Journal of Algebraic Combinatorics

, Volume 47, Issue 2, pp 319–343 | Cite as

Linked systems of symmetric group divisible designs

  • Hadi Kharaghani
  • Sho Suda


We introduce the concept of linked systems of symmetric group divisible designs. The connection with association schemes is established, and as a consequence we obtain an upper bound on the number of symmetric group divisible designs which are linked. Several examples of linked systems of symmetric group divisible designs are provided.


Symmetric group divisible design Association scheme Mutually orthogonal Latin square Mutually UFS Latin square Generalized Hadamard matrix 



The authors are grateful to anonymous referees for many suggestions and corrections which have improved the presentation of the paper very much. Hadi Kharaghani is supported by an NSERC Discovery Grant. Sho Suda is supported by JSPS KAKENHI Grant Number 15K21075.


  1. 1.
    Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. Benjamin/Cummings, Menlo Park, CA (1984)MATHGoogle Scholar
  2. 2.
    Bose, R.C.: Symmetric group divisible designs with the dual property. J. Stat. Plan. Inference 1, 87–101 (1977)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cameron, P.J.: On groups with several doubly-transitive permutation representations. Math. Z 128, 1–14 (1972)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Higman, D.G.: Rank \(5\) association schemes and triality. Linear Algebra Appl. 226–228, 197–222 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Holzmann, W.H., Kharaghani, H., Orrick, W.: On the real unbiased Hadamard matrices. Contemp. Math. Am. Math. Soc. 531, 243–250 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Holzmann, W.H., Kharaghani, H., Suda, S.: Mutually unbiased biangular vectors and association schemes. In: Colbourn, C.J. (eds.) Algebraic Design Theory and Hadamard Matrices, Volume 133 of Springer Proceedings in Mathematics Statistics, pp. 149–157. Springer International Publishing (2015)Google Scholar
  7. 7.
    Kharaghani, H.: New class of weighing matrices. Ars. Comb. 19, 69–72 (1985)MathSciNetMATHGoogle Scholar
  8. 8.
    Kharaghani, H., Sasani, S., Suda, S.: Mutually unbiased Bush-type Hadamard matrices and association schemes. Electron. J. Comb. 22, p3.10 (2015)MathSciNetMATHGoogle Scholar
  9. 9.
    LeCompte, N., Martin, W.J., Owens, W.: On the equivalence between real mutually unbiased bases and a certain class of association schemes. Eur. J. Comb. 31, 1499–1512 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Martin, W.J., Muzychuk, M., Williford, J.: Imprimitive cometric association schemes: constructions and analysis. J. Algebraic Comb. 25, 399–415 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mathon, R.: The systems of linked \(2\)-\((16,6,2)\) designs. Ars Comb. 11, 131–148 (1981)MathSciNetMATHGoogle Scholar
  12. 12.
    van Dam, E.: Three-class association schemes. J. Algebraic Comb. 10, 69–107 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    van Dam, E., Martin, W., Muzychuk, M.: Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems. J. Combin. Theory Ser. A 120, 1401–1439 (2013)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of LethbridgeLethbridgeCanada
  2. 2.Department of Mathematics EducationAichi University of EducationKariyaJapan

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