On Bruen chains

John Bamberg; Jesse Lansdown; Geertrui Van de Voorde

doi:10.1016/j.ffa.2024.102491

On Bruen chains

John Bamberg, Jesse Lansdown, Geertrui Van de Voorde

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

It is known that a Bruen chain of the three-dimensional projective space PG(3,q) exists for every odd prime power q at most 37, except for q=29. It was shown by Cardinali et al. (2005) that Bruen chains do not exist for 41⩽q⩽49. We develop a model, based on finite fields, which allows us to extend this result to 41⩽q⩽97, thereby adding more evidence to the conjecture that Bruen chains do not exist for q>37. Furthermore, we show that Bruen chains can be realised precisely as the (q+1)/2-cliques of a two related, yet distinct, undirected simple graphs.

Original language	English
Article number	102491
Number of pages	23
Journal	Finite Fields and Their Applications
Volume	99
Early online date	14 Aug 2024
DOIs	https://doi.org/10.1016/j.ffa.2024.102491
Publication status	E-pub ahead of print - 14 Aug 2024

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10.1016/j.ffa.2024.102491Licence: CC BY

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title = "On Bruen chains",

abstract = "It is known that a Bruen chain of the three-dimensional projective space PG(3,q) exists for every odd prime power q at most 37, except for q=29. It was shown by Cardinali et al. (2005) that Bruen chains do not exist for 41⩽q⩽49. We develop a model, based on finite fields, which allows us to extend this result to 41⩽q⩽97, thereby adding more evidence to the conjecture that Bruen chains do not exist for q>37. Furthermore, we show that Bruen chains can be realised precisely as the (q+1)/2-cliques of a two related, yet distinct, undirected simple graphs.",

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T1 - On Bruen chains

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AU - Lansdown, Jesse

AU - Van de Voorde, Geertrui

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N2 - It is known that a Bruen chain of the three-dimensional projective space PG(3,q) exists for every odd prime power q at most 37, except for q=29. It was shown by Cardinali et al. (2005) that Bruen chains do not exist for 41⩽q⩽49. We develop a model, based on finite fields, which allows us to extend this result to 41⩽q⩽97, thereby adding more evidence to the conjecture that Bruen chains do not exist for q>37. Furthermore, we show that Bruen chains can be realised precisely as the (q+1)/2-cliques of a two related, yet distinct, undirected simple graphs.

AB - It is known that a Bruen chain of the three-dimensional projective space PG(3,q) exists for every odd prime power q at most 37, except for q=29. It was shown by Cardinali et al. (2005) that Bruen chains do not exist for 41⩽q⩽49. We develop a model, based on finite fields, which allows us to extend this result to 41⩽q⩽97, thereby adding more evidence to the conjecture that Bruen chains do not exist for q>37. Furthermore, we show that Bruen chains can be realised precisely as the (q+1)/2-cliques of a two related, yet distinct, undirected simple graphs.

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KW - Projective space

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JO - Finite Fields and Their Applications

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ER -

On Bruen chains

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The synchronisation hierarchy of permutation groups

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On Bruen chains

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The synchronisation hierarchy of permutation groups

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