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Optimal codes in the Stiefel manifold

John Jasper, Nathan Mankovich, Dustin G. Mixon

TL;DR

This work addresses coding on the Stiefel manifold with chordal distance, aiming to maximize the minimum Frobenius distance among $n$ codewords in $\operatorname{St}_F(d,r)$. It extends Rankin's spherical bounds to the Stiefel setting, deriving the Stiefel simplex bound and the Stiefel orthoplex bound, and then constructs numerous optimal codes that achieve equality in these bounds. The contributions include a suite of SSC and SOC constructions, leveraging lifting, Kronecker products, resolvable BIBDs, group-orbit actions, and Hadamard/Hamming-based methods, with explicit results in small dimensions and broad existence results in general. These developments connect spherical code theory to Stiefel-manifold codes, with potential implications for MIMO communications and high-dimensional signal processing.

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.

Optimal codes in the Stiefel manifold

TL;DR

This work addresses coding on the Stiefel manifold with chordal distance, aiming to maximize the minimum Frobenius distance among codewords in . It extends Rankin's spherical bounds to the Stiefel setting, deriving the Stiefel simplex bound and the Stiefel orthoplex bound, and then constructs numerous optimal codes that achieve equality in these bounds. The contributions include a suite of SSC and SOC constructions, leveraging lifting, Kronecker products, resolvable BIBDs, group-orbit actions, and Hadamard/Hamming-based methods, with explicit results in small dimensions and broad existence results in general. These developments connect spherical code theory to Stiefel-manifold codes, with potential implications for MIMO communications and high-dimensional signal processing.

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.
Paper Structure (5 sections, 13 theorems, 32 equations)

This paper contains 5 sections, 13 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Optimal codes in small dimensions
  3. Stiefel simplex codes
  4. Stiefel orthoplex codes
  5. Discussion

Key Result

Proposition 8

For every $x_1,\ldots,x_n\in S^{d-1}$, it holds that with equality precisely when $\{x_i\}_{i\in[n]}$ form the vertices of a regular simplex centered at the origin. Furthermore, such a code exists only if $n\leq d+1$.

Theorems & Definitions (29)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Proposition 8: Rankin's simplex bound
  • Proposition 9: Rankin's orthoplex bound
  • Theorem 10: Stiefel simplex bound
  • ...and 19 more