  A set of vectors of equal norm in $\mathbb{C}^d$ 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 $\mathbb{C}^d$ for every $d \geq 2$. 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.
