# Constructing Subspace Packings from Other Packings

## TL;DR

This paper explores methods to construct subspace packings from existing configurations, analyzing their algebraic and geometric properties, and extends the theory to include various optimal packings like equiisoclinic and orthoplex-bound arrangements.

## Contribution

It generalizes known construction techniques for subspace packings, characterizes preserved properties, and analyzes new configurations including those outside traditional parameter regimes.

## Key findings

- Provides a unified framework for constructing subspace packings.
- Characterizes algebraic and geometric properties preserved in constructions.
- Includes analysis of optimal packings like orthoplex-bound arrangements.

## Abstract

The desirable properties when constructing collections of subspaces often include the algebraic constraint that the projections onto the subspaces yield a resolution of the identity like the projections onto lines spanned by vectors of an orthonormal basis (the so-called tightness condition) and the geometric constraint that the subspaces form an optimal packing of the Grassmannian, again like the one-dimensional subspaces spanned by vectors in an orthonormal basis. In this article a generalization of related constructions which use known packings to build new configurations and which appear in numerous forms in the literature is given, as well as the characterization of a long list of desirable algebraic and geometric properties which the construction preserves. Another construction based on subspace complementation is similarly analyzed. While many papers on subspace packings focus only on so-called equiisoclinic or equichordal arrangements, attention is also given to other configurations like those which saturate the orthoplex bound and thus are optimal but lie outside of the parameter regime where equiisoclinic and equichordal packings can occur. Keywords: fusion frame, Grassmannian packing, simplex bound, orthoplex bound, equichordal, strongly simplicial, equiisoclinic