Two-graphs and skew two-graphs in finite geometries

Moorhouse, G. Eric

doi:10.1016/0024-3795(95)00242-j

Cited by 21 publications

(23 citation statements)

References 21 publications

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“…The fingerprint is a combinatorial invariant, originally due to Conway that can be computed for any projective plane (see Charnes [5] for a description of this, as Conway has not published). The fingerprint of a general projective plane is rather cumbersome, but for a translation plane a much more compact version can be derived and Moorhouse [18] describes a very simple way of computing the fingerprint. This is based on the representation of a spread of P G (3, 7) as a spread-set.…”

Section: Spread Sets and Fingerprintsmentioning

confidence: 99%

The translation planes of order 49

Mathon

Royle²

1995

Des Codes Crypt

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Section: Spread Sets and Fingerprintsmentioning

confidence: 99%

The translation planes of order 49

Mathon

Royle²

1995

Des Codes Crypt

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“…As shown in [4] , every 4-subset of ᏻ contains an even number of triples from ⌬ ; thus ⌬ is a two -graph with point set ᏻ . We next show that any 2-subset of ᏻ is contained in exactly 1 -2 ( q n Ϫ 1 ϩ 1)( q Ϫ 1) triples from ⌬ .…”

Section: T He R Esultmentioning

confidence: 99%

“…The O Ϫ 4 ( q ) quadric embeds as an ovoid in O 5 ( q ) and , for q odd , the resulting regular two-graph of degree 1 -2 ( q 2 Ϫ 1) on q 2 ϩ 1 vertices is of Paley type [4] . However , more examples of ovoids in O 5 ( q ) are known [3] .…”

Section: C Oncluding R Emarksmentioning

confidence: 99%

The Non-existence of Ovoids inO9(q)

Gunawardena

Moorhouse

1997

European Journal of Combinatorics

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“…In another paper, Charnes and Dempwolff [8] use fingerprints as an aid to classify O + 8 (5) ovoids. Moorhouse in [24] describes a simple way of computing fingerprints which has O(r 3 ) time complexity for an r−cap and shows its connection to two-graphs of ovoids. The ovoids with regular two-graphs have the same fingerprint.…”

Section: A Survey Of Invariantsmentioning

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,…”

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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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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Two-graphs and skew two-graphs in finite geometries

Cited by 21 publications

References 21 publications

The translation planes of order 49

The translation planes of order 49

The Non-existence of Ovoids inO9(q)

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

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