除均匀睫状体切开术外

IF 1.2 3区 数学 Q1 MATHEMATICS
Sophie Huczynska , Laura Johnson , Maura B. Paterson
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引用次数: 0

摘要

环切术,即对环切类和环切数的研究,是由高斯首先研究的数论领域。它在离散数学和信息论中有天然的应用。尽管有这么长的历史,有什么是已知的明确的睫状体切开术数显著的局限性,这限制了在应用中使用睫状体切开术。可用的主要明确工具是1982年由Baumert, Mills和Ward引入的均匀睫状体切开术。本文给出了一致环切的推广,给出了对任意素数幂q和n≥2求(qn−1)/(q−1)阶的GF(qn)上的所有环切数的直接方法,该方法不使用特征理论,也不使用域内的直接计算。这允许直接评估许多其他方法未知或不切实际的环切数,扩展目前有限的工具组合,以与环切数工作。我们的方法利用了环切术、辛格差分集和有限几何之间的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Beyond uniform cyclotomy
Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area of number theory first studied by Gauss. It has natural applications in discrete mathematics and information theory. Despite this long history, there are significant limitations to what is known explicitly about cyclotomic numbers, which limits the use of cyclotomy in applications. The main explicit tool available is that of uniform cyclotomy, introduced by Baumert, Mills and Ward in 1982. In this paper, we present an extension of uniform cyclotomy which gives a direct method for evaluating all cyclotomic numbers over GF(qn) of order dividing (qn1)/(q1), for any prime power q and n2, which does not use character theory nor direct calculation in the field. This allows the straightforward evaluation of many cyclotomic numbers for which other methods are unknown or impractical, extending the currently limited portfolio of tools to work with cyclotomic numbers. Our methods exploit connections between cyclotomy, Singer difference sets and finite geometry.
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来源期刊
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.
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