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

## Daniel Hughes and Shayne Waldron

## Abstract:

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

## Availability: