Equiangular tight frames from Paley tournaments

We prove the existence of equiangular tight frames having n=2d-1 elements drawn from either Cd or Cd-1 whenever n is either 2k-1 for k∈N, or a power of a prime such that . We also find a simple explicit expression for the prime power case by establishing a connection to a 2d-element equiangular tight frame based on quadratic residues.