Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 57–65 | Cite as

Sets of mutually orthogonal Sudoku frequency squares

  • John T. Ethier
  • Gary L. Mullen


We discuss sets of mutually orthogonal frequency Sudoku squares. In particular, we provide upper bounds for the maximum number of such mutually orthogonal squares. In addition, we provide constructions for sets of such squares. We also briefly discuss an extension of these ideas to sets of higher dimensional mutually orthogonal frequency hypercubes.


Latin square Frequency square MOFS MOSFS Sudoku 

Mathematics Subject Classification




The authors would like to thank the referees for their helpful comments.


  1. 1.
    Bailey R.A., Cameron P.J., Connelly R.: Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes. Am. Math. Mon. 115, 383–404 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bremigan R., Lorch J.: Mutually orthogonal rectangular gerechte designs. Linear Algebra Appl. 497, 44–61 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Colbourn C.J., Dinitz J.H. (eds.): Handbook of Combinatorial Designs, 2nd edn. CRC Press, Boca Raton (2007).Google Scholar
  4. 4.
    Dénes J., Keedwell A.D.: Latin Squares and Their Applications. Academic Press, New York (1974).zbMATHGoogle Scholar
  5. 5.
    D’haeseleer J., Metsch K., Storme L., Van de Voorde G.: On the maximality of a set of mutually orthogonal Sudoku latin squares. Des. Codes Cryptogr. 84, 143–152 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Laywine C.F., Mullen G.L.: Discrete Mathematics Using Latin Squares. Wiley, New York (1998).zbMATHGoogle Scholar
  7. 7.
    Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics and Its Applications, 2nd edn. Cambridge University Press, Cambridge (1997).Google Scholar
  8. 8.
    Mullen G.L.: Polynomial representation of complete sets of frequency squares of prime power order. Discret. Math. 69, 79–84 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pederson R., Vis T.: Sets of mutually orthogonal Sudoku Latin Squares. Coll. Math. J. 40(3), 174–180 (2009).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Street A.P., Street D.J.: Combinatorics of Experimental Design. Oxford University Press, New York (1987).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical and Computer SciencesMetropolitan State University of DenverDenverUSA
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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