Putatively optimal projective spherical
designs with little apparent symmetry
Alex Elzenaar and Shayne Waldron
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.
One example of interest is a 12-point spherical (2, 2)-design for R 4 given by four
Mercedes-Benz frames lying 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.
Keywords:
spherical t-designs, spherical half-designs, tight spherical designs, finite
tight frames, integration rules, cubature rules, cubature rules for the sphere, numerical
optimisation, Manopt software, real algebraic variety
Math Review Classification:
Primary 05B30, 65D30, 65K10, 49Q12, 65H14;
Secondary 14Q10, 14Q65, 42C15, 94B25.
Length: 24 Pages
Last Updated: 31 May 2024
Availability: