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arXiv · 2004.07131

Latin Hypercubes and Cellular Automata

Maximilien Gadouleau·Luca Mariot
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Abstract

Latin squares and hypercubes are combinatorial designs with several applications in statistics, cryptography and coding theory. In this paper, we generalize a construction of Latin squares based on bipermutive cellular automata (CA) to the case of Latin hypercubes of dimension $k>2$. In particular, we prove that linear bipermutive CA (LBCA) yielding Latin hypercubes of dimension $k>2$ are defined by sequences of invertible Toeplitz matrices with partially overlapping coefficients, which can be described by a specific kind of regular de Bruijn graph induced by the support of the determinant function. Further, we derive the number of $k$-dimensional Latin hypercubes generated by LBCA by counting the number of paths of length $k-3$ on this de Bruijn graph.

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Maximilien Gadouleau, Luca Mariot. 2020-04-15. Latin Hypercubes and Cellular Automata. https://arxiv.org/abs/2004.07131

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