On Relative $t$-Designs in Polynomial Association Schemes

  • Eiichi Bannai Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
  • Etsuko Bannai Misakigaoka 2-8-21, Itoshima 819-1136, Japan
  • Sho Suda Department of Mathematics Education, Aichi University of Education, Kariya 448-8542, Japan
  • Hajime Tanaka Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
Keywords: Relative $t$-design, Fisher type inequality, Terwilliger algebra


Motivated by the similarities between the theory of spherical $t$-designs and that of $t$-designs in $Q$-polynomial association schemes, we study two versions of relative $t$-designs, the counterparts of Euclidean $t$-designs for $P$- and/or $Q$-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple $\mathbb{C}$-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative $t$-designs, assuming that certain irreducible modules behave nicely. The two versions of relative $t$-designs turn out to be equivalent in the case of the Hamming schemes. From this point of view, we establish a new algebraic characterization of the Hamming schemes.

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