Partial and semipartial geometries: an update

Clerck, Frank De

doi:10.1016/s0012-365x(02)00604-0

Cited by 23 publications

(20 citation statements)

References 29 publications

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“…An incidence structure S = (P, L, I) has order (s, t) if I satisfies the following axioms: (a) Any two distinct points are incident with at most one line; (b) Each line is incident with exactly s + 1 points; (c) Each point is incident with exactly t + 1 lines. If both s, t ≥ 1 then S is called a partial linear space of order (s, t) [12]. Otherwise S is called a trivial structure.…”

Section: Q(λx) = λ 2 Q(x)mentioning

confidence: 99%

“…Note that the isomorphism invariant |P| is sufficient to identify the ovoids in O 5 (q) where q = 9, 25 or 27. (2,27), (3, 60), (7, 3)} {(1, 395), (2,27), (3, 60), (7, 3)} O 13 928 464 464 {(1, 348), (2, 84), (3,12), (4,11), (6, 9)} {(1, 358), (2,66), (3,9), (4,24), (5,6) (2,104), (3,13), (4,8), (5, 1), (6, 2)} {(1, 354), (2,104), (3,13), (4,8), (5, 1), (6, 2)} O 17 796 398 398 {(1, 265), (2,54), (3,48), (4,24), (5,6), (7, 1)} {(1, 265), (2,54), (3,48), (4,24), (5,6), (7, 1)} O 18 688 344 344 {(1, 144), (2,144), (3,…”

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confidence: 99%

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On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

Fuelberth

Gunawardena

Shaffer

2009

Des. Codes Cryptogr.

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The revisions were made according to the referees' suggestions as mentioned in the bold and italic statements below.Reviewer #1: You may have missed the recent paper A classification of transitive ovoids, spreads and m-systems of polar spaces, Forum Mathematicum 21(2009), 181-216 which classifies primitive ovoids in all classical polar spaces provided they do not have cardinality a prime number. I recommend that you add it to your list of references, and refer to it in Section 1 (after discussing Kleidman's result) and/or in Section 5, when referring to your own result about primitive ovoids from [15].Added to the list of references. Refered to it in Section 1 and Section 5.Typos : voids for ovoids (page 13, line 9), bipartitie for bipartite (page 6, line 21); there is no Tits ovoid over the field of order 27 (page 13, line 28), but they should have occurred two sentences earlier in the list of known ovoids. Page 13, line 27, you need two definite articles added : For q=25 the only... For q=27 the only... Typos were corrected. Reviewer 2The authors introduce new invariants of ovoids in finite orthogonal spaces. This paper is suitable for publication in DCC provided the following changes are made: p3 l 49 reference to [19] should be replaced by: Charnes and Dempwolff [CD7], and this reference should be included in the bibliography. It should also be mentioned that fingerprints give an effective method of calculating the automorphism groups of ovoids. The following typos were corrected. Abstract. We study the point-line incidence structures of nonsingular points and hyperbolic secant lines associated with ovoids in finite orthogonal spaces. We show that these incidence structures frequently produce partial linear spaces and the parameters of the bipartite graphs (called ovoidal graphs) associated with these structures produce simple and effective isomorphism invariants to distinguish non-isomorphic ovoids. We prove explicit formulas for these isomorphism invariants for a number of infinite families of 2-transitive ovoids. Abstract. We study the point-line incidence structures of nonsingular points and hyperbolic secant lines associated with ovoids in finite orthogonal spaces. We show that these incidence structures frequently produce partial linear spaces and the parameters of the bipartite graphs (called ovoidal graphs) associated with these structures produce simple and effective isomorphism invariants to distinguish non-isomorphic ovoids. We prove explicit formulas for these isomorphism invariants for a number of infinite families of 2-transitive ovoids.

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Section: Q(λx) = λ 2 Q(x)mentioning

confidence: 99%

mentioning

confidence: 99%

On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

Fuelberth

Gunawardena

Shaffer

2009

Des. Codes Cryptogr.

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“…Semipartial geometries generalize at the same time the partial quadrangles and the partial geometries. See for instance [6] for more information on generalized quadrangles and [3,4] for more information on partial and semipartial geometries.…”

Section: S225mentioning

confidence: 99%

“…Dual partial quadrangles embedded in PG(3, q) Assume that #=(i+l) 3 ; then n = μ+ ί = ^-^4-r = r(r 2 + 2ί + 2), μ = (/Η-1) 2 ί. The line graph of «5^ is a strongly regular graph with ή = (ί+1).…”

Section: B-l = ( Q+ L)t(l+^)=(p» Andmentioning

confidence: 99%

Dual partial quadrangles embedded in PG(3,q)

Clerck¹,

Durante²,

Thas³

2003

Advg

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The prqjective full embeddings of partial geometries are known. So are the projective full embeddings of semipartial and dual semipartial geometries in case of α > 1. If α = 1, a semipartial geometry is known as a partial quadrangle. No projective full embedding of a proper partial quadrangle is known. However besides a unique example for q = 2, there is one example known of a dual partial quadrangle fully embedded in a PG(3, q), any q. In this paper we will prove that if the dual of a proper partial quadrangle £f is fully embedded in PG(3, q), then μ ^ q --^ . If equality holds, then 5^ is uniquely defined.

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“…Such a graph is geometric if it is the point graph of a pg(s, t, α). For a survey on partial geometries we refer to [3]; an update is made in [2].…”

Section: Introductionmentioning

confidence: 99%

A Geometric Construction of Partial Geometries with a Hermitian Point Graph

Kuijken

2002

European Journal of Combinatorics

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Partial and semipartial geometries: an update

Cited by 23 publications

References 29 publications

On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

Dual partial quadrangles embedded in PG(3,q)

A Geometric Construction of Partial Geometries with a Hermitian Point Graph

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