Explicit construction of spherical $5$- and $7$-designs
Ryutaro Misawa
TL;DR
The paper constructs explicit spherical $5$-designs in all dimensions with $|X_d|=\mathcal{O}(d^3)$ and, in even dimensions, spherical $7$-designs with $|X_d|=\mathcal{O}(d^6)$ under prime-power thinning. Central to the approach is a lifting framework: high-dimensional spherical designs are built from tight $t$-fusion frames and lower-dimensional designs, with a core simplex $3$-design realized as $S_d$-orbits driving the construction. The authors explicitly realize simplex $3$-designs, build tight fusion frames on Grassmannians, and fuse projective toric and complex projective designs to obtain ordinary spherical designs via a sequence of explicit steps, including Rabau–Bajnok liftings for odd dimensions. The work combines combinatorial and geometric design theory to deliver implementable, scalable constructions that advance explicit design theory and have potential applications in numerical integration on spheres and related areas.
Abstract
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 $|X|=\mathcal{O}(d^3)$. 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 $\mathcal{O}(d^6)$.