Spherical designs in four dimensions

A preliminary report on our investigations into the existence of spherical t-designs on the unit sphere Ω₄ in 4-dimensional Euclidean space. Tables are given of the putatively best t-designs with up to 100 points. Some general constructions are proposed and explicit constructions are given for N-point strength t designs with N=4p and t=min{p-1,5}, N=6p and t=min{p-1,7}, and N=12p and t=min{p-1,11}, for all p ≥ 1.