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arXiv · 2506.19605

De Bruijn Tori Without Zeros: A Field-Theoretic Perspective

Ming Hsuan Kang·Yu Hsuan Hsieh
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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 \( \mathbb{F}_p \)-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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Ming Hsuan Kang, Yu Hsuan Hsieh. 2025-06-24. De Bruijn Tori Without Zeros: A Field-Theoretic Perspective. https://arxiv.org/abs/2506.19605

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