Submitted to Linear Algebra and its Applications.
In a recent paper, Holmes and Paulsen established a necessary condition for the existence of an N -vector equiangular tight frame in a d-dimensional real Euclidean space. This article develops much stronger necessary conditions using a combina- tion of field theory and graph theory. This investigation rules out many possibilities admitted by the work of Holmes and Paulsen. Using a new one-to-one correspon- dence between equivalence classes of real equiangular tight frames and strongly regular graphs of a certain type, it has been verified that a real equiangular tight frame exists for each pair (d, N ) with N ² 100 that meets the new conditions. The arguments also extend to deliver novel necessary conditions for the existence of equiangular tight frames whose Gram matrices have entries drawn from a discrete set of complex numbers.