De Bruijn Tori Without Zeros: A Field-Theoretic Perspective

TL;DR

This paper introduces an algebraic method for constructing nonzero De Bruijn tori over finite fields, utilizing field trace functions and basis properties to efficiently generate and analyze these combinatorial structures.

Contribution

It provides a novel algebraic construction of nonzero De Bruijn tori using finite field theory and trace functions, with a focus on efficient computation and pattern characterization.

Findings

Constructs nonzero De Bruijn tori over finite fields using algebraic methods.

Characterizes sampling patterns as subsets forming an _p-basis.

Establishes recursive update rules for efficient computation.

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.