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

The University of Western Australia
University of Canterbury

Research output: Contribution to a Journal (Peer & Non Peer) › 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
Journal	Finite Fields and their Applications
Volume	99
DOIs	https://doi.org/10.1016/j.ffa.2024.102491
Publication status	Published - Oct 2024
Externally published	Yes

Keywords

Bruen chain
Finite field
Projective space

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10.1016/j.ffa.2024.102491

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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.",

keywords = "Bruen chain, Finite field, Projective space",

author = "John Bamberg and Jesse Lansdown and \{Van de Voorde\}, Geertrui",

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doi = "10.1016/j.ffa.2024.102491",

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AU - Bamberg, John

AU - Lansdown, Jesse

AU - Van de Voorde, Geertrui

PY - 2024/10

Y1 - 2024/10

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.

KW - Bruen chain

KW - Finite field

KW - Projective space

UR - https://www.scopus.com/pages/publications/85201183446

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DO - 10.1016/j.ffa.2024.102491

M3 - Article

AN - SCOPUS:85201183446

SN - 1071-5797

VL - 99

JO - Finite Fields and their Applications

JF - Finite Fields and their Applications

M1 - 102491

ER -

On Bruen chains

Abstract

Keywords

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