Latin Hypercubes and Cellular Automata
- 80 Downloads
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.
KeywordsLatin 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”.
- 1.Blakley, G.R.: Safeguarding cryptographic keys. In: Managing Requirements Knowledge, International Workshop on, pp. 313–317 (1979)Google Scholar