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 $\mathbb{R}^4$.