Radon-Hurwitz Grassmannian codes
Matthew Fickus, Enrique Gomez-Leos, Joseph W. Iverson
TL;DR
The paper develops a Radon–Hurwitz–driven framework for equi-isoclinic tight fusion frames with subspace dimension $d=2r$, providing an explicit isometry form and a complete existence criterion $n \le \rho_{\mathbb{F}}(r)+2$ via $\rho$-spaces and $\rho$-simplexes. It shows that all Radon–Hurwitz EITFFs possess at least alternating symmetry and identifies broad conditions under which total symmetry occurs, including new infinite families constructed from RH data. The results unite Grassmannian code optimality with classical Radon–Hurwitz theory, yielding deterministic, explicitly constructible codes with minimal block coherence and potential benefits for compressed sensing. The work also clarifies how symmetry properties constrain parameter regimes and opens avenues to finish real-case total-symmetry classifications and to integrate with existing symmetric EITFF constructions.
Abstract
Every equi-isoclinic tight fusion frame (EITFF) is a type of optimal code in a Grassmannian, consisting of subspaces of a finite-dimensional Hilbert space for which the smallest principal angle between any pair of them is as large as possible. EITFFs yield dictionaries with minimal block coherence and so are ideal for certain types of compressed sensing. By refining classical work of Lemmens and Seidel based on Radon-Hurwitz theory, we fully characterize EITFFs in the special case where the dimension of the subspaces is exactly one-half of that of the ambient space. We moreover show that each such "Radon-Hurwitz EITFF" is highly symmetric, where every even permutation is an automorphism.