On frames from abelian group codes

Designing low coherence matrices and low-correlation frames is a point of interest in many fields including compressed sensing, MIMO communications and quantum measurements. The challenge is that one must control the (ⁿ₂) pairwise inner products between the frame elements. In this paper, we exploit 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. We demonstrate how to efficiently find optimal representations of cyclic groups, and we show how basic abelian groups can be used to construct tight frames that have the same dimensions and inner products as those arising from certain more complex nonabelian groups. We support our work with theoretical bounds and simulations.