Table of Perfect Lattices
Keywords: tables, perfect lattices, quadratic forms
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).
Last modified Jun 29, 2001
Perfect Lattices
This table of perfect lattices is complete through 7 dimensions.
- 7 dimensions
P7.1 = E7,
P7.2 = E7*,
P7.3,
P7.4 = D7,
P7.5,
P7.6,
P7.7,
P7.8,
P7.9,
P7.10,
P7.11,
P7.12,
P7.13,
P7.14,
P7.15,
P7.16,
P7.17,
P7.18,
P7.19,
P7.20,
P7.21,
P7.22,
P7.23,
P7.24,
P7.25,
P7.26,
P7.27,
P7.28,
P7.29,
P7.30,
P7.31,
P7.32,
P7.33 = A7.
- 8 dimensions The list of perfect forms
perfect-forms-dim8.txt
as given by Dutour-Sikric, Schuermann, Vallentin.
Remarks
Dutour Sikric, Schuermann and Vallentin proved the
completeness of the list of perfect forms in 8 dimensions.
The 10916 perfect forms are available in a human readable format
(output created by DS,S,V):
perfect-forms-dim8.txt.
Only 2408 of these perfect forms are eutactic (C. Riener) and
hence define local maxima of the density function (extreme lattices).
References
- J. H. Conway and N. J. A. Sloane,
Low-Dimensional Lattices III: Perfect Forms,
Proc. Royal Soc. London, Series A, volume 418, pages 43-80, 1988.
- D.-O. Jaquet-Chiffelle,
Enume'ration comple`te des classes de formes parfaites
en dimension 7, Ann. Inst. Fourier,
43 (1993), 21-55,
showed that the
known list of perfect 7-dim lattices is indeed complete.
- J. Martinet, Les Re'seaux Parfaits des Espaces Euclidiens,
Masson, Paris, 1996.
- J. Martinet,
Home page
(among other things, lists all known 8-dimensional perfect lattices).
- Mathieu Dutour Sikiric, Achill Schuermann, Frank Vallentin
Classification of eight dimensional perfect forms
- Cordian Riener,
On extreme forms in dimension 8.
J. Théor. Nombres Bordeaux 18 (2006), no. 3, 677--682.
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