Bounds on Equiangular Lines and on Related Spherical Codes

An L-spherical code is a set of Euclidean unit vectors whose pairwise inner products belong to the set L. We show, for a fixed 0<α,β<1, that the size of any [-1,-β]∪\α\-spherical code is at most linear in the dimension. In particular, this bound applies to sets of lines such that every two are at a fixed angle to each another.