A Simple Construction of Complex Equiangular Lines

A set of vectors of equal norm in C-d represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is d(2), and it is conjectured that sets of this maximum size exist in C-d for every d >= 2. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.