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Primitive Ovoids in O+8(q)

2000
*
Journal of combinatorial theory. Series A
*

Let

doi:10.1006/jcta.1999.3004
fatcat:luzazmk5j5gmhl6y2ri6tt4pee
*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. ...##
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Generalizing flocks of Q+(3,q)

2001
*
Advances in Geometry
*

We define flocks of Segre varieties S n; n as a generalization of flocks of

doi:10.1515/advg.2001.019
fatcat:szzvbnowpbdlzezv3cjkzwixve
*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 ...##
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On m-ovoids of W3(q)

2008
*
Finite Fields and Their Applications
*

For

doi:10.1016/j.ffa.2006.04.001
fatcat:li4rxceb6fggnkxb3uvfpg4qcm
*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. ...##
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Linear codes associated with the Desarguesian ovoids in Q^+(7,q)
[article]

2022
*
arXiv
*
pre-print

The shortest PGL(2,

arXiv:2208.12919v1
fatcat:oqjp4i5xbrayrorcqp5eeabfhy
*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 . ...##
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New families of Q-polynomial association schemes

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 . ...

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

2021
*
Annals of Combinatorics
*

We also obtain a computer classification of all locally Hermitian

doi:10.1007/s00026-021-00538-3
fatcat:s5e3oavw4neptl47g4gtascxj4
*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 ...##
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Common point reguli of different generalized hexagons on Q(6,q)

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. ...

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

2011
*
Journal of combinatorial theory. Series A
*

Specifically, we show that for a graph AT4(qs,

doi:10.1016/j.jcta.2010.10.001
fatcat:mzh7g2onifa7lcxbgf24kp3co4
*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 ...##
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Transitive ovoids of the Hermitian surface of PG(3,q2), q even

2003
*
Journal of combinatorial theory. Series A
*

A 89 (2000) 70) who determined the

doi:10.1016/s0097-3165(02)00021-3
fatcat:shihiwhma5hkxh4hhvcb7anxcm
*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. ...##
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On Twisted Tensor Product Group Embeddings and the Spin Representation of Symplectic Groups: The Case q Odd

2011
*
ISRN Geometry
*

It is the full stabilizer of a complete partial

doi:10.5402/2011/694605
fatcat:od37ka3obfdu5mrrbiejuc3btu
*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. ...##
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On putative q-Analogues of the Fano Plane and Related Combinatorial Structures
[article]

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. ...

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

2019
*
Finite Fields and Their Applications
*

Let

doi:10.1016/j.ffa.2018.11.002
fatcat:nszabsrc4rgwhddfe53tm65tua
*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*...##
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Hyperovals on H(3,q2)

2011
*
Journal of combinatorial theory. Series A
*

A construction of an irreducible 2-

doi:10.1016/j.jcta.2010.11.016
fatcat:r6kcr7k5yzc6bmeccgjaup4tn4
*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. ...##
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Embedding of orthogonal Buekenhout-Metz unitals in the Desarguesian plane of order q^2

2019
*
Ars Mathematica Contemporanea
*

A unital, that is a 2-(

doi:10.26493/1855-3974.1681.4ec
fatcat:ys5jiqbaazfcfcmsqfr5c3qsdu
*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*. ...##
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A unified construction of finite geometries associated with q-clans in characteristic 2

2003
*
Advances in Geometry
*

Examples are rare

doi:10.1515/advg.2003.002
fatcat:ry4om5lsyjedfizbuxk6pw4cuu
*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. ...
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