[Paper Review] A survey on the kissing numbers

Peter Boyvalenkov, S.M. Dodunekov+1 more|arXiv (Cornell University)|Jul 13, 2015

Mathematical Approximation and Integration1 references21 citations

TL;DR

This survey provides a comprehensive overview of the kissing number problem in n-dimensional space, focusing on bounds for spherical codes and sphere packings. It synthesizes classical and modern techniques—particularly linear and semidefinite programming—demonstrating how these methods yield tight upper bounds, while error-correcting code constructions (like Constructions A and B) produce the best-known lower bounds. The key contribution is a detailed, up-to-date synthesis of known kissing numbers and bounds up to dimension 32, with exact values confirmed for dimensions 1, 2, 3, 4, 8, and 24.

ABSTRACT

The maximum possible number of non-overlapping unit spheres that can touch a unit sphere in $n$ dimensions is called kissing number. The problem for finding kissing numbers is closely connected to the more general problems of finding bounds for spherical codes and sphere packings. We survey old and recent results on the kissing numbers keeping the generality of spherical codes.

Motivation & Objective

• To survey known results on the kissing number problem in n-dimensional space, emphasizing connections to spherical codes and sphere packings.
• To analyze and compare upper and lower bounds for kissing numbers in dimensions up to 32, with a focus on recent advances.
• To explain the role of linear programming and semidefinite programming in deriving tight upper bounds on spherical code sizes.
• To present and evaluate constructions based on error-correcting codes (e.g., Constructions A and B) that yield strong lower bounds for kissing numbers.
• To provide a comprehensive, up-to-date reference table of best-known lower and upper bounds for kissing numbers in dimensions 1–32 as of July 2012.

Proposed method

• Utilizes the Delsarte-Goethals-Seidel linear programming theorem, which bounds spherical code size using polynomials satisfying specific non-negativity and sign conditions.
• Applies the Kabatiansky-Levenshtein asymptotic upper bound and compares it with the best known lower bounds via error-correcting codes.
• Employs the Schläfli function and Coxeter-Böröczky bound as foundational tools for estimating spherical code parameters.
• Introduces semidefinite programming extensions (Bachoc-Vallentin, Mittelmann-Vallentin) to strengthen linear programming bounds, especially in dimensions 4–10.
• Uses three-point distance distributions to refine bounds by incorporating higher-order correlation structures in spherical codes.
• Applies code-based constructions (A and B) using binary codes with specific minimum distance and weight properties to generate dense sphere packings and estimate kissing numbers.

Experimental results

Research questions

• RQ1 What are the best-known upper and lower bounds for the kissing number in dimensions 5 through 32, and how have they evolved?
• RQ2 How do linear programming and semidefinite programming techniques improve upon classical bounds like Fejes Tóth and Coxeter-Böröczky?
• RQ3 To what extent do constructions based on error-correcting codes (e.g., Constructions A and B) achieve the best-known kissing numbers in low and medium dimensions?
• RQ4 Why are the kissing numbers in dimensions 8 and 24 exactly known, and what structural properties of these spaces enable such exact solutions?
• RQ5 How do three-point distance distributions and higher-order correlation structures contribute to tighter bounds in intermediate dimensions?

Key findings

• The kissing number is exactly known in dimensions 1, 2, 3, 4, 8, and 24, with values 2, 6, 12, 24, 240, and 196560, respectively.
• For dimension 8, the upper bound of 240 matches the lower bound from the E8 lattice, confirming the exact value via linear programming.
• For dimension 24, the upper and lower bounds both equal 196560, confirming the exact value using the Leech lattice and semidefinite programming techniques.
• In dimension 4, the upper bound of 24 matches the lower bound from the 600-cell, confirming τ₄ = 24.
• For dimensions 5–7 and 9–23, the best-known upper bounds are derived from semidefinite programming (Mittelmann-Vallentin), improving upon earlier linear programming results.
• New configurations in dimensions 25–31, discovered after 1982, improved upon the laminated lattice records, suggesting ongoing progress beyond the 1982 baseline.