DeepAI AI Chat
Log In Sign Up

Harmonic Grassmannian codes

12/28/2021
by   Matthew Fickus, et al.
0

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 ℂ^Q for each prime power Q≥ 4, of Q-1 planes in ℂ^Q for each odd prime power Q, and of 11 three-dimensional subspaces in 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

03/21/2019

Harmonic equiangular tight frames comprised of regular simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a...
02/17/2023

Doubly transitive equiangular tight frames that contain regular simplices

An equiangular tight frame (ETF) is a finite sequence of equal norm vect...
10/29/2022

Harmonic Tutte polynomials of matroids II

In this work, we introduce the harmonic generalization of the m-tuple we...
08/31/2021

Upper bounds on the length function for covering codes

The length function ℓ_q(r,R) is the smallest length of a q-ary linear co...
01/28/2021

A note on tight projective 2-designs

We study tight projective 2-designs in three different settings. In the ...
03/27/2018

Minimal Linear Codes over Finite Fields

As a special class of linear codes, minimal linear codes have important ...
12/23/2022

Equi-isoclinic subspaces, covers of the complete graph, and complex conference matrices

In 1992, Godsil and Hensel published a ground-breaking study of distance...