An equiangular tight frame (ETF) is a sequence of equal-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 are formed by restricting the characters of a finite abelian group to a difference set. Recently, it was shown that some harmonic ETFs are themselves comprised of regular simplices. In this thesis, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame, which is a type of optimal packing in a Grassmannian space. It turns out that such ETFs yield complex circulant conference matrices; this is remarkable since real examples of such matrices are known to not exist. We further show that some of these ETFs yield mutually unbiased simplices, which are a natural generalization of the quantum-information-theoretic concept of mutually unbiased bases. Finally, we provide infinite families of ETFs that have all of these properties.
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