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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
DOI: https://doi.org/10.1090/S1079-6762-04-00130-1
Published electronically: June 17, 2004
MathSciNet review: 2075897
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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, Philos. Trans. Roy. Soc. London. Ser. A. 249 (1957), 461–506. MR 0086833, https://doi.org/10.1098/rsta.1957.0005
  • [Bl] H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39 (1935), 1-15.
  • [C] Henry Cohn, New upper bounds on sphere packings. II, Geom. Topol. 6 (2002), 329–353. MR 1914571, https://doi.org/10.2140/gt.2002.6.329
  • [CE] Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. MR 1973059, https://doi.org/10.4007/annals.2003.157.689
  • [CK] H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, preprint, 2003, arXiv:math.MG/0403263.
  • [CS] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447
  • [DGS] P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363–388. MR 0485471
  • [E] Noam D. Elkies, Lattices, linear codes, and invariants. I, Notices Amer. Math. Soc. 47 (2000), no. 10, 1238–1245. MR 1784239
  • [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 http://gdz.sub.uni-goettingen.de.
  • [GL] P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. MR 893813
  • [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. Levenšteĭn, Boundaries for packings in 𝑛-dimensional Euclidean space, Dokl. Akad. Nauk SSSR 245 (1979), no. 6, 1299–1303 (Russian). MR 529659
  • [M] Jacques Martinet, Perfect lattices in Euclidean spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 327, Springer-Verlag, Berlin, 2003. MR 1957723
  • [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, J. Combin. Theory Ser. A 26 (1979), no. 2, 210–214. MR 530296, https://doi.org/10.1016/0097-3165(79)90074-8
  • [V] N. M. Vetčinkin, Uniqueness of classes of positive quadratic forms, on which values of Hermite constants are reached for 6≤𝑛≤8, Trudy Mat. Inst. Steklov. 152 (1980), 34–86, 237 (Russian). Geometry of positive quadratic forms. MR 603814

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Additional Information

Henry Cohn
Affiliation: Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399
Email: cohn@microsoft.com

Abhinav Kumar
Affiliation: Department of Mathematics, Harvard University, Cambridge, MA 02138
Email: abhinav@math.harvard.edu

DOI: https://doi.org/10.1090/S1079-6762-04-00130-1
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.