Optimal codes in the Stiefel manifold

TL;DR

This paper investigates optimal coding strategies in the Stiefel manifold using geometric bounds, providing new upper bounds and explicit code constructions that achieve these bounds.

Contribution

It introduces upper bounds on the minimum distance of Stiefel codes using Rankin's bounds and constructs codes that attain these bounds.

Findings

Derived upper bounds on code minimum distance in Stiefel manifold

Constructed explicit codes achieving the bounds

Extended results to various low-dimensional cases

Abstract

We consider the coding problem in the Stiefel manifold with chordal distance. After considering various low-dimensional instances of this problem, we use Rankin's bounds on spherical codes to prove upper bounds on the minimum distance of a Stiefel code, and then we construct several examples of codes that achieve equality in these bounds.