Harmonic Grassmannian codes

TL;DR

This paper introduces new infinite families of equi-isoclinic tight fusion frames (EITFFs) with applications in compressed sensing, constructed via harmonic analysis and properties of Gauss sums, expanding the known classes of these codes.

Contribution

The paper constructs novel infinite families of harmonic EITFFs that are not tensor-sized, generalizing previous known examples and linking harmonic ETFs with difference sets.

Findings

Constructed EITFFs of Q planes in ^Q for prime powers Qa0ea0

Developed EITFFs of Q-1 planes in ^Q for odd prime powers Q

Created 11 three-dimensional subspaces in ^{11}

Abstract

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.