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


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