Equiangular frames, k-angle frames, and their duals
Equiangular tights frames (ETFs) are highly desirable objects arising in many different areas as in communications, quantum information processing, and coding theory. The vertices of a regular simplex will give an ETF of d+1 vectors in a d dimensional space. However, aside from this, ETFs do not exist in many cases, and are hard to construct when they do exist. ETFs are frames which are tight and the angle between any two vectors is the same (equiangular). In the absence of ETFs, one can consider generalizations of ETFs called k-angle tight frames which are tight frames with few (k) distinct angles between pairs of vectors. In this talk, we first give an explicit construction of ETFs of size d+1 in a d dimensional space. Using this, we discuss the construction of tight frames for which the number of distinct angles between vectors can be made less than k, for a given k. Looking at duals of frames, the canonical dual of an ETF is also an ETF and hence equiangular. Considering equiangular frames that are not necessarily tight, we discuss conditions under which the canonical dual will have a small number of distinct angles between pairs even if it cannot be equiangular.