In this paper we study the normaliser of $H$, which we view as a group of symmetries of the equations that determine a Heisenberg frame. This allows us to simplify the equations for a Heisenberg frame. From these simplified equations we are able construct analytic solutions for $d=5,7$, and make conjectures about the form of a solution when $d$ is odd. Most notably, it appears that solutions for $d$ odd are eigenvectors of some element in the normaliser which has (scalar) order $3$.
Keywords: complex spherical 2-design, equiangular lines, equiangular tight frame, Grassmannian frame, Heisenberg frame, isometric embeddings, discrete Heisenberg group modulo d, SICPOVM (symmetric informationally--complete positive operator--valued measure)
Math Review Classification: Primary 05B30, 42C15, 65D30, 81P15; Secondary 51F25, 81R05
Length: 27 pages
Last Updated: 25 June 2007