Exact optimality of the simplex codebook

Let $\mathbb{S}^{d-1}$ be the unit sphere, and consider codebooks consisting of points on this sphere. A simplex codebook is a maximally separated codebook; the paper also considers rotationally symmetric codebooks. Simplex-codebook conjecture. The maximally separated codebook (simplex codebook) is exact-optimal because it provides the most effective coverage of the space $\mathbb{S}^{d-1}$. The paper proves that an exact-optimal codebook must be rotationally symmetric, but leaves the stronger simplex-codebook claim for future work.