A notion of optimal packings of subspaces with mixed-rank and solutions

TL;DR

This paper introduces a new framework for optimal packings of subspaces with mixed ranks, resolving a longstanding problem by reformulating Grassmannian fusion frames and constructing infinite solutions using combinatorial designs.

Contribution

It reformulates Grassmannian fusion frames for mixed dimensions, introduces a classical embedding for comparison, and constructs infinite solutions using mutually unbiased bases and Hadamard 3-designs.

Findings

Resolved a longstanding open problem in subspace packings.

Constructed infinite families of solutions using combinatorial designs.

Showed that maximal orthoplectic fusion frames relate to Hadamard 3-designs.

Abstract

We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to compare elements of mixed dimension, we use a classical embedding to send all fusion frame elements to points on a higher dimensional Euclidean sphere, where they are given "equal footing". Over the embedded images -- a compact subset in the higher dimensional embedded sphere -- we define optimality in terms of the corresponding restricted coding problem. We then construct infinite families of solutions to the problem by using maximal sets of mutually unbiased bases and block designs. Finally, we show that using Hadamard 3-designs in this construction leads to infinite examples of maximal orthoplectic fusion frames of constant-rank. Moreover, any such fusion frames constructed by this method must come from Hadamard 3-designs.