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Conference Papers Year : 2020

Latin Hypercubes and Cellular Automata

Maximilien Gadouleau
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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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hal-03659462 , version 1 (05-05-2022)

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Maximilien Gadouleau, Luca Mariot. Latin Hypercubes and Cellular Automata. 26th International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA), Aug 2020, Stockholm, Sweden. pp.139-151, ⟨10.1007/978-3-030-61588-8_11⟩. ⟨hal-03659462⟩
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