Kissing numbers, sphere packings, and some unexpected proofs

Notices of the American Mathematical Society

Volumn 51, Issue 8, 2004, Pages 873-883

  Pfender, Florian ^a   Ziegler, Günter M ^b  

^a Deutsche Forschungsgemeinschaft   (Germany)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 24344439521     PISSN: 00029920     EISSN: 10889477     Source Type: Journal    
DOI: None     Document Type: Review

References (29)

1. M. AIGNER, Gitter, Codes und Graphen, Lecture Notes, FU Berlin, 1994/95.
2. G. E. ANDREWS, R. ASKEY, AND R. ROY, Special Functions, Encyclopedia Math. Appl., vol. 71, Cambridge University Press, Cambridge, 1999.
3. 4, Math. Notes 68 (2000), 419-435.
4. M. R. BEST, Optimal codes, Packing and Covering in Combinatorics (A. Schrijver, ed.), Mathematical Centre Tracts, vol. 106, CWI, Amsterdam, 1979, pp. 119-140.
5. H. COHN AND N. ELKIES, New upper bounds on sphere packings. I, Ann. of Math. 157 (2003), 689-714.
6. H. COHN AND A. KUMAR, Optimality and uniqueness of the Leech lattice among lattices, preprint, March 2004, 39 pages; arXiv:math.MG/0403263.
7. H. COHN AND A. KUMAR, The densest lattice in twenty-four dimensions, Elec. Res. Ann. 10 (2004), 58-67.
8. J. H. CONWAY AND N. J. A. SIOANE, Sphere Packings, Lattices and Groups, Grundlehren Math. Wiss., vol. 290 Springer-Verlag, New York, third ed., 1993.
9. P. DELSARTE, Bounds for unrestricted codes, by linear programming, Philips Res. Rep. 27 (1972), 272-289.
10. P. DELSARTE, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. (1973), vi+97.
11. P. DELSARTE, J. M. GOETHALS, AND J. J. SEIDEL, Spherical codes and designs, Geom. Dedicata 6 (1977), 363-388.
12. N. D. ELKIES, Lattices, linear codes, and invariants. I/II, Notices Amer. Math. Soc. 47 (2000), 1238-1245 and 1382-1391.
13. E. GAWRILOW AND M. JOSWIG, Polymake: A software package for analyzing convex polytopes, http://www. math.tu-berlin.de/diskregeom/polymake/.
14. T. C. HALES, Cannonballs and honeycombs, Notices Amer. Math. Soc. 47 (2000), 440-449.
15. _, A computer verification of the Kepler conjecture, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 795-804; math.MG/0305012.
16. G. A. KABATJANSKIǏ AND V. I. LEVENŠTEǏN, Bounds for packings on the sphere and in space, Problemy Peredači Informatsii 14 (1978), 3-25.
17. J. C. LAGARLIS, J. A. REEDS, M. H. WRIGHT, AND P. E. WRIGHT, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim. 9 (1999), 112-147.
18. J. LEECH, The problem of the thirteen spheres, The Math. Gazette 40 (1956), 22-23.
19. V. I. LEVENŠTEǏN, On bounds for packings in n-dimensional euclidean space Soviet Math. Doklady 20 (1979), 417-421.
20. K. I. M. MCKINNON, Convergence of the Nelder-Mead simplex algorithm to a non-stationary point, SIAM J. Optim. 9 (1999), 148-158.
21. C. MÜLLER, Spherical Harmonics, Lecture Notes in Math., vol. 17, Springer-Verlag, Berlin and Heidelberg, 1966.
22. O. R. MUSIN, The kissing number in four dimensions, preprint, September 2003, 22 pages; math. MG/0309430.
23. O. R. MUSIN, The problem of the twenty-five spheres, Russian Math. Surveys 58 (2003), 794-795.
24. 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), 210-214.
25. W. H. PRESS, S. A. TEUKOLSKY, W. T. VETTERLING, AND B. P. FLANNERY, Numerical Recipes in C. The Art of Scientific Computing, Cambridge University Press, Cambridge, second ed., 1992.
26. I. J. SCHOEKBERG, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96-107.
27. K. SCHÜTTE AND B. L. VAN DER WAERDEN, Das Problem der dreizehn Kugeln, Math. Ann. 53 (1953), pp. 325-334.
28. G. G. SZPIRO, Kepler's Conjecture. How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World, Wiley, 2003.
29. T. M. THOMPSON, From Error-Correcting Codes through Sphere Packings to Simple Groups, Carus Math. Monogr., vol. 21, Math. Assoc. America, Washington, DC, 1983.