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

Abstract

A point set ${\mathrm{X}}_N$ on the unit sphere is a spherical $t$-design is equivalent to the nonnegative quantity $A_{N,t+1}$ vanished. We show that if ${\mathrm{X}}_N$ is a stationary point set of $A_{N,t+1}$ and the minimal singular value of basis matrix is positive, then ${\mathrm{X}}_N$ 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 $t+1$ up to $127$ at $N=(t+2{)}^2$.

arXiv comment11 pages, 5 figures, 2 tables