HEXAGONbeta

Improved Kissing Numbers in Nineteen and Twenty-One Dimensions

Primarily AI-generated textHuman understanding: all partsmath.MG — Metric Geometry

Contributed by Kevin Li ↗

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

We prove that the kissing numbers in 19 and 21 dimensions are at least 12460 and 31560, improving the best published bounds 11948 (Ho) and 29768 (Cohn and Li). Both constructions start from the odd-sign configurations of Cohn and Li, which leave room around the sign vectors they add. We use this room to reshape the added vectors so that codewords closer than n/4 can be used together. In 21 dimensions each added vector has entries ±2 on one line of the projective plane of order 4 formed by the weight-5 words of the punctured Golay code, and ±3 elsewhere; a linear labelling of the codewords by lines gives 3840 new points, and the proof is by hand. In 19 dimensions we cut the 20-dimensional configuration by a hyperplane and project the sign vectors of the 20-dimensional code into it. The projections have a 16-coordinate head and a three-dimensional tail, the tails separate words whose heads are close, and one point for each of the 1792 heads with a nonzero tail can be added, where Ho adds 1280.

Provenance statement

GPT-6 Astra Pro, Claude Opus 5.5 and Claude Fable 5.1 were used to find these constructions, prepare supplementary data and draft this paper. The manuscript was proofread and lightly revised by the author. The author welcomes detailed questions regarding the use of AI in this work.

Tools used

OpenAI
ChatGPTVersion GPT-6 Astra Pro
Anthropic
Claude OpusVersion 5.5
Anthropic
Claude FableVersion 5.1

References

  1. H. Cohn, Kissing numbers , table, https://cohn.mit.edu/kissing-numbers/, accessed 4 October 2026.link
  2. H. Cohn and A. Li, Improved kissing numbers in seventeen through twenty-one dimensions , preprint, 2024, version 3, 2026. arXiv:2411.04916v3 (https://arxiv.org/abs/2411.04916v3).arXiv
  3. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups , third edition, Grundlehren der mathematischen Wissenschaften 290, Springer, New York, 1999.
  4. B. S. Ho, A new lower bound for the kissing number in 19 dimensions , preprint, 2026. arXiv:2603.10425v2 (https://arxiv.org/abs/2603.10425v2).arXiv
  5. J. Leech, Notes on sphere packings , Canadian J. Math. 19 (1967), 251--267.
  6. J. Leech and N. J. A. Sloane, Sphere packings and error-correcting codes , Canadian J. Math. 23 (1971), no. 4, 718--745.
  7. N. Leijenhorst and D. de Laat, Solving clustered low-rank semidefinite programs arising from polynomial optimization , Math. Program. Comput. 16 (2024), no. 3, 503--534.
  8. F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes , North-Holland, Amsterdam, 1977.

Version history

  1. v1Submitted by Kevin LiInitial depositCurrentOct 14, 2026