# Numerical computation of triangular complex spherical designs with small mesh ratio

## TL;DR

This paper develops methods for constructing triangular complex spherical designs with optimal point counts on complex spheres, emphasizing numerical computation and geometric efficiency, and provides explicit examples for dimensions 2 to 6.

## Contribution

It introduces a variational approach for computing triangular complex spherical designs with small mesh ratios, establishing their existence and providing explicit numerical examples.

## Key findings

- Existence of triangular and square complex spherical t-designs with optimal point counts.
- Numerical methods for computing designs with good geometric properties.
- Explicit examples of designs for dimensions 2 to 6.

## Abstract

This paper provides triangular spherical designs for the complex unit sphere $\Omega^d$ by exploiting the natural correspondence between the complex unit sphere in $d$ dimensions and the real unit sphere in $2d-1$. The existence of triangular and square complex spherical $t$-designs with the optimal order number of points is established. A variational characterization of triangular complex designs is provided, with particular emphasis on numerical computation of efficient triangular complex designs with good geometric properties as measured by their mesh ratio. We give numerical examples of triangular spherical $t$-designs on complex unit spheres of dimension $d=2$ to $6$.