On Bruen chains

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-08-14 DOI:10.1016/j.ffa.2024.102491

John Bamberg , Jesse Lansdown , Geertrui Van de Voorde

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引用次数: 0

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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关于布鲁恩链条

众所周知，除了 q=29 以外，三维投影空间 PG(3,q) 的布伦链对于每个奇素数幂 q 至多 37 都是存在的。Cardinali 等人（2005 年）的研究表明，41⩽q⩽49 的布伦链并不存在。我们建立了一个基于有限域的模型，使我们能够将这一结果扩展到 41⩽q⩽97，从而为布伦链不存在于 q>37 的猜想增添了更多证据。此外，我们还证明了布伦链可以精确地实现为两个相关但不同的无向简单图的 (q+1)/2-cliques 。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.