Nearly orthogonal vectors and small antipodal spherical codes

Abstract

How can d+k vectors in ℝ^d be arranged so that they are as close to orthogonal as possible? In particular, define θ(d, k) := min_X max_x≠y∈X |〈x, y〉 | where the minimum is taken over all collections of d + k unit vectors X ⊆ ℝ^d. In this paper, we focus on the case here k is fixed and d → ∞. In establishing bounds on θ(d, k), we find an intimate connection to the existence of systems of $\left(\begin{array}{c}k+1\\ 2\end{array}\right)$ equiangular lines in ℝ^k. Using this connection, we are able to pin down θ(d, k) whenever k ∈ {1, 2, 3, 7, 23} and establish asymptotics for general k. The main tool is an upper bound on $\mathbb{E}_{x,y\sim\mu}|\langle{x,y}\rangle|$ whenever μ is an isotropic probability mass on ℝ^k, which may be of independent interest. Our results translate naturally to the analogous question in ℂ^d. In this case, the question relates to the existence of systems of k² equiangular lines in ℂ^k, also known as SIC-POVM in physics literature.