Nearly orthogonal vectors and small antipodal spherical codes

How can d+k vectors in R be arranged so that they are as close to orthogonal as possible? In particular, define θ(d, k) := minX maxx 6=y∈X |〈x, y〉| where the minimum is taken over all collections of d + k unit vectors X ⊆ R. In this paper, we focus on the case where k is fixed and d → ∞. In establishing bounds on θ(d, k), we find an intimate connection to the existence of systems of ( k+1 2 ) equiangular lines in R. 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 Ex,y∼μ|〈x, y〉| whenever μ is an isotropic probability mass on R, which may be of independent interest. Our results translate naturally to the analogous question in C. In this case, the question relates to the existence of systems of k equiangular lines in C, also known as SIC-POVM in physics literature.