Highest Kissing Numbers

Table of the Highest Kissing Numbers Presently Known

Keywords: tables, kissing number, lattices, quadratic forms, packings

Part of the Catalogue of Lattices which is a joint project of Gabriele Nebe, University of Ulm (nebe@mathematik.uni-ulm.de) and Neil J. A. Sloane, AT&T Labs-Research (njas@research.att.com).

Last modified Jul 10, 2004

**Highest kissing numbers presently known in dimensions up to 128.**
The entry gives the lattice with the highest kissing number known
(and if a **nonlattice** with a higher kissing number is known this appears in parentheses)
The kissing number in a nonlattice packing may vary from sphere to sphere -- we give the largest value.
For further details about any of the packings mentioned that don't have links, see SPLAG 3rd edition
Dim	Kissing number for a lattice (nonlattice)	Lattice (resp. nonlattice)
1	2	LAMBDA₁ = A₁ = Z
2	6	LAMBDA₂ = A₂
3	12	LAMBDA₃ = A₃ = D3
4	24	LAMBDA₄ = D₄
5	40	LAMBDA₅ = D₅
6	72	LAMBDA₆ = E₆
7	126	LAMBDA₇ = E₇
8	240	LAMBDA₈ = E₈
9	272 (306)*	LAMBDA₉ ( P_9a )*
10	336 (500)*	LAMBDA₁₀ ( P_10b )*
11	438 (582)*	LAMBDA₁₁ ( P_11c )*
12	756 (840)*	KAPPA₁₂ = K₁₂ (the Coxeter-Todd lattice) ( P_12a )*
13	918 (1130)*	KAPPA₁₃ ( P_13a )*
14	1422 (1582)*	LAMBDA₁₄ ( P_14b )*
15	2340 (2564)*	LAMBDA₁₅ ( P_15a )*
16	4320	LAMBDA₁₆
17	5346	LAMBDA₁₇
18	7398	LAMBDA₁₈
19	10668	LAMBDA₁₉
20	17400	LAMBDA₂₀
21	27720	LAMBDA₂₁
22	49896	LAMBDA₂₂
23	93150	LAMBDA₂₃
24	196560	Leech lattice LAMBDA₂₄
25	196656	LAMBDA₂₅
26	196848	LAMBDA₂₆
27	197142	LAMBDA₂₇
28	197736	LAMBDA₂₈
29	198506	LAMBDA₂₉
30	200046	LAMBDA₃₀
31	202692	LAMBDA₃₁
32	261120 (276032)*	Q₃₂ and others (Nonlattice [EdRS98])*
33	262272 (294592)*	Q₃₃ (Nonlattice [EdRS98])*
34	264576 (318020)*	Q₃₄ (Nonlattice [EdRS98])*
35	268032 (370892)*	Q₃₅ (Nonlattice [EdRS98])*
36	274944 (438872)*	Q₃₆ (Nonlattice [EdRS98])*
37	284160 (439016)*	Q₃₇ (Nonlattice [EdRS98])*
38	302592 (566652)*	Q₃₈ (Nonlattice [EdRS98])*
39	333696 (714184)*	Q₃₉ (Nonlattice [EdRS98])*
40	399360 (991792)*	Q₄₀ (Nonlattice [EdRS98])*
42	(1196788)*	(Nonlattice [EdRS98])*
44	2708112 (2948552)*	MW₄₄ (Nonlattice [EdRS98])*
48	52416000	P_48n, P_48p, P_48q
64	138458880 (331737984)*	Ne₆₄ [Nebe98], [Nebe98b] ([EdRS98])*
80	1250172000 (1368532064)*	L₈₀ of Bachoc-Nebe [BacoNe98] ([EdRS98])*
128	218044170240 (8863556495104)*	MW₁₂₈ [Elkies98] ([EdRS98])*

__________
* Nonlattice packing.

References (Other than those already given in SPLAG)

[EdRS98] Y. Edel, E. M. Rains and N. J. A. Sloane, On kissing numbers in dimensions 32 to 128, Electronic J. Combin., to appear, 1998.
[Elkies98] N. D. Elkies, personal communication, Feb. 11, 1998.

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