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

## TL;DR

This paper introduces a numerical method using the Barzilai-Borwein algorithm to construct spherical t-designs on the unit sphere, achieving designs with high degrees up to t+1=127.

## Contribution

It demonstrates that stationary points of a specific quantity, combined with basis matrix conditions, can produce spherical t-designs via an efficient numerical approach.

## Key findings

- Successfully constructed spherical t-designs with t+1 up to 127.
- Validated the method's effectiveness for designs with N=(t+2)^2 points.
- Established theoretical conditions linking stationary points to t-designs.

## 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 $N=(t+2)^2$.