Design Theory from the Viewpoint of Algebraic Combinatorics

Abstract

We give a survey on various design theories from the viewpoint of algebraic combinatorics. We will start with the following themes.

  1. (i)

    The similarity between spherical t-designs and combinatorial t-designs, as well as t-designs in Q-polynomial association schemes.

  2. (ii)

    Euclidean t-designs as a two-step generalization of spherical t-designs.

  3. (iii)

    Relative t-designs as a two-step generalization of t-designs in Q-polynomial association schemes, and the similarity with Euclidean t-designs.

  4. (iv)

    Fisher type lower bounds for the sizes of these designs as well as the classification problems of some of tight t-designs and/or tight relative t-designs.

Our emphasis will be focused on the proposal to study relative t-designs, mostly tight relative t-designs, in known classical examples of P- and Q-polynomial association schemes. We relate our study with the representation theoretical aspect of the relevant association schemes and permutation groups, due to Charles Dunkl and Dennis Stanton and others. We propose several open problems, which seem to play a key role in this research direction. We also put emphasis on the future research directions in this research area and not on presenting the details of the established results on the study of design theory. In particular, we propose to study t-designs in each shell of these classical P- and Q-polynomial association schemes. In general, each shell is not Q-polynomial. It may be even non-commutative in some cases. The importance of the use of Terwilliger algebras in the study of relative t-designs in such association schemes will also be highlighted.

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Notes

  1. 1.

    We would like to record the following comment by a referee. “The authors allow any weight function for t-wise balanced design, but I think that normally when we consider t-wise balanced designs, it allows only weight functions with positive integer values. So it would be better to mention about it”.

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Correspondence to Eiichi Bannai.

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E. Bannai was supported in part by NSFC Grant No. 11271257.

H. Tanaka was supported in part by JSPS KAKENHI Grant No. 25400034.

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Bannai, E., Bannai, E., Tanaka, H. et al. Design Theory from the Viewpoint of Algebraic Combinatorics. Graphs and Combinatorics 33, 1–41 (2017). https://doi.org/10.1007/s00373-016-1739-2

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Keywords

  • Association scheme
  • t-design
  • Relative t-design
  • Spherical design
  • Euclidean design
  • Tight design
  • Terwilliger algebra