Spherical (t,t)-designs with a small number of vectors

Daniel Hughes and Shayne Waldron


For t=1,2,... fixed, a natural class of spherical designs is given by the vectors v_1,...,v_n in F^d=R^d,C^d (not all zero) which give equality in the bound $$ \sum_{j=1}^n \sum_{k=1}^n |\inpro{v_j,v_k}|^{2t} \ge c_t(\Fd) \Bigl(\sum_{\ell=1}^n \norm{v_\ell}^{2t}\Bigr)^2, $$ where $c_t(\Fd)$ is a known constant. These spherical (t,t)-designs integrate a space of homogeneous polynomials of degree 2t, and are variously known as real spherical half-designs of order 2t, complex (projective) t-designs, complex spherical semi-designs, and as tight frames when t=1. Little is known about the minimal number of vectors n for such a design.
Here we report on the results of a numerical search for (t,t)-designs with a minimal number of vectors. In some cases, we obtain the designs explicitly as an orbit of a unitary action of a finite group on the sphere. We also list all the currently known (t,t)-designs. It is shown that many of these belong to a family of designs which we construct from the complex reflection groups. This family includes several new spherical (t,t)-designs with a small number of vectors.

Keywords: (weighted) spherical designs, integration (cubature) rules for the sphere, spherical (t,t)-designs, spherical half-designs, finite tight frames, tight spherical designs, MUBs (mutually unbiased bases), SICs (symmetric informationally complete positive operator valued measures), equiangular lines, highly symmetric tight frames, complex reflection groups.

Math Review Classification: Primary 05B30, 20F55, 31C20, 65D30, 65D32: Secondary 42C15, 51F15, 94A12.

Length: 21 pages

Last Updated: 5 August 2018