Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimality of spherical codes via exact semidefinite programming bounds

Henry Cohn, David de Laat, Nando Leijenhorst

TL;DR

The paper advances the theory of spherical codes by proving exact optimality for several high-dimensional configurations using three-point semidefinite programming bounds, notably the spectral embeddings of triangle-free strongly regular graphs and Kerdock-based constructions. It develops a fast, rigorous rounding pipeline to convert floating SDP solutions into exact algebraic solutions, enabling certified optimality proofs for complex codes and universal-optimality results for the 288-point Nordstrom-Robinson code in $\mathbb{R}^{16}$. The authors also establish uniqueness for key constructions (e.g., block-length $64$ Kerdock codes) and provide comprehensive computational data, including a public codebase, to validate their bounds. Collectively, the work broadens the catalog of provably optimal spherical codes and demonstrates that three-point SDP bounds can be sharp in substantially more cases than previously known, with implications for energy minimization on spheres and related combinatorial designs.

Abstract

We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are $56$ points in $20$ dimensions, $50$ points in $21$ dimensions, and $77$ points in $21$ dimensions), as are certain mutually unbiased basis arrangements constructed using Kerdock codes in up to $1024$ dimensions (namely, $2^{4k} + 2^{2k+1}$ points in $2^{2k}$ dimensions for $2 \le k \le 5$). As a consequence of the latter, we obtain optimality of the Kerdock binary codes of block length $64$, $256$, and $1024$, as well as uniqueness for block length $64$. We also prove universal optimality for $288$ points on a sphere in $16$ dimensions. To prove these results, we use three-point semidefinite programming bounds, for which only a few sharp cases were known previously. To obtain rigorous results, we develop improved techniques for rounding approximate solutions of semidefinite programs to produce exact optimal solutions.

Optimality of spherical codes via exact semidefinite programming bounds

TL;DR

The paper advances the theory of spherical codes by proving exact optimality for several high-dimensional configurations using three-point semidefinite programming bounds, notably the spectral embeddings of triangle-free strongly regular graphs and Kerdock-based constructions. It develops a fast, rigorous rounding pipeline to convert floating SDP solutions into exact algebraic solutions, enabling certified optimality proofs for complex codes and universal-optimality results for the 288-point Nordstrom-Robinson code in . The authors also establish uniqueness for key constructions (e.g., block-length Kerdock codes) and provide comprehensive computational data, including a public codebase, to validate their bounds. Collectively, the work broadens the catalog of provably optimal spherical codes and demonstrates that three-point SDP bounds can be sharp in substantially more cases than previously known, with implications for energy minimization on spheres and related combinatorial designs.

Abstract

We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are points in dimensions, points in dimensions, and points in dimensions), as are certain mutually unbiased basis arrangements constructed using Kerdock codes in up to dimensions (namely, points in dimensions for ). As a consequence of the latter, we obtain optimality of the Kerdock binary codes of block length , , and , as well as uniqueness for block length . We also prove universal optimality for points on a sphere in dimensions. To prove these results, we use three-point semidefinite programming bounds, for which only a few sharp cases were known previously. To obtain rigorous results, we develop improved techniques for rounding approximate solutions of semidefinite programs to produce exact optimal solutions.
Paper Structure (25 sections, 6 theorems, 67 equations, 1 figure, 7 tables)

This paper contains 25 sections, 6 theorems, 67 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Triangle-free strongly regular graphs
  3. Kerdock codes
  4. Other spherical codes
  5. Universal optimality
  6. Organization of the paper
  7. Rounding procedure
  8. Describing the optimal face
  9. Transforming the problem
  10. Rounding the solution
  11. Rounding to algebraic numbers
  12. Finding the algebraic number field
  13. General applicability of the rounding procedure
  14. Implementation
  15. Computations
  16. ...and 10 more sections

Key Result

Theorem 1.1

Each of the spherical codes listed in Table table:newcases is optimal, and those in at most $64$ dimensions are unique up to isometry. Furthermore, the optimal $288$-point spherical code in $\mathbb{R}^{16}$ is universally optimal.

Figures (1)

  • Figure 1.1: This packing of $180$ spherical caps on a transparent sphere corresponds to a conjecturally optimal spherical code in $\mathbb{R}^3$ with chiral icosahedral symmetry T83. Its maximal inner product is the root $0.9621287940857901\dots$ of the polynomial $448505x^{18} + 1214430x^{17} + 1867769x^{16} + 1054048x^{15} - 1540076x^{14} - 3472264x^{13} - 2273628x^{12} + 709568x^{11} + 2115198x^{10} + 712036x^9 - 369522x^8 - 221472x^7 + 20772x^6 + 3416x^5 - 6956x^4 + 256x^3 + 81x^2 - 18x + 1$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Theorem 4.1
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • ...and 2 more