Frames, Group Codes, and Subgroups of (Z/pZ)×

The problem of designing low coherence matrices and low-correlation frames arises in a variety of fields, including compressed sensing, MIMO communications and quantum measurements. The challenge is that one must control the (^n_2) pairwise inner products of the columns of the matrix. In this paper, we follow the group code approach of David Slepian [1], which constructs frames using unitary group representations and which in general reduces the number of distinct inner products to n-1. When n is a prime p, we present a carefully chosen representation which reduces the number of distinct inner products further to ^(n-1)/_m, where m is the number of rows in the matrix. The resulting frames have superior performance to many earlier frame constructions and, in some cases, yield frames with optimally low coherence. We further expand a connection between frames and difference sets noted first in [2] to find bounds on the coherence when ^(n-1)/_m = 2 and 3.