Constructing high order spherical designs as a union of two of lower order

We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical \((t,t)\)-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a \(32\) point \((9,9)\)-design for \(\mathbb{C}^2\), a \(48\) point \((4,4)\)-design for \(\mathbb{C}^3\), and a \(400\) point \((5,5)\)-design for \(\mathbb{C}^4\).From a real reflection group, we construct a \(360\) point \((9,9)\)-design for \(\mathbb{R}^4\) (spherical half-design of order \(18\)), i.e., a \(720\) point spherical \(19\)-design for \(\mathbb{R}^4\).