Finite Generalized Quadrangles

Payne, Sam; Thas, J. A.

doi:10.4171/066

Cited by 535 publications

(562 citation statements)

References 122 publications

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“…Note that this proof is very similar to the one in [14]. We give it here for sake of completeness and as an illustration of the usefulness of tight sets and m-ovoids.…”

Section: Corollary 67 Let G Be a Generalised Quadrangle Of Order (Smentioning

confidence: 53%

“…The following corollary also follows from [13,II.4]. It applies in particular to W(3, q), since for all pairs {x, y} of non-collinear points |{x, y} ⊥⊥ | = q + 1 (cf., [14, 3.3…”

Section: Theorem 44 Let X and Y Be Two Non-collinear Points Of A Genmentioning

confidence: 89%

“…A set of points T of a generalised quadrangle of order (s, t) is an i-tight set if for every point p in T , there are s + i points of T collinear with p, and for every point p not in T , there are i points of T collinear with p (see [13]). So…”

Section: Tight Setsmentioning

confidence: 99%

“…Thas [16] and S.E. Payne [13], respectively. An m-ovoid and an i-tight set intersect in mi points [3,Theorem 4.3], and from this observation, one can prove or reprove interesting results about generalised quadrangles.…”

Section: Introductionmentioning

confidence: 99%

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Weighted intriguing sets of finite generalised quadrangles

2011

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We construct and analyse interesting integer valued functions on the points of a generalised quadrangle which lie in the orthogonal complement of a principal eigenspace of the collinearity relation. These functions generalise the intriguing sets introduced by Bamberg et al. (Combinatorica 29(1):1-17, 2009), and they provide the extra machinery to give new proofs of old results and to establish new insight into the existence of certain configurations of generalised quadrangles. In particular, we give a geometric characterisation of Payne's tight sets, we give a new proof of Thas' result that an m-ovoid of a generalised quadrangle of order (s, s 2 ) is a hemisystem, and we give a bound on the values of m for which it is possible for an m-ovoid of the four dimensional Hermitian variety to exist.

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“…Note that this proof is very similar to the one in [14]. We give it here for sake of completeness and as an illustration of the usefulness of tight sets and m-ovoids.…”

Section: Corollary 67 Let G Be a Generalised Quadrangle Of Order (Smentioning

confidence: 53%

“…The following corollary also follows from [13,II.4]. It applies in particular to W(3, q), since for all pairs {x, y} of non-collinear points |{x, y} ⊥⊥ | = q + 1 (cf., [14, 3.3…”

Section: Theorem 44 Let X and Y Be Two Non-collinear Points Of A Genmentioning

confidence: 89%

Section: Tight Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weighted intriguing sets of finite generalised quadrangles

2011

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“…Tight sets in generalized quadrangles were introduced by Payne in [14]. These are subsets of points such that the number of pairs of collinear points in it reaches a certain upper bound.…”

Section: The Regular Semilattice Associated With the Polar Spacementioning

confidence: 99%

Antidesigns and regularity of partial spreads in dual polar graphs

Vanhove

2010

J. Combin. Designs

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We give several examples of designs and antidesigns in classical finite polar spaces. These types of subsets of maximal totally isotropic subspaces generalize the dualization of the concepts of m-ovoids and tight sets of points in generalized quadrangles. We also consider regularity of partial spreads and spreads. The techniques that we apply were developed by Delsarte. In some polar spaces of small rank, some of these subsets turn out to be completely regular codes.

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Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

Metsch¹,

Storme²

2007

J of Combinatorial Designs

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Maximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3, q 2 ) and H(4, q 2 ) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4, q 2 ). We also construct in H(3, q 2 ), q = 2 2h+1 , h ≥ 1, maximal partial spreads of size smaller than the size q 2 + 1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4, q 2 ) of small positive deficiency δ.

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Finite Generalized Quadrangles

Cited by 535 publications

References 122 publications

Weighted intriguing sets of finite generalised quadrangles

Weighted intriguing sets of finite generalised quadrangles

Antidesigns and regularity of partial spreads in dual polar graphs

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

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