Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved kissing numbers in seventeen through twenty-one dimensions

Henry Cohn, Anqi Li

Abstract

We prove that the kissing numbers in 17, 18, 19, 20, and 21 dimensions are at least 5730, 7654, 11692, 19448, and 29768, respectively. The previous records were set by Leech in 1967, and we improve on them by 384, 256, 1024, 2048, and 2048. Unlike the previous constructions, the new configurations are not cross sections of the Leech lattice minimal vectors. Instead, they are constructed by modifying the signs in the lattice vectors to open up more space for additional spheres.

Improved kissing numbers in seventeen through twenty-one dimensions

Abstract

We prove that the kissing numbers in 17, 18, 19, 20, and 21 dimensions are at least 5730, 7654, 11692, 19448, and 29768, respectively. The previous records were set by Leech in 1967, and we improve on them by 384, 256, 1024, 2048, and 2048. Unlike the previous constructions, the new configurations are not cross sections of the Leech lattice minimal vectors. Instead, they are constructed by modifying the signs in the lattice vectors to open up more space for additional spheres.

Paper Structure

This paper contains 5 sections, 2 theorems, 32 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Dimensions 19, 20, and 21
  3. Dimension 17
  4. Dimension 18
  5. Potential generalizations

Key Result

Theorem 1.1

The kissing numbers in $\mathbb{R}^{17}$, $\mathbb{R}^{18}$, $\mathbb{R}^{19}$, $\mathbb{R}^{20}$, and $\mathbb{R}^{21}$ are at least $5730$, $7654$, $11692$, $19448$, and $29768$, respectively.

Theorems & Definitions (3)

  • Theorem 1.1
  • Lemma 3.1
  • proof