Inversive geometry and some new translation planes, I

Bruen, Aiden A.

doi:10.1007/bf00181353

Cited by 24 publications

(38 citation statements)

References 3 publications

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“…This last case does not occur in PG (3,7). LEMMA 3.1 Every spread in PG(3,7) contains at least four lines from a regulus.…”

Section: Finding the Spreadsmentioning

confidence: 90%

“…Let G be the intersection graph of the lines of PG (3,7). Then G has 2850 vertices and a spread of PG (3,7) is an independent set of size 50 inside G. The automorphism group of G is not P FL (4, 7) so independent sets that are equivalent under Aut(G) are not necessarily equivalent under PFL (4,7).…”

Section: Proof (By Computer)mentioning

confidence: 99%

“…It is very easy to find spreads in PG (3,7); by far the hardest task is to determine whether two spreads are isomorphic. The problem of finding the spreads was treated as a setcovering problem on the 400 points of PG(3, 7) while the isomorph rejection was treated as a graph-isomorphism problem solved using Brendan McKay's nauty [17].…”

Section: The Exhaustive Searchmentioning

confidence: 99%

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The translation planes of order 49

Mathon

Royle²

1995

Des Codes Crypt

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“…This last case does not occur in PG (3,7). LEMMA 3.1 Every spread in PG(3,7) contains at least four lines from a regulus.…”

Section: Finding the Spreadsmentioning

confidence: 90%

Section: Proof (By Computer)mentioning

confidence: 99%

Section: The Exhaustive Searchmentioning

confidence: 99%

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The translation planes of order 49

Mathon

Royle²

1995

Des Codes Crypt

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“…It turns out that this spread contains 7 reguli. 4 ͘ accordingly as q ϵ 1 (mod 4) or q ϵ 3 (mod 4). Our previous work shows that D is contained in the translation complement of ȏ.…”

Section: Using the Above Notation Either T Ker T B Or T B Is An Elmentioning

confidence: 99%

“…By Theorem 1 the only circles contained in the point set Ũ are the circles of Ñ , and hence Aut( 4 , where Z (qϩ1)/2 denotes the cyclic group of order (q ϩ 1)/2 and K 4 denotes the Klein 4-group. In practice, G ϭ Aut(Ñ ), as has been verified for q Յ 19.…”

Section: Theorem 2 (Schmidtmentioning

confidence: 99%

Filling the Nest Gaps

Baker

Ebert

1996

Finite Fields and Their Applications

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This paper presents a robust construction technique for two-dimensional translation planes of odd order using the notion of nesting previously developed by the authors. More precisely, nonisomorphic planes of order q 2 are obtained by constructing replaceable -nests for roughly between (3/4)q Ϫ ͙q/2 and (3/4)q ϩ ͙q/2. In all cases the translation complement of any such plane contains a direct product of two cyclic homology groups of order (q ϩ 1)/2 with affine axes.

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A Bruen chain forq=19

Baker

Ebert

1994

Des Codes Crypt

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Abstract. The one-to-one correspondence between the class of two-dimensional translation planes of order q2 and the collection of spreads of PG(3, q) has long provided a natural context for describing new planes. The method often used for constructing "interesting" spreads is to start with a regular spread, corresponding to a desarguesian plane, and then replace some "nice" subset of lines by another partial spread covering the same set of points. Indeed the first approach was replacing the lines of a regulus by the lines of its opposite regulus, or doing this process for a set of disjoint reguli. Nontrivial generalizations of this idea include the chains of Bruen and the nests of Baker and Ebert. In this paper we construct a replaceable subset of a regular spread of PG(3, 19) which is the union of 11 reguli double covering the lines in their union, hence is a chain in the terminology of Bruen or a I l-nest in the Baker-Ebert terminology.

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Inversive geometry and some new translation planes, I

Cited by 24 publications

References 3 publications

The translation planes of order 49

The translation planes of order 49

Filling the Nest Gaps

A Bruen chain forq=19

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