Latin Hypercubes and Cellular Automata

  • Maximilien Gadouleau
  • Luca MariotEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12286)


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.


Latin squares Latin hypercubes Cellular automata Bipermutivity Toeplitz matrices De bruijn graphs 



This work has been partially supported by COST Action IC1405, “Reversible Computation – Extending the Horizons of Computing”.


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© IFIP International Federation for Information Processing 2020

Authors and Affiliations

  1. 1.Department of Computer ScienceDurham UniversityDurhamUK
  2. 2.Cyber Security Research GroupDelft University of TechnologyDelftThe Netherlands

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