Journal of Geometry

, 109:5 | Cite as

The number of different reduced complete sets of MOLS corresponding to PG \(\varvec{(2,q)}\)

  • K. H. Hicks
  • G. L. Mullen
  • L. Storme
  • J. Vanpoucke


In the first part of this article we determine the exact number of different reduced complete sets of mutually orthogonal latin squares (MOLS) of order q, for \(q = p^d\), p prime, \(d \ge 1\), corresponding to the Desarguesian projective planes PG(2, q). In the second part we provide some computational results and enumerate the maximal sets of reduced latin squares of order n as part of a set containing exactly r MOLS.


Computations Desarguesian projective planes latin squares mutually orthogonal latin squares 

Mathematics Subject Classification

05B15 05B25 51E15 


  1. 1.
    Berger, T.: Classification des groupes de permutations d’un corps fini contenant le groupe affine. C. R. Acad. des Sciences de Paris Ser. I 319, 117–119 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bose, R.C.: On the application of the properties of Galois fields to the construction of hyper-Graeco-Latin squares. Sankhy\(\bar{\text{a}}\) 3, 323–338 (1938)Google Scholar
  3. 3.
    Bruen, A.A.: Unembeddable nets of small deficiency. Pac. J. Math. 43, 51–54 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs. CRC Press, Taylor and Francis Group, Boca Raton (2007)zbMATHGoogle Scholar
  5. 5.
    Danziger, P., Wanless, I.M., Webb, B.S.: Monogamous latin squares. J. Combin. Theory Ser. A 118, 796–807 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deńes, J., Keedwell, A.D.: Latin Squares and Their Applications. Academic Press, New York (1974)zbMATHGoogle Scholar
  7. 7.
    Drake, D.A., Myrvold, W.: The non-existence of maximal sets of four mutually orthogonal latin squares of order 8. Des. Codes Cryptogr. 33, 63–69 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drake, D.A., van Rees, G.H.J., Wallis, W.D.: Maximal sets of mutually orthogonal latin squares. Discrete Math. 194, 87–94 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Egan, J., Wanless, I.M.: Enumeration of MOLS of small order. Math. Comput. 85, 799–824 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hedayat, A., Federer, W.T.: On embedding and enumeration of orthogonal latin squares. Ann. Math. Stat. 42, 509–516 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lam, C.W.H., Kolesova, G., Thiel, L.: A computer search for finite projective planes of order 9. Discrete Math. 92, 187–195 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Laywine, C.F., Mullen, G.L.: Discrete Mathematics Using Latin Squares. Wiley, New York (1998)zbMATHGoogle Scholar
  13. 13.
    Liebeck, M.W., Praeger, C.E., Saxl, J.: A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111, 365–383 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McKay, B.D., Wanless, I.M.: On the number of latin squares. Ann. Comb. 9, 335–344 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsOhio UniversityAthensUSA
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  3. 3.Department of MathematicsGhent UniversityGhentBelgium
  4. 4.Department of Mathematics, Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium

Personalised recommendations