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A new lower bound for the kissing number in 19 dimensions

Boon Suan Ho

Abstract

We prove that the kissing number in 19 dimensions is at least 11948, improving the bound of Cohn and Li by 256. The proof combines Cohn and Li's odd-sign construction with an explicit nonlinear binary code of length 19, size 1280, and minimum distance 5 inside a 5-punctured extended binary Golay code. The construction makes use of nested codes $M\le K\le D$: quotienting a particular graph on $K$ by $M$ yields the Clebsch graph, then an independent set of size 5 in that quotient lifts to a 320-word code in $K$, and finally the four cosets of $K$ in $D$ give the full 1280-word code.

A new lower bound for the kissing number in 19 dimensions

Abstract

We prove that the kissing number in 19 dimensions is at least 11948, improving the bound of Cohn and Li by 256. The proof combines Cohn and Li's odd-sign construction with an explicit nonlinear binary code of length 19, size 1280, and minimum distance 5 inside a 5-punctured extended binary Golay code. The construction makes use of nested codes : quotienting a particular graph on by yields the Clebsch graph, then an independent set of size 5 in that quotient lifts to a 320-word code in , and finally the four cosets of in give the full 1280-word code.
Paper Structure (3 sections, 2 theorems, 22 equations, 1 table)

This paper contains 3 sections, 2 theorems, 22 equations, 1 table.

Table of Contents

  1. Introduction
  2. An explicit 1280-word code
  3. Consequence for kissing numbers

Key Result

Theorem 1

The kissing number in 19 dimensions satisfies

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 2
  • proof
  • Remark