Frames from generalized group fourier transforms and SL2(Fq)

We explore the problem of deterministically constructing frames and matrices with low coherence, which arises in areas such as compressive sensing, spherical codes, and MIMO communications. In particular, we present a generalization of the familiar harmonic frame by selecting a subset of rows of the generalized discrete Fourier transform matrix over finite groups. We apply our methods to the group SL
₂
(F
_q
) and show how to produce frames with remarkably low coherence, for which we provide upper bounds.