  Spherical t-designs are Chebyshev-type averaging sets on the d-sphere S^d
which are exact for polynomials of degree at most t. This concept was
introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the
minimum possible size of such designs, in particular, that the number of points
in a 3-design on S^d must be at least n>=2d+2. In this paper we give explicit
constructions for spherical 3-designs on S^d consisting of n points for d=1 and
n>=4; d=2 and n=6; 8; >= 10; d=3 and n=8; >=10; d = 4 and n = 10; 12; >= 14;
d>=5 and n>=5(d+1)/2 odd or n>=2d+2 even. We also provide some evidence that
3-designs of other sizes do not exist.
