A construction of spherical $5$-designs with $O(d^2)$ points

For every $d\ge 1$ we give an explicit equal-weight spherical $5$-design in ${\mathbb{S}}^{d-1}\subset {\mathbb{R}}^d$ with at most $72d^2$ points. Our approach utilizes recent construction of complex projective $2$-designs based on Sidon sets.