  An equi-isoclinic tight fusion frame (EITFF) is a type of Grassmannian code,
being a sequence of subspaces of a finite-dimensional Hilbert space of a given
dimension with the property that the smallest spectral distance between any
pair of them is as large as possible. EITFFs arise in compressed sensing,
yielding dictionaries with minimal block coherence. Their existence remains
poorly characterized. Most known EITFFs have parameters that match those of one
that arose from an equiangular tight frame (ETF) in a rudimentary,
direct-sum-based way. In this paper, we construct new infinite families of
non-"tensor-sized" EITFFs in a way that generalizes the one previously known
infinite family of them as well as the celebrated equivalence between harmonic
ETFs and difference sets for finite abelian groups. In particular, we construct
EITFFs consisting of $Q$ planes in $\mathbb{C}^Q$ for each prime power $Q\geq
4$, of $Q-1$ planes in $\mathbb{C}^Q$ for each odd prime power $Q$, and of $11$
three-dimensional subspaces in $\mathbb{R}^{11}$. The key idea is that every
harmonic EITFF -- one that is the orbit of a single subspace under the action
of a unitary representation of a finite abelian group -- arises from a smaller
tight fusion frame with a nicely behaved "Fourier transform." Our particular
constructions of harmonic EITFFs exploit the properties of Gauss sums over
finite fields.
