Mutually orthogonal latin squares based on cellular automata

  • Luca MariotEmail author
  • Maximilien Gadouleau
  • Enrico Formenti
  • Alberto Leporati


We investigate sets of mutually orthogonal latin squares (MOLS) generated by cellular automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order \(q^{d-1}\), we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field \(\mathbb {F}_q\) are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over \(\mathbb {F}_q\). Finally, we present a construction for families of MOLS based on LBCA, and prove that their cardinality corresponds to the maximum number of pairwise coprime polynomials with nonzero constant term. Although our construction does not yield all such families of MOLS, we show that the resulting lower bound is asymptotically close to their actual number.


Mutually orthogonal latin squares Cellular automata Sylvester matrices Polynomials 

Mathematics Subject Classification

05B15 68Q80 11T06 



The authors wish to thank Arthur Benjamin, Curtis Bennett and Igor Shparlinski for their insightful suggestions on how to count the number of pairs of coprime polynomials with nonzero constant term. Further, the authors thank the anonymous reviewers for their useful comments to improve the readability of the paper.


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Authors and Affiliations

  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-BicoccaMilanItaly
  2. 2.Department of Computer ScienceDurham UniversityDurhamUK
  3. 3.Laboratoire d’Informatique, Signaux et Systèmes de Sophia-Antipolis (I3S)Université Côte d’AzurSophia AntipolisFrance

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