Numerical construction of spherical t-designs by Barzilai-Borwein method

Abstract

A point set XN on the unit sphere is a spherical t-design is equivalent to the nonnegative quantity AN,t+1 vanished. We show that if XN is a stationary point set of AN,t+1 and the minimal singular value of basis matrix is positive, then XN is a spherical t-design. Moreover, the numerical construction of spherical t-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical t-designs with N=(t+2)2 points for t+1 up to 127.