On Weyl-Heisenberg orbits of equiangular lines

Abstract

An element z ℂℙd-1 is called fiducial if {gz:g ε G} is a set of lines with only one angle between each pair, where G ≅ ℤd ×ℤ d is the one-dimensional finite Weyl-Heisenberg group modulo its centre. We give a new characterization of fiducial vectors. Using this characterization, we show that the existence of almost flat fiducial vectors implies the existence of certain cyclic difference sets. We also prove that the construction of fiducial vectors in prime dimensions 7 and 19 due to Appleby (J. Math. Phys. 46(5):052107, 2005) does not generalize to other prime dimensions (except for possibly a set with density zero). Finally, we use our new characterization to construct fiducial vectors in dimension 7 and 19 whose coordinates are real. © 2007 Springer Science+Business Media, LLC.