Nearly orthogonal vectors and small antipodal spherical codes
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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):=_X_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 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^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 E_x,y∼μ|〈 x,y〉| 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 k^2 equiangular lines in C^k, also known as SIC-POVM in physics literature.
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