q = 4 和 q = 5 时避免仿射超平面的有限域 Fqr 的基元

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-03-26 DOI:10.1016/j.ffa.2024.102416

Philipp A. Grzywaczyk , Arne Winterhof

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引用次数: 0

摘要

对于固定 q 和 r 足够大的有限域 Fqr，我们证明了在 q=4 和 q=5 时，在一组 r 个仿射超平面之外存在一个基元。这是对费尔南德斯和雷斯早先关于 q≥7 的结果的补充。对于 q=3 ，类似的结果可以从 Iyer 和 Shparlinski 最新的特征和约束中推导出来。对于 q=2 ，集合只包含一个元素，因此不可能得出这样的结果。

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Primitive elements of finite fields Fqr avoiding affine hyperplanes for q = 4 and q = 5

For a finite field $F_{q^{r}}$ with fixed q and r sufficiently large, we prove the existence of a primitive element outside of a set of r many affine hyperplanes for $q = 4$ and $q = 5$ . This complements earlier results by Fernandes and Reis for $q \geq 7$ . For $q = 3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q = 2$ the set consists only of a single element, and such a result is thus not possible.

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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.