Efficient lifts into odd dimensions of Spherical $5$- and $7$-designs with Balanced Halves

We propose constructions of spherical $5$- and $7$-designs in odd dimensions by lifting balanced halves of even-dimensional designs to several layers of suitable latitudes. For every odd dimension, we obtain an explicit spherical $5$-design with at most $60(d^2-1)$ points, attaining the optimal quadratic order in the dimension. Assuming Zauner’s conjecture, this bound improves to $7.5(d-1{)}^2$. For every odd $d\ge 9$, we also construct a spherical $7$-design with $110M$ points from a complex projective $3$-design with $M$ points in ${\mathbb{C}\mathbb{P}}^{(d-3)\mathrm{/}2}$.