  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 $S^1$.
