On computing all harmonic frames of n vectors in Cd

There are a finite number of inequivalent isometric frames (equal-norm tight frames) of n vectors for Cd which are generated from a single vector by applying an Abelian group G of symmetries. Each of these so-called harmonic frames can be obtained by taking d rows of the character table of G; often in many different ways, which may even include using different Abelian groups. Using an algorithm implemented in the algebra package Magma, we determine which of these are equivalent. The resulting list of all harmonic frames for various choices of n and d is freely available, and it includes many properties of the frames such as: a simple description, which Abelian groups generate it, identification of the full group of symmetries, the minimum, average and maximum distance between vectors in the frame, and whether it is real or complex, lifted or unlifted. Additional attributes aimed at specific applications include: a measure of the cross correlation (Grassmannian frames), the number of erasures (robustness to erasures), and the diversity product of the full group of its symmetries (multiple-antenna code design). Some outstanding frames are identified and discussed, and a number of questions are answered by considering the examples on the list.