Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials

We characterize the cases of existence of spherical designs of an odd strength attaining the Fazekas–Levenshtein bound for covering and prove some of their properties. We also find all universal minima of the potential of regular spherical configurations in two new cases: the demihypercube on \(S^d\) $S^d$ , \(d\ge 4\) $d\ge 4$ , and the \(2_{41}\) $2_{41}$ polytope on \(S^7\) $S^7$ (which is dual to the \(E_8\) $E_8$ lattice).