Optimal size of fixed-strength spherical designs

Determine whether, for every fixed positive integer t, there exist spherical 2t-designs in R^d with O_t(d^t) points.

Background

The paper studies the asymptotic regime in which the strength t is fixed and the dimension d tends to infinity. The Delsarte–Goethals–Seidel lower bound gives an Omega_t(d^t) lower bound for spherical 2t-designs, while the paper establishes constructions of signed designs of this order and weighted designs with comparable upper bounds.

The unresolved issue is whether the same O_t(d^t) upper bound can be achieved by ordinary, unweighted spherical designs. The paper notes that the answer is known for strengths 4 and 8 in the notation 2t, corresponding to t=2 and t=4, but leaves the general case open.