A survey on tight Euclidean t-designs and tight relative t-designs in certain association schemes

Abstract

It is known that there is a close analogy between the two relations “Euclidean t-designs vs. spherical t-designs” and “relative t-designs in binary Hamming association schemes vs. combinatorial t-designs.” We first look at this analogy and survey the known results, putting emphasis on the study of tight relative t-designs in certain Q-polynomial association schemes. We then specifically study tight relative 2-designs on two shells in binary Hamming association schemes H(n, 2) and Johnson association schemes J(v, k). The purpose of this paper is to convince the reader that there is a rich theory even for these special cases and that the time is ripe to study tight relative t-designs more systematically for general Q-polynomial association schemes.

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Correspondence to Eiichi Bannai.

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Bannai, E., Bannai, E. & Zhu, Y. A survey on tight Euclidean t-designs and tight relative t-designs in certain association schemes. Proc. Steklov Inst. Math. 288, 189–202 (2015). https://doi.org/10.1134/S0081543815010149

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Keywords

  • Weight Function
  • Orthonormal Basis
  • STEKLOV Institute
  • Association Scheme
  • Cubature Formula