On Tight 4-Designs in Hamming Association Schemes


We complete the classification of tight 4-designs in Hamming association schemes H(n,q), i.e., that of tight orthogonal arrays of strength 4, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight 4-design in H(n,q) and analyze its triple intersection numbers to conclude the non-existence in all open cases.

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  1. [1]

    E. Bannai: On tight designs, Quart. J. Math. Oxford Ser. (2)28 (1977), 433–448.

    MathSciNet  Article  Google Scholar 

  2. [2]

    E. Bannai, E. Bannai and T. Ito: Introduction to Algebraic Combinatorics, Kyoritsu Shuppan, Tokyo, 2016.

    Google Scholar 

  3. [3]

    E. Bannai and R. M. Damerell: Tight spherical designs. I, J. Math. Soc. Japan31 (1979), 199–207.

    MathSciNet  Article  Google Scholar 

  4. [4]

    E. Bannai and R. M. Damerell: Tight spherical designs. II, J. London Math. Soc. (2)21 (1980), 13–30.

    MathSciNet  Article  Google Scholar 

  5. [5]

    E. Bannai and T. Ito: Algebraic combinatorics I: Association schemes, The Benjamin/Cummings Publishing Co., Inc., 1984.

    Google Scholar 

  6. [6]

    A. E. Brouwer, A. M. Cohen and A. Neumaier: Distance-regular graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1989.

    Google Scholar 

  7. [7]

    P. J. Cameron, J.-M. Goethals and J. J. Seidel: Strongly regular graphs having strongly regular subconstituents, J. Algebra55 (1978), 257–280.

    MathSciNet  Article  Google Scholar 

  8. [8]

    K. Coolsaet and A. Jurisic: Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs, J. Combin. Theory Ser. A115 (2008), 1086–1095.

    MathSciNet  Article  Google Scholar 

  9. [9]

    E. Van Dam, W. Martin and M. Muzychuk: Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems, J. Combin. Theory Ser. A120 (2013), 1401–1439.

    MathSciNet  Article  Google Scholar 

  10. [10]

    P. Delsarte: An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppi, (10):vi+97, 1973.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    P. Delsarte, J.-M. Goethals and J. J. Seidel: Spherical codes and designs, Geometriae Dedicata6 (1977), 363–388.

    MathSciNet  Article  Google Scholar 

  12. [12]

    A. L. Gavrilyuk and J. H. Koolen: The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to pseudo-partition graphs, Linear Algebra Appl.466 (2015), 117–140.

    MathSciNet  Article  Google Scholar 

  13. [13]

    A. S. Hedayat, N. J. A. Sloane and J. Stufken: Orthogonal arrays. Theory and applications, Springer Series in Statistics. Springer, New York, 1999.

    Google Scholar 

  14. [14]

    Y. Hong: On the nonexistence of nontrivial perfect e-codes and tight 2e-designs in Hamming schemes H(n,q) with e>3 and g>3, Graphs Combin.2 (1986), 145–164.

    MathSciNet  Article  Google Scholar 

  15. [15]

    A. Jurisic, J. Koolen and P. Terwilliger: Tight distance-regular graphs, J. Algebraic Combin.12 (2000), 163–197.

    MathSciNet  Article  Google Scholar 

  16. [16]

    A. Jurisic and J. Vidali: Extremal 1-codes in distance-regular graphs of diameter 3, Des. Codes Cryptogr.65 (2012), 29–47.

    MathSciNet  Article  Google Scholar 

  17. [17]

    A. Jurisic and J. Vidali: Restrictions on classical distance-regular graphs, J. Algebraic Combin.46 (2017), 571–588.

    MathSciNet  Article  Google Scholar 

  18. [18]

    W. J. Martin, M. Muzychuk and J. Williford: Imprimitive cometric association schemes: constructions and analysis, J. Algebraic Combin.25 (2007), 399–415.

    MathSciNet  Article  Google Scholar 

  19. [19]

    R. Mukerjee and S. Kageyama: On existence of two symbol complete orthogonal arrays, J. Combin. Theory Ser. A66 (1994), 176–181.

    MathSciNet  Article  Google Scholar 

  20. [20]

    R. Noda: On orthogonal arrays of strength 4 achieving Rao’s bound, J. London Math. Soc. (2)19 (1979), 385–390.

    MathSciNet  Article  Google Scholar 

  21. [21]

    R. C. Rao: Factorial experiments derivable from combinatorial arrangements of arrays, Suppl. J. Roy. Statist. Soc.9 (1947), 128–139.

    MathSciNet  Article  Google Scholar 

  22. [22]

    D. K. Ray-Chaudhuri and R. M. Wilson: On i-designs, Osaka J. Math. 12 (1975), 737–744, http://projecteuclid.org/euclid.ojm/1200758175.

    MathSciNet  MATH  Google Scholar 

  23. [23]

    The Sage Developers: SageMath, the Sage Mathematics Software System (Version 8.8), 2019, http://www.sagemath.org.

    Google Scholar 

  24. [24]

    D. R. Stinson: Bounds for orthogonal arrays with repeated rows, 2018, arXiv:1812.05145.

    Google Scholar 

  25. [25]

    S. Sijda: Coherent configurations and triply regular association schemes obtained from spherical designs, J. Combin. Theory Ser. A117 (2010), 1178–1194.

    MathSciNet  Article  Google Scholar 

  26. [26]

    M. Urlep: Triple intersection numbers of Q-polynomial distance-regular graphs, European J. Combin.33 (2012), 1246–1252.

    MathSciNet  Article  Google Scholar 

  27. [27]

    J. Vidali: Using symbolic computation to prove nonexistence of distance-regular graphs, Electron. J. Combin. 25 (2018), P4.21. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i4p21

    MathSciNet  Article  Google Scholar 

  28. [28]

    J. Vidali: jaanos/sage-drg: sage-drg, Sage package v0.9, 2019, https://github.com/jaanos/sage-drg/.

    Google Scholar 

  29. [29]

    R. M. Wilson and D. K. Ray-Chaudhuri: Generalization of Fisher’s inequality to i-designs, Notices Amer. Math. Soc. 18 (1971), 805.

    Google Scholar 

  30. [30]

    Z. Xiang: Nonexistence of nontrivial tight 8-designs, J. Algebraic Combin.47 (2018), 301–318.

    MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Janoš Vidali.

Additional information

Alexander L. Gavrilyuk is supported by BK21plus Center for Math Research and Education at Pusan National University, and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number NRF-2018R1D1A1B07047427). Sho Suda is supported by JSPS KAKENHI Grant Number 18K03395. Janos Vidali is supported by the Slovenian Research Agency (research program Pl-0285 and project J1-8130).

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Gavrilyuk, A.L., Suda, S. & Vidali, J. On Tight 4-Designs in Hamming Association Schemes. Combinatorica 40, 345–362 (2020). https://doi.org/10.1007/s00493-019-4115-z

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Mathematics Subject Classification (2010)

  • 05E30
  • 05B15