De Bruijn tori without zeros: a field-theoretic perspective

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2026-06-01 Epub Date: 2026-01-15 DOI:10.1016/j.ffa.2026.102790

Ming Hsuan Kang, Yu Hsuan Hsieh

引用次数: 0

Abstract

We present an algebraic construction of trace-based De Bruijn tori over finite fields, focusing on the nonzero variant that omits the all-zero pattern. The construction arranges nonzero field elements on a toroidal grid using two multiplicatively independent generators, with values obtained by applying a fixed linear map, typically the field trace.

We characterize sampling patterns as subsets whose associated field elements form an

F_{p}

-basis, and show that column structures correspond to cyclic shifts of De Bruijn sequences determined by irreducible polynomials over subfields. Recursive update rules based on multiplicative translations enable efficient computation.

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没有零的德布鲁因托里：场理论视角

我们给出了有限域上基于迹的德布鲁因托里的代数构造，重点讨论了忽略全零模式的非零变体。该构造使用两个相乘独立的生成器在环面网格上排列非零场元素，其值通过应用固定的线性映射（通常是场迹）获得。我们将采样模式描述为子集，其相关的域元素形成一个fp基，并表明列结构对应于子域上由不可约多项式决定的De Bruijn序列的循环移位。基于乘法转换的递归更新规则实现了高效的计算。

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.