Putatively Optimal Projective Spherical Designs With Little Apparent Symmetry

Abstract

We give some new explicit examples of putatively optimal projective spherical designs, that is, 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 $\left(3,3\right)$-design for $R^3$, and a 12-point spherical $\left(2,2\right)$-design for $R^4$ given by four Mercedes-Benz frames that lie on equi-isoclinic planes. The latter example shows that the set of optimal spherical designs can be uncountable. 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.