Latin Hypercubes and Cellular Automata

Conference paper (2020)

Authors

Maximilien Gadouleau Durham University
L. Mariot Cyber Security - EEMCS
Research Group
Cyber Security (EEMCS) (TU Delft)
Copyright: © 2020 Maximilien Gadouleau, L. Mariot
DOI: https://doi.org/10.1007/978-3-030-61588-8_11
Cellular automata Bipermutivity De bruijn graphs Latin hypercubes Latin squares Toeplitz matrices
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Published Date

2020

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Intelligent Systems

Research Group

Cyber Security

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. In particular, we prove that linear bipermutive CA (LBCA) yielding Latin hypercubes of dimension 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 on this de Bruijn graph.