Hemisystems of the Hermitian surface

Korchmáros, Gábor; Nagy, Gábor P.; Speziali, Pietro

doi:10.1016/j.jcta.2018.11.008

Cited by 17 publications

(11 citation statements)

References 14 publications

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“…Therefore mregular systems of Q − (5, q) and m-ovoids of H(3, q 2 ) are equivalent objects and m-regular systems of H(3, q 2 ) and m-ovoids of Q − (5, q) are equivalent objects. It is a well known result that regular systems of H(3, q 2 ) are hemisystems and several examples are known [4,8,18,19,40]. On the other hand, m-ovoids of H(3, q 2 ) exist for all possible m. Indeed, H(3, q 2 ) admits a fan, i.e., a partition of the pointset into ovoids.…”

Section: Chains Of Regular Systemsmentioning

confidence: 99%

“…If k = 1 and the rank d of the polar space is at least three, then some constructions are known on the parabolic quadric Q(2d, q) [18,41] and on the Hermitian variey H(2d − 1, q) [10,20]. Finally since polar spaces of rank two are generalized quadrangles, several examples of regular systems arise from m-ovoids by using duality, see [2,4,8,7,15,16,18,19,33,34,40,44,43]. However, in the case of generalized quadrangles, many questions are still unsolved.…”

Section: Introductionmentioning

confidence: 99%

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On regular systems of finite classical polar spaces

Cossidente¹,

Marino²,

Pavese³

et al. 2021

Preprint

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Let P be a finite classical polar space of rank d. An m-regular system with respect to (k − 1)-dimensional projective spaces of P, 1 ≤ k ≤ d − 1, is a set R of generators of P with the property that every (k − 1)-dimensional projective space of P lies on exactly m generators of R. Regular systems of polar spaces are investigated. Some non-existence results about certain 1-regular systems of polar spaces with low rank are proved and a procedure to obtain m ′ -regular systems from a given m-regular system is described. Finally, three different construction methods of regular systems w.r.t. points of various polar spaces are discussed.

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Section: Chains Of Regular Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On regular systems of finite classical polar spaces

Cossidente¹,

Marino²,

Pavese³

et al. 2021

Preprint

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“…Such a set of lines in H(3, q 2 ) is called a hemisystem, which was first studied by Segre [91]. Constructions of hemisystems can be found in [4,7,29,63].…”

Section: 5mentioning

confidence: 99%

9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

Momihara¹,

Wang²,

Xiang³

2019

Combinatorics and Finite Fields

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In this paper, we survey constructions of and nonexistence results on combinatorial/geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on difference sets related to cyclotomy, and cyclotomic constructions of sequences with low correlation. We also give an extensive survey of recent results on constructions of strongly regular Cayley graphs and related geometric substructures such as m-ovoids and i-tight sets in classical polar spaces.

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“…In [7], by extending Segre's result, it was shown that Q − (5, q), q even, possesses no non-trivial m-ovoids, see also [18,Section 19.3]. Several constructions of hemisystems of Q − (5, q) have been presented in the literature in the last fifteen years [1,2,4,9,10,23]. However, not much is known about m-ovoids of Q − (2r + 1, q) for r > 2.…”

mentioning

confidence: 93%

A modular equality for $m$-ovoids of elliptic quadrics

Gavrilyuk¹,

Metsch²,

Pavese³

2021

Preprint

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An m-ovoid of a finite polar space P is a set O of points such that every maximal subspace of P contains exactly m points of O. In the case when P is an elliptic quadric Q − (2r + 1, q) of rank r in F 2r+2 q , we prove that an m-ovoid exists only if m satisfies a certain modular equality, which depends on q and r. This condition rules out many of the possible values of m. Previously, only a lower bound on m was known, which we slightly improve as a byproduct of our method. We also obtain a characterization of the m-ovoids of Q − (7, q) for q = 2 and (m, q) = (4, 3).

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Hemisystems of the Hermitian surface

Cited by 17 publications

References 14 publications

On regular systems of finite classical polar spaces

On regular systems of finite classical polar spaces

9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

A modular equality for $m$-ovoids of elliptic quadrics

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