Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



The densest lattice in twenty-four dimensions

Authors: Henry Cohn and Abhinav Kumar
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 58-67
MSC (2000): Primary 11H31, 52C15; Secondary 05B40, 11H55
Published electronically: June 17, 2004
MathSciNet review: 2075897
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

References [Enhancements On Off] (What's this?)

  • [Ba] E. S. Barnes, The complete enumeration of extreme senary forms, Phil. Transact. Royal Soc. London A 249 (1957), 461-506. MR 0086833 (19:251d)
  • [Bl] H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39 (1935), 1-15.
  • [C] H. Cohn, New upper bounds on sphere packings II, Geom. Topol. 6 (2002), 329-353, arXiv:math.MG/0110010. MR 1914571 (2004b:52032)
  • [CE] H. Cohn and N. Elkies, New upper bounds on sphere packings I, Annals of Mathematics 157 (2003), 689-714, arXiv:math.MG/0110009. MR 1973059 (2004b:11096)
  • [CK] H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, preprint, 2003, arXiv:math.MG/0403263.
  • [CS] J. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Springer-Verlag, 1999. MR 1662447 (2000b:11077)
  • [DGS] P. Delsarte, J. Goethals, and J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), 363-388. MR 0485471 (58:5302)
  • [E] N. Elkies, Lattices, linear codes, and invariants, Notices Amer. Math. Soc. 47 (2000), 1238-1245 and 1382-1391. MR 1784239 (2001g:11110)
  • [G] C. F. Gauss, Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, Göttingische gelehrte Anzeigen, July 9, 1831. Reprinted in Werke, Vol. 2, Königliche Gesellschaft der Wissenschaften, Göttingen, 1863, 188-196. Available from the Göttinger Digitalisierungszentrum at
  • [GL] P. Gruber and C. Lekkerkerker, Geometry of Numbers, second edition, Elsevier Science Publishers, 1987. MR 0893813 (88j:11034)
  • [KZ1] A. Korkine and G. Zolotareff, Sur les formes quadratiques, Math. Ann. 6 (1873), 366-389.
  • [KZ2] A. Korkine and G. Zolotareff, Sur les formes quadratiques positives, Math. Ann. 11 (1877), 242-292.
  • [Lev] V. I. Levenshtein, On bounds for packings in $n$-dimensional Euclidean space, Soviet Mathematics Doklady 20 (1979), 417-421. MR 0529659 (80d:52017)
  • [M] J. Martinet, Perfect Lattices in Euclidean Space, Springer-Verlag, 2003. MR 1957723 (2003m:11099)
  • [OS] 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 A 26 (1979), 210-214. MR 0530296 (81d:52010)
  • [V] N. M. Vetcinkin, Uniqueness of classes of positive quadratic forms on which values of the Hermite constant are attained for $6 \le n \le 8$, in Geometry of positive quadratic forms, Trudy Math. Inst. Steklov. 152 (1980), 34-86. English translation in Proc. Steklov Inst. Math. 152 (1982), 37-95. MR 0603814 (82f:10040)

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 11H31, 52C15, 05B40, 11H55

Retrieve articles in all journals with MSC (2000): 11H31, 52C15, 05B40, 11H55

Additional Information

Henry Cohn
Affiliation: Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399

Abhinav Kumar
Affiliation: Department of Mathematics, Harvard University, Cambridge, MA 02138

Received by editor(s): April 14, 2004
Published electronically: June 17, 2004
Additional Notes: Kumar was supported by a summer internship in the Theory Group at Microsoft Research.
Communicated by: Brian Conrey
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society