# Harmonic equiangular tight frames comprised of regular simplices

## TL;DR

This paper investigates harmonic equiangular tight frames made of regular simplices, characterizing when their subspaces form optimal packings, and constructs new complex circulant conference matrices from difference sets.

## Contribution

It characterizes conditions for subspaces of harmonic ETFs to form equi-isoclinic tight fusion frames and introduces new constructions of complex circulant conference matrices using difference sets.

## Key findings

- Characterization of when subspaces form EITFFs
- Connection between difference sets and complex circulant conference matrices
- Explicit infinite families of ETFs satisfying the theory

## Abstract

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies.