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

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.