A low coherence frame is a set of n unit norm vectors of dimension m<n, where
the maximum of the absolute value of the pairwise inner products is as small as
possible. Low coherence frames appear in many applications: MIMO communications,
spherical codes, quantum measurements, compressed sensing, etc. The challenge in
designing low coherence frames is that one needs to control "n-choose-2", i.e.,
O(n^2), pairwise inner products. We provide a grou-theoretic approach which, for
n prime, requires us to control only (n-1)/m inner products. As a result, the
frames thereby constructed have superior coherence over many existing
constructions and in some cases can be proven to have optimal coherence. We
remark on generalizations to non-prime n and present various simulation results.