Putatively optimal projective spherical designs with little apparent symmetry

TL;DR

This paper introduces new explicit examples of minimal-size projective spherical designs with little symmetry, using novel construction techniques, and investigates their algebraic variety through extensive numerical analysis.

Contribution

It provides the first explicit constructions of certain minimal projective spherical designs lacking symmetry, and explores their algebraic structure.

Findings

New 11-point spherical (3, 3)-design in R^3

New 12-point spherical (2, 2)-design in R^4 with Mercedes-Benz frames

Numerical evidence on the algebraic variety of optimal designs

Abstract

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in general, which requires the introduction of new techniques for their construction. New examples of interest include an 11-point spherical (3, 3)-design for R 3 , and a 12-point spherical (2, 2)-design for R 4 given by four Mercedes-Benz frames that lie on equi-isoclinic planes. We also give results of an extensive numerical study to determine the nature of the real algebraic variety of optimal projective real spherical designs, and in particular when it is a single point (a unique design) or corresponds to an infinite family of designs.