Nearly orthogonal vectors

How can d+k vectors be arranged in Rd so that they are as
close to orthogonal as possible? In particular, define θ(d,k) =
minX∈(Rd)d+k maxi̸=j |⟨xi,xj⟩|. In this talk, we will explore results
from a paper by Boris Bukh and Christopher Cox, Nearly orthogonal
vectors and small antipodal spherical codes, to establish a lower
bound on θ(d, k). The main tool is an upper bound on Ex,y∼μ|⟨x,y⟩|
whenever μ is an isotropic probability mass on Rk. The results
naturally translate to the analogous question for Cd.