## 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, RWTH Aachen university (nebe@math.rwth-aachen.de) and Neil J. A. Sloane, (njasloane@gmail.com).

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
DimKissing number for a lattice (nonlattice)Lattice (resp. nonlattice)
12LAMBDA1 = A1 = Z
26LAMBDA2 = A2
312LAMBDA3 = A3 = D3
424LAMBDA4 = D4
540LAMBDA5 = D5
672LAMBDA6 = E6
7126LAMBDA7 = E7
8240LAMBDA8 = E8
9272 (306)*LAMBDA9 ( P9a )*
10336 (500)*LAMBDA10 ( P10b )*
11438 (582)*LAMBDA11 ( P11c )*
12756 (840)*KAPPA12 = K12 (the Coxeter-Todd lattice) ( P12a )*
13918 (1154)*KAPPA13 ( [ZE99] )*
141422 (1606)*LAMBDA14( [ZE99])*
152340 (2564)LAMBDA15 (P15a)*
164320LAMBDA16
175346LAMBDA17
187398LAMBDA18
1910668LAMBDA19
2017400LAMBDA20
2127720LAMBDA21
2249896LAMBDA22
2393150LAMBDA23
24196560Leech lattice LAMBDA24
25196656 (197048)*LAMBDA25 ([CJKT2011])*
26196848 (198512)*LAMBDA26 ([CJKT2011])*
27197142 (199976)* LAMBDA27 ([CJKT2011])*
28197736 (204368)* LAMBDA28 ([CJKT2011])*
29198506 (208272)* LAMBDA29 ([CJKT2011])*
30200046 (219984)* LAMBDA30 ([CJKT2011])*
31202692 (232874)* LAMBDA31 ([CJKT2011])*
32261120 (276032)*Q32 and others (Nonlattice [EdRS98])*
33262272 (294592)*Q33 (Nonlattice [EdRS98])*
34264576 (318020)*Q34 (Nonlattice [EdRS98])*
35268032 (370892)*Q35 (Nonlattice [EdRS98])*
36274944 (484568)*Q36 (Nonlattice [EdRS98] update by Brouwer using A(36,8,8) >= 2742)*
37284160 (439016)*Q37 (Nonlattice [EdRS98])*
38302592 (566652)*Q38 (Nonlattice [EdRS98])*
39333696 (714184)*Q39 (Nonlattice [EdRS98])*
40399360 (1063216)*Q40 (Nonlattice [EdRS98] update by Brouwer using A(40,8,8) >= 3674)*
42              (1196788)*      (Nonlattice [EdRS98])*
442708112 (2948552)*MW44 (Nonlattice [EdRS98])*
4852416000P48n, P48p, P48q
64138458880 (331737984)*Ne64 [Nebe98], [Nebe98b] ([EdRS98])*
726218175600Gamma72 Nebe 2010
806218175600+240 Gamma72perp E8
128218044170240 (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. 1998.
• [Elkies98] N. D. Elkies, personal communication, Feb. 11, 1998.
• [ZE99] Zinoviev and Ericson, New lower bounds for kissing numbers in small dimensions, Problems Inform. Transmission 35 (1999) 287-294
• [CJKT2011] Henry Cohn, Yang Jiao, Abhinav Kumar, Salvatore Torquato, Rigidity of spherical codes, Preprint 2011.
• [KKW2016] Kenz Kallal, Tomoka Kan, Eric Wang, Improved Lower Bounds for Kissing Numbers in Dimensions 25 Through 31. SIAM J. Discrete Math., 31(3), 1895–1908.