Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N = 24, 26, ≥ 28; 7-designs for N = 24, 30, 32, 34, ≥ 36; 8-designs for N = 36, 40, 42, ≥ 44; 9-designs for N = 48, 50, 52, ≥ 54; 10-designs for N = 60, 62, ≥ 64; 11-designs for N = 70, 72, ≥ 74; and 12-designs for N = 84, ≥ 86. The existence of some of these designs is established analytical...