Nearly orthogonal vectors and small antipodal spherical codes

How can $d+k$ vectors in $mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $theta(d,k):=min_Xmax_{x eq yin X}|langle x,yrangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $Xsubseteqmathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $dtoinfty$. In establishing bounds on $theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1choose 2}$ equiangular lines in $mathbb{R}^k$. Using this connection, we are able to pin down $theta(d,k)$ whenever $kin{1,2,3,7,23}$ and establish asymptotics for general $k$. The main tool is an upper bound on $mathbb{E}_{x,ysimmu}|langle x,yrangle|$ whenever $mu$ is an isotropic probability mass on $mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $mathbb{C}^k$, also known as SIC-POVM in physics literature.