The concept of a spherical design is due to Delsarte, Goethals, and Seidel,[1] although these objects were understood as particular examples of cubature formulas earlier.
[2] Shortly thereafter, Seymour and Zaslavsky[3] proved that such designs exist of all sufficiently large sizes; that is, given positive integers d and t, there is a number N(d,t) such that for every N ≥ N(d,t) there exists a spherical t-design of N points in dimension d. However, their proof gave no idea of how big N(d,t) is.
Mimura constructively found conditions in terms of the number of points and the dimension which characterize exactly when spherical 2-designs exist.
There are many sporadic small spherical designs; many of them are related to finite group actions on the sphere.
for all positive integers d and t. This asymptotically matches the lower bound given originally by Delsarte, Goethals, and Seidel.