Geodesic
The densest lattice in twenty-four dimensions
Cohn, Henry1 ; Kumar, Abhinav2
1 Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399
2 Department of Mathematics, Harvard University, Cambridge, MA 02138
Electronic research announcements of the American Mathematical Society, Tome 10 (2004), pp. 58-67.

Voir la notice de l'article provenant de la source American Mathematical Society

In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in $\mathbb {R}^{24}$. Complete details will appear elsewhere, but here we illustrate our techniques by applying them to the case of lattice packings in $\mathbb {R}^2$, and we discuss the obstacles that arise in higher dimensions.
DOI : 10.1090/S1079-6762-04-00130-1

Cohn, Henry 1 ; Kumar, Abhinav 2

1 Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399
2 Department of Mathematics, Harvard University, Cambridge, MA 02138 
@article{ERAAMS_2004_10_a6,
     author = {Cohn, Henry and Kumar, Abhinav},
     title = {The densest lattice in twenty-four dimensions},
     journal = {Electronic research announcements of the American Mathematical Society},
     pages = {58--67},
     publisher = {mathdoc},
     volume = {10},
     year = {2004},
     doi = {10.1090/S1079-6762-04-00130-1},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-04-00130-1/}
}
TY  - JOUR
AU  - Cohn, Henry
AU  - Kumar, Abhinav
TI  - The densest lattice in twenty-four dimensions
JO  - Electronic research announcements of the American Mathematical Society
PY  - 2004
SP  - 58
EP  - 67
VL  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-04-00130-1/
DO  - 10.1090/S1079-6762-04-00130-1
ID  - ERAAMS_2004_10_a6
ER  - 
%0 Journal Article
%A Cohn, Henry
%A Kumar, Abhinav
%T The densest lattice in twenty-four dimensions
%J Electronic research announcements of the American Mathematical Society
%D 2004
%P 58-67
%V 10
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-04-00130-1/
%R 10.1090/S1079-6762-04-00130-1
%F ERAAMS_2004_10_a6
Cohn, Henry; Kumar, Abhinav. The densest lattice in twenty-four dimensions. Electronic research announcements of the American Mathematical Society, Tome 10 (2004), pp. 58-67. doi : 10.1090/S1079-6762-04-00130-1. http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-04-00130-1/

[1] Barnes, E. S. The complete enumeration of extreme senary forms Philos. Trans. Roy. Soc. London Ser. A 1957 461 506

[2] Cohn, Henry New upper bounds on sphere packings. II Geom. Topol. 2002 329 353

[3] Cohn, Henry, Elkies, Noam New upper bounds on sphere packings. I Ann. of Math. (2) 2003 689 714

[4] Conway, J. H., Sloane, N. J. A. Sphere packings, lattices and groups 1999

[5] Delsarte, P., Goethals, J. M., Seidel, J. J. Spherical codes and designs Geometriae Dedicata 1977 363 388

[6] Elkies, Noam D. Lattices, linear codes, and invariants. I Notices Amer. Math. Soc. 2000 1238 1245

[7] Gruber, P. M., Lekkerkerker, C. G. Geometry of numbers 1987

[8] Levenšteĭn, V. I. Boundaries for packings in 𝑛-dimensional Euclidean space Dokl. Akad. Nauk SSSR 1979 1299 1303

[9] Martinet, Jacques Perfect lattices in Euclidean spaces 2003

[10] Odlyzko, A. M., Sloane, N. J. A. New bounds on the number of unit spheres that can touch a unit sphere in 𝑛 dimensions J. Combin. Theory Ser. A 1979 210 214

[11] Vetčinkin, N. M. Uniqueness of classes of positive quadratic forms, on which values of Hermite constants are reached for 6≤𝑛≤8 Trudy Mat. Inst. Steklov. 1980

Cité par Sources :