Title

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

Document Type

Article

Publication Date

2-1-2020

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.

Comments

The "Link to Full Text" button on this page loads the open access e-print at arXiv:1903.09177 [math.FA];
Date of arXiv submission: 21 Mar 2019, updated 4 Oct 2019,

The publisher version of record is hosted at ScienceDirect as a subscription article. A citation is noted below.

DOI

10.1016/j.laa.2019.10.019

Source Publication

Linear Algebra and Its Applications

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