Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

Beule, Jan De; Govaerts, Patrick; Hallez, Anja; Storme, Leo

doi:10.1007/s10623-008-9223-5

Cited by 14 publications

(14 citation statements)

References 24 publications

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“…In this section we present the characterization of i-tight sets in the Hermitian variety H(2r + 1, q) and in the symplectic polar space W (2r + 1, q) when i < (q 2/3 − 1)/2. This result is the improvement of the result from [13] where the upper bound on i was q 5/8 / √ 2 + 1. Combining the generalized version of known techniques [26] with recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q 2/3 − 1)/2.…”

Section: Tight Sets In Finite Classical Polar Spacesmentioning

confidence: 48%

Tight sets in finite classical polar spaces

Nakić

Storme

2017

Advances in Geometry

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We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries $\text{PG}(2r+1,\,\sqrt{q})$ and generators of H(2r + 1, q), if q ≥ 81 is an odd square and i < (q2/3 − 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Δ, Δ⊥}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Δ, Δ⊥} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity ⊥ of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where $i \lt q^{5/8} / \sqrt{2} +1$ was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q2/3 − 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).

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Section: Tight Sets In Finite Classical Polar Spacesmentioning

confidence: 48%

Tight sets in finite classical polar spaces

Nakić

Storme

2017

Advances in Geometry

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“…Result 1 has more applications. We mention the following one, which improves [4,Corollary 5.10]. For this we consider a map w of the line set of Q (4, q) to the set of non-negative integers.…”

Section: Another Application Of the Results On Blocking Sets In Quadricsmentioning

confidence: 99%

The non-existence of Cameron–Liebler line classes with parameter 2 <x≤q

Metsch

2010

Bull. London Math. Soc.

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“…In [19], it has been shown that an i-tight set T of W (2r +1, q), Q + (2r +1, q), or H(2r+1, q) is a set of iθ r points intersecting every hyperplane in at least iθ r−1 points, hence T is an (iθ r , iθ r−1 ; 2r + 1, q)-minihyper. Using characterization results about minihypers of [19] and [29], it is possible to characterize i-tight sets in the aforementioned polar spaces.…”

Section: Minihypers and I-tight Setsmentioning

confidence: 99%

“…Using characterization results about minihypers of [19] and [29], it is possible to characterize i-tight sets in the aforementioned polar spaces.…”

Section: Minihypers and I-tight Setsmentioning

confidence: 99%

The Use of Blocking Sets in Galois Geometries and in Related Research Areas

Pepe

Storme

2011

Buildings, Finite Geometries and Groups

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Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems. Definitions and introductory resultsA set B of points of PG(n, q) is called a k-blocking set if every (n − k)-dimensional subspace of PG(n, q) has a non-empty intersection with B, and a set B of points of PG(n, q) is called a t-fold k-blocking set if every (n − k)-dimensional subspace contains at least t points of B. In PG(2, q), the 1-blocking sets are simply called blocking sets. Two subspaces Σ 1 and Σ 2 of PG(n, q) of dimension k and n − k always have a non-empty intersection, hence a set B containing a k-dimensional subspace is called a trivial k-blocking set. Moreover, a k-blocking set B is called minimal if it is minimal with respect to the containment relation, i.e. B \ {P } is not a k-blocking set for every P ∈ B, and it is called small if |B| < 3(q k +1) 2 . The blocking sets of PG(2, q) have been extensively studied in the last years and there are plenty of results about characterizations (of small ones) and about the bounds on the size. A trivial blocking set of PG(2, q) is a set of points containing a line. The first examples of non-trivial blocking sets of small size of PG(2, q) have been given in the following: Theorem 1. In PG(2, q), q odd, there exists a projective triangle of side q+3 2 that is a minimal blocking set of size[14]. In PG(2, q), q even, there exists a projective triad of side q+2 2 that is a minimal blocking set of size 3q+2 2[40].This motivates the choice of the boundfor the size of a small blocking set in PG(2, q) and then generalized asfor the k-blocking sets of PG(n, q).

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Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

Cited by 14 publications

References 24 publications

Tight sets in finite classical polar spaces

Tight sets in finite classical polar spaces

The non-existence of Cameron–Liebler line classes with parameter 2 <x≤q

The Use of Blocking Sets in Galois Geometries and in Related Research Areas

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