Relation between spherical designs through a Hopf map

Cohn--Conway--Elkies--Kumar [Experiment. Math. (2007)] described that one can construct a family of designs on $S^{2n-1}$ from a design on $\mathbb{CP}^{n-1}$. In this paper, we prove their claim for the case where $n=2$. That is, we give an algorithm to construct $2t$-designs on $S^{3}$ as products through a Hopf map $S^3 \rightarrow S^2$ of a $t$-design on $S^2$ and a $2t$-design on $S^1$.