Explicit construction of spherical $5$- and $7$-designs

This paper develops an explicit and implementable framework for constructing spherical designs by lifting point sets from tight fusion frames. By combining existing ingredients, we obtain, in every dimension, explicit spherical $5$-designs with $\mathrm{\mid }X\mathrm{\mid }=\mathcal{O}(d^3)$. As a core component of the method, we give an explicit construction of simplex $3$-designs realized as orbits of the symmetric group. Using these simplex designs as input, we further construct spherical $7$-designs in arbitrary even dimensions; more precisely, for every even integer $d\ge 6$ we obtain spherical $7$-designs in dimension $d$, and if $\frac{d}{2}-1$ is a prime power then the number of points is $\mathcal{O}(d^6)$.