Bounds on partial ovoids and spreads in classical generalized quadrangles

Winter, Stefaan De; Thas, Koen

doi:10.2140/iig.2010.11.19

Cited by 5 publications

(8 citation statements)

References 14 publications

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“…We note that in [3], where a completely different approach is used, a comparable conclusion on the pairs point-antipode is made. Finally, we also mention the work in [6], where the non-existence for larger q is shown under the extra assumption that (q 2 − 1) 2 divides the order of the automorphism group of the maximal partial ovoid.…”

Section: Proofmentioning

confidence: 97%

On large maximal partial ovoids of the parabolic quadric Q(4, q)

Beule

2012

Des. Codes Cryptogr.

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We use the representation T 2 (Ø) for Q(4, q) to show that maximal partial ovoids of Q(4, q) of size q 2 − 1, q = p h , p odd prime, h > 1, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for q odd prime. keywords: maximal partial ovoid, generalized quadrangle, parabolic quadric.MSC (2010): 05B25, 51D20, 51E12, 51E20, 51E21.

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Section: Proofmentioning

confidence: 97%

On large maximal partial ovoids of the parabolic quadric Q(4, q)

Beule

2012

Des. Codes Cryptogr.

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“…Suppose there exists a point R ∈ D \ T . Then each of the θ 2r−1, √ q Baer subplanes in D through the line P R intersects at most one good line in its unique Baer subline of holes, and therefore (15) γ ≤ θ 2r−1, √ q .…”

Section: Lemma 23 [1]mentioning

confidence: 99%

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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“…When d = 2, 3, 5, 7 or 11, there exist extremely exotic examples of unextendible sets of Pauli classes of size d 2 − 1 in C d 2 (details, constructions and references can be found in [14]). We propose calling the corresponding sets of MUBs "Galois MUBs" because they are all related to exotic two-transitive representations of special linear groups, as was first noted by Galois (see also [14]).…”

Section: "Galois Mubs"mentioning

confidence: 99%

Unextendible Mutually Unbiased Bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

Thas

2016

Entropy

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Abstract:We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of "unextendible mutually unbiased bases." We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set of nonidentity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d a prime, and the geometry of non-degenerate alternating bilinear forms of rank N over finite fields F d . We then supply alternative and short proofs of results obtained in Mandayam et al. (2014), as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of Mandayam et al. (2014) and speculate on variations of this conjecture.

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Bounds on partial ovoids and spreads in classical generalized quadrangles

Abstract: We present an improvement on a recent bound for small maximal partial ovoids of W(q 3 ). We also classify maximal partial ovoids of size (q 2 − 1) of Q(4, q) which allow a certain large automorphism group, and discuss examples for small q.

Cited by 5 publications

References 14 publications

On large maximal partial ovoids of the parabolic quadric Q(4, q)

On large maximal partial ovoids of the parabolic quadric Q(4, q)

Tight sets in finite classical polar spaces

Unextendible Mutually Unbiased Bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

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