# On the Existence of Equiangular Tight Frames

### Authors:

M. A. Sustik,

J. A. Tropp,

I.
Dhillon, and

R. W.
Heath, Jr.
### Reference:

Submitted to *Linear Algebra and its Applications*.

### Abstract:

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.

This paper is available on .IEEE Xplore . The final version is available at Linear Algebra and its Applications.