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Linked systems of symmetric group divisible designs of type II

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Abstract

Linked systems of symmetric group divisible designs of type II are introduced, and several examples are obtained from affine resolvable designs and a variant of mutually orthogonal Latin squares. Furthermore, an equivalence between such symmetric group divisible designs and some association schemes with 5-classes is provided.

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Notes

  1. If \(k=\lambda _1\), then its incidence matrix A satisfies that \(A=\bar{A}\otimes J_n\) for some incidence matrix \(\bar{A}\) of a symmetric design.

  2. In [11] the theorems are valid under the assumption \(k>\lambda _1\). If \(k=\lambda _1\), then linked systems of symmetric group divisible designs are \(A_{i,j}\otimes J_n\) where \(A_{i,j}\)’s are linked systems of symmetric designs.

References

  1. Bannai E.: Subschemes of some association schemes. J. Algebra 144, 167–188 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  2. Bannai E., Ito T.: Algebraic Combinatorics I: Association Schemes. Benjamin/Cummings, Menlo Park, CA (1984).

    MATH  Google Scholar 

  3. Bose R.C.: Symmetric group divisible designs with the dual property. J. Stat. Plan. Inference 1, 87–101 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  4. Colbourn Charles J., Dinitz Jeffrey H. (eds.): Handbook of Combinatorial Designs, 2nd edn, Part IV. Chapman & Hall/CRC, Boca Raton (2006).

  5. van Dam E.: Three-class association schemes. J. Algebraic Comb. 10(1), 69–107 (1999).

    Google Scholar 

  6. van Dam E., Martin W., Muzychuk M.: Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems. J. Comb. Theory Ser. A 120, 1401–1439 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. Holzmann W.H., Kharaghani H., Orrick W.: On the Real Unbiased Hadamard Matrices. Combinatorics and Graphs. Contemporary Mathematics, vol. 531, pp. 243–250. American Mathematical Society, Providence, RI (2010).

  8. Holzmann W.H., Kharaghani H., Suda S.: Mutually unbiased biangular vectors and association schemes. In: Colbourn C.J. (ed.) Algebraic Design Theory and Hadamard Matrices, vol. 133, pp. 149–157. Springer Proceedings in Mathematics Statistics. Springer, Berlin (2015).

  9. Ivanov N.V.: Affine planes, ternary rings, and examples of non-Desarguesian planes. arXiv:1604.04945.

  10. Kharaghani H., Sasani S., Suda S.: Mutually unbiased Bush-type Hadamard matrices and association schemes. Electron. J. Comb. 22, \( {\rm {\# }}\)P3.10 (2015).

  11. Kharaghani H., Suda S.: Linked systems of symmetric group divisible designs. J. Algebraic Comb. 47, 319–343 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  12. Kharaghani H., Suda S.: Commutative association schemes obtained from twin prime powers, Fermat primes, Mersenne primes, preprint. arXiv:1709.08150.

  13. Kharaghani H., Suda S.: Disjoint weighing matrices, in preparation.

  14. Kharaghani H., Torabi R.: On a decomposition of complete graphs. Gr. Comb. 19(4), 519–526 (2003).

    Article  MATH  Google Scholar 

  15. Koukouvinos C.: Some new orthogonal designs in linear regression models. Stat. Probab. Lett. 29(2), 125–129 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  16. 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).

    Article  MathSciNet  MATH  Google Scholar 

  17. Mathon R.: The systems of linked 2-(16,6,2) designs. Ars Comb. 11, 131–148 (1981).

    MathSciNet  MATH  Google Scholar 

  18. Muzychuk M.E.: V-rings of permutation groups with invariant metric, Ph.D. thesis, Kiev State University (1987).

  19. Qiao Z., Du S.F., Koolen J.H.: \(2\)-Walk regular dihedrants from group divisible designs. Electron. J. Comb. 23, \( {\rm {\# }}\)P2.51 (2016).

  20. Robinson P.J.: Using product designs to construct orthogonal designs. Bull. Aust. Math. Soc. 16, 297–305 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  21. Stinson D.R.: Combinatorial Designs: Constructions and Analysis. Springer, New York (2004).

    MATH  Google Scholar 

  22. Wallis W.D.: Construction of strongly regular graphs using affine designs. Bull. Aust. Math. Soc. 4, 41–49 (1971).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for many useful comments, especially for suggesting a change that significantly shortened the proof of Proposition 3.2. Hadi Kharaghani is supported by an NSERC Discovery Grant. Sho Suda is supported by JSPS KAKENHI Grant Number 15K21075, 18K03395.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada

    Hadi Kharaghani

  2. Department of Mathematics Education, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi, 448-8542, Japan

    Sho Suda

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  1. Hadi Kharaghani
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Correspondence to Sho Suda.

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Communicated by M. Buratti.

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Kharaghani, H., Suda, S. Linked systems of symmetric group divisible designs of type II. Des. Codes Cryptogr. 87, 2341–2360 (2019). https://doi.org/10.1007/s10623-019-00622-z

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