Kissing numbers

Improve the bounds on kissing numbers in dimensions up to 48, starting with dimension 5, where 40 ≤ τ(5) ≤ 44.

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State the problem, from Kissing numbers and spherical codes (Henry Cohn)

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	1	+# Kissing numbers
	2	+
	3	+How many unit spheres can touch a central unit sphere without overlapping? The answer is known exactly only in dimensions 1, 2, 3, 4, 8 and 24. Henry Cohn's table records the best bounds up to dimension 48, and since 2024 its lower bounds have been improved in more than a dozen dimensions by explicit configurations that can be verified exactly.
	4	+
	5	+## Problem
	6	+
	7	+The kissing number $\tau(n)$ is the largest number of non-overlapping unit balls in $\mathbb{R}^n$ that can touch a fixed unit ball. Equivalently, it is the largest size of a set of unit vectors in $\mathbb{R}^n$ in which every two distinct vectors $x, y$ satisfy $\langle x, y \rangle \le 1/2$, that is, form an angle of at least $60^\circ$.
	8	+
	9	+Task: improve the lower or upper bound on $\tau(n)$ for some $n \le 48$, starting with the smallest open dimensions 5, 6, 7, 9 and 10.
	10	+
	11	+## Background
	12	+
	13	+In three dimensions the problem was the subject of a famous 1694 discussion between Newton and Gregory, and Schütte and van der Waerden proved $\tau(3) = 12$ in 1953. Upper bounds come from Delsarte's linear programming method and its semidefinite strengthenings, which gave the exact values $\tau(8) = 240$ and $\tau(24) = 196560$ (Odlyzko and Sloane; Levenshtein; 1979) and $\tau(4) = 24$ (Musin, 2008). Lower bounds come from lattices, codes and, increasingly, computer-found configurations. Kissing configurations are spherical codes with minimal angle $60^\circ$ and are tied to sphere packing.
	14	+
	15	+## What is known
	16	+
	17	+From the table (October 2026):
	18	+
	19	+- Exact values: $\tau(1) = 2$, $\tau(2) = 6$, $\tau(3) = 12$, $\tau(4) = 24$, $\tau(8) = 240$, $\tau(24) = 196560$.
	20	+- Smallest open dimensions: $40 \le \tau(5) \le 44$, $72 \le \tau(6) \le 77$, $126 \le \tau(7) \le 134$ (lower bounds from Korkine and Zolotareff, 1873; upper bounds from Mittelmann and Vallentin, 2010, and for $n = 6$ from de Laat, Leijenhorst and de Muinck Keizer, 2024); $306 \le \tau(9) \le 363$; $510 \le \tau(10) \le 553$.
	21	+- Recent lower bounds: $\tau(11) \ge 604$ (Bianchi, Kwon, Pappu and Zou, 2026), $\tau(12) \ge 841$ (Takhanov, Assylbekov and Yun, 2026), and new constructions in dimensions 10 and 14 (Ganzhinov), 17 to 21 (Cohn and Li; Kravatskiy; Li) and most dimensions from 25 to 47 (several groups, 2025–2026).
	22	+- Upper bounds for $n \ge 11$ come from large semidefinite programs solved by Leijenhorst and de Laat (2024).
	23	+
	24	+## What would count as progress
	25	+
	26	+- A configuration of unit vectors in $\mathbb{R}^n$ with more points than the table's lower bound. Given exact coordinates (rational, or in a small number field), checking that all pairwise inner products are at most $1/2$ takes seconds to minutes even for tens of thousands of points.
	27	+- A smaller upper bound for some dimension, typically from a semidefinite program; it counts once the dual solution is rounded to an exact certificate that others can check.
	28	+- The dimension 5 case, $40 \le \tau(5) \le 44$, is a natural first target on both sides.
	29	+
	30	+## References
	31	+
	32	+1. K. Schütte and B. L. van der Waerden, "Das Problem der dreizehn Kugeln", Mathematische Annalen 125 (1953), 325–334. [doi:10.1007/BF01343127](https://doi.org/10.1007/BF01343127)
	33	+2. A. M. Odlyzko and N. J. A. Sloane, "New bounds on the number of unit spheres that can touch a unit sphere in n dimensions", Journal of Combinatorial Theory, Series A 26 (1979), 210–214. [doi:10.1016/0097-3165(79)90074-8](https://doi.org/10.1016/0097-3165(79)90074-8)
	34	+3. O. R. Musin, "The kissing number in four dimensions", Annals of Mathematics 168 (2008), 1–32. [doi:10.4007/annals.2008.168.1](https://doi.org/10.4007/annals.2008.168.1)
	35	+4. H. D. Mittelmann and F. Vallentin, "High-accuracy semidefinite programming bounds for kissing numbers", Experimental Mathematics 19 (2010), 174–178. [arXiv:0902.1105](https://arxiv.org/abs/0902.1105)
	36	+5. N. Leijenhorst and D. de Laat, "Solving clustered low-rank semidefinite programs arising from polynomial optimization", Mathematical Programming Computation 16 (2024), 503–534. [arXiv:2202.12077](https://arxiv.org/abs/2202.12077)
	37	+6. H. Cohn and A. Li, "Improved kissing numbers in seventeen through twenty-one dimensions" (2024). [arXiv:2411.04916](https://arxiv.org/abs/2411.04916)
	38	+
	39	+## Source
	40	+
	41	+Imported from [Kissing numbers, table by Henry Cohn](https://cohn.mit.edu/kissing-numbers), curated by Henry Cohn. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.