizers are determined, and some computer results are given.  ...  We show that the generalized quadrangle W 3 (q) for odd q has exponentially many 1 2 (q + 1)-ovoids, thus implying that the generalized quadrangle Q(4, q) has exponentially many hemisystems for odd q.  ...  While some of the independently obtained results in this paper overlap work done in [2] , the approaches are completely different.  ... 
doi:10.1016/j.ffa.2006.04.001 fatcat:li4rxceb6fggnkxb3uvfpg4qcm

Linear codes associated with the Desarguesian ovoids in Q^+(7,q) [article]

Tao Feng, Michael Kiermaier, Peixian Lin, Kai-Uwe Schmidt
2022 arXiv   pre-print
The shortest PGL(2,q^3)-orbit O gives the Desarguesian ovoid in Q^+(7,q) for even q and it is known to give a complete partial ovoid of the symplectic polar space W(7,q) for odd q.  ...  The Desarguesian ovoids in the orthogonal polar space Q^+(7,q) with q even have first been introduced by Kantor by examining the 8-dimensional absolutely irreducible modular representations of PGL(2,q^  ...  x q+q 2 ), N(x)) ⊤ with θ being a primitive element of F q 3 .  ... 
arXiv:2208.12919v1 fatcat:oqjp4i5xbrayrorcqp5eeabfhy

New families of Q-polynomial association schemes

Tim Penttila, Jason Williford
2011 Journal of combinatorial theory. Series A  
In this paper, we construct the first known infinite family of primitive Q -polynomial schemes which are not generated by distanceregular graphs.  ...  The first result (Corollary 2.2 in [2] ) is that the size of the intersection of two subtended ovoids O x and O y is either r 2 + 1, 1 or r + 1 depending only on whether x and y subtend the same ovoid  ...  in S. • R 4 : We have (x, y) ∈ R 4 if and only if O x = O y .  ... 
doi:10.1016/j.jcta.2010.08.001 fatcat:dkobd7srsfdlxhtipthsgobmlm

On Ovoids of the Generalized Quadrangle $$H(3,q^2)$$

Bart De Bruyn
2021 Annals of Combinatorics  
We also obtain a computer classification of all locally Hermitian ovoids of H(3, q 2 ) for q ≤ 4, and compare the obtained classification for q = 3 with the classification of all ovoids of H(3, 9) which  ...  We construct examples and families of locally Hermitian ovoids of the generalized quadrangle H(3, q 2 ).  ...  + x2The ovoids of S q , q ≤ 4 Ovoid q # Intersection Pattern f (x) O 1 2 8 0 3 1 4 4 1 αx O 2 3 54 0 8 1 45 9 1 βx O 3 3 108 0 24 1 18 3 12 βx 3 O 4 3 486 0 24 1 12 2 16 5 2 β 2 x 5 O 5 4 192 0 15 1 176  ... 
doi:10.1007/s00026-021-00538-3 fatcat:s5e3oavw4neptl47g4gtascxj4

Common point reguli of different generalized hexagons on Q(6,q)

A. De Wispelaere, H. Van Maldeghem
2007 European journal of combinatorics (Print)  
In this paper, we consider any two split Cayley generalized hexagons represented on the parabolic quadric Q(6, q) and determine their common point reguli.  ...  Van Maldeghem, Ovoids and spreads of the generalized hexagon H(3), Discrete Math. 305 (1-3) (2005) 299-311], are a spread of some hexagon on this quadric.  ...  Dually, one defines a dual ovoidal subspace. An ovoid, short for distance-3 ovoid, of H(q) is a set of q 3 + 1 opposite points (Proposition 7.2.3 in [8] ). Dually one defines a spread.  ... 
doi:10.1016/j.ejc.2006.08.005 fatcat:c46yga3ugnak7d2h4gnnp7oo64

Classification of the family AT4(qs,q,q) of antipodal tight graphs

Aleksandar Jurišić, Jack Koolen
2011 Journal of combinatorial theory. Series A  
Specifically, we show that for a graph AT4(qs, q, q) there are exactly five possibilities for the pair (s, q), with an example for each: the Johnson graph J (8, 4) for (1, 2), the halved 8-cube for (2,  ...  2), the 3.O − 6 (3) graph for (1, 3) , the Meixner2 graph for (2, 4) and the 3.O 7 (3) graph for (3, 3).  ...  The Meixner2 graph is one of the examples of AT4(8, 4, 4) in (ii) and the 3.O 7 (3) graph is one of the examples of AT4(9, 3, 3Case  ... 
doi:10.1016/j.jcta.2010.10.001 fatcat:mzh7g2onifa7lcxbgf24kp3co4

Transitive ovoids of the Hermitian surface of PG(3,q2), q even

A. Cossidente, G. Korchmáros
2003 Journal of combinatorial theory. Series A  
A 89 (2000) 70) who determined the primitive ovoids of the quadric O þ 8 ðqÞ: Transitive ovoids of the classical polar space arising from the Hermitian surface Hð3; q 2 Þ of PGð3; q 2 Þ with even q are  ...  Both are linearly transitive in the sense that the subgroup of PGUð4; q 2 Þ preserving the ovoid is still transitive on it.  ...  Transitive ovoids of other classical polar spaces have been investigated in [8] , [14] , and [23] . Theorem 1.1.  ... 
doi:10.1016/s0097-3165(02)00021-3 fatcat:shihiwhma5hkxh4hhvcb7anxcm

On Twisted Tensor Product Group Embeddings and the Spin Representation of Symplectic Groups: The Case q Odd

Antonio Cossidente
2011 ISRN Geometry  
It is the full stabilizer of a complete partial ovoid and of a complete partial 3-spread of 𝒲7(q).  ...  The group PSp8(q), q odd, has a maximal subgroup isomorphic to 3.PSp2(q3) belonging to the Aschbacher class 𝒞9.  ...  Moreover, N P Ω 8 q PSp 2 q 3 is the stabilizer of O in P Ω 8 q and it is a maximal subgroup of PSp 6 q . Remark 3.2.  ... 
doi:10.5402/2011/694605 fatcat:od37ka3obfdu5mrrbiejuc3btu

On putative q-Analogues of the Fano Plane and Related Combinatorial Structures [article]

Thomas Honold, Michael Kiermaier
2015 arXiv   pre-print
in precisely one member of F_q.  ...  The existence problem for such q-analogues remains unsolved for every single value of q.  ...  It follows that each plane E is tangent to a unique ovoid in O and meets the remaining q ovoids in q + 1 points.  ... 
arXiv:1504.06688v1 fatcat:7ej37ev3yzewdbfmnmjfzgvimm

On minimum size blocking sets of the outer tangents to a hyperbolic quadric in PG(3,q)

Bart De Bruyn, Binod Kumar Sahoo
2019 Finite Fields and Their Applications  
Let Q + (3, q) be a hyperbolic quadric in PG(3, q) and T 1 be the set of all lines of PG(3, q) meeting Q + (3, q) in singletons (the so-called outer tangents).  ...  If k is the minimum size of a T 1 -blocking set in PG(3, q), then we prove that k ≥ q 2 − 1.  ...  Acknowledgements The first author would like to thank the National Institute of Science Education and Research, Bhubaneswar for the kind hospitality provided during his visit to the School of Mathematical Sciences in  ... 
doi:10.1016/j.ffa.2018.11.002 fatcat:nszabsrc4rgwhddfe53tm65tua

Hyperovals on H(3,q2)

Antonio Cossidente
2011 Journal of combinatorial theory. Series A  
A construction of an irreducible 2-ovoid of H(3, 25) and some results on packings of the Hermitian curve are given.  ...  Infinite families of hyperovals of the generalized quadrangle H(3, q 2 ) are constructed. Some sporadic examples are also presented.  ...  Given a classical ovoid O of H(3, q 2 ), choose two distinct points P 1 and P 2 on O. Then the line through P 1 and P 2 meets O at q + 1 points.  ... 
doi:10.1016/j.jcta.2010.11.016 fatcat:r6kcr7k5yzc6bmeccgjaup4tn4

Embedding of orthogonal Buekenhout-Metz unitals in the Desarguesian plane of order q^2

Gábor Korchmáros, Alessandro Siciliano
2019 Ars Mathematica Contemporanea  
A unital, that is a 2-(q 3 + 1, q + 1, 1) block-design, is embedded in a projective plane π of order q 2 if its points are points of π and its blocks are subsets of lines of π, the point-block incidences  ...  being the same as in π.  ...  By definition, the Buekenhout representation U of U is a cone that project an ovoid O from a point of φ(P ∞ ) not in O.  ... 
doi:10.26493/1855-3974.1681.4ec fatcat:ys5jiqbaazfcfcmsqfr5c3qsdu

A unified construction of finite geometries associated with q-clans in characteristic 2

William E. Cherowitzo, Christine M. O'Keefe, Tim Penttila
2003 Advances in Geometry  
Examples are rare in the case of characteristic two, and it is the purpose of this paper to contribute a fifth infinite family.  ...  Flocks of Laguerre planes, generalized quadrangles, translation planes, ovals, BLTsets, and the deep connections between them, are at the core of a developing theory in the area of geometry over finite  ...  V Q þ ð5; qÞ is an ovoid of Q þ ð5; qÞ, and we let S be the spread of PGð3; qÞ associated with W via the Klein correspondence.  ... 
doi:10.1515/advg.2003.002 fatcat:ry4om5lsyjedfizbuxk6pw4cuu
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