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Experimental Mathematics Volume 27, 2018 - Issue 3
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Original Articles

Improving the Semidefinite Programming Bound for the Kissing Number by Exploiting Polynomial Symmetry

Fabrício Caluza MachadoInstituto de Matemática e Estatística, Universidade de Sao Paulo, São Paulo, BrazilCorrespondence[email protected]
&
Fernando Mário de Oliveira FilhoInstituto de Matemática e Estatística, Universidade de Sao Paulo, São Paulo, Brazil
Pages 362-369 | Published online: 02 Mar 2017

References

  • [Bachoc et al. 12] C. Bachoc, D. C. Gijswijt, A. Schrijver, and F. Vallentin. “Invariant Semidefinite Programs.” In Handbook on Semidefinite, Conic, and Polynomial Optimization, edited by M. F. Anjos and J. B. Lasserre, pp. 219–269. New York: Springer, 2012.
  • [Bachoc and Vallentin 08] C. Bachoc and F. Vallentin. “New Upper Bounds for Kissing Numbers from Semidefinite Programming.” J. Am. Math. Soc. 21 (2008), 909–924.
  • [Bannai and Sloane 81] E. Bannai and N. J. A. Sloane. “Uniqueness of Certain Spherical Codes.” Can. J. Math. 33 (1981), 437–449.
  • [Conway and Sloane 88] J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups, Grundlehren der mathematischen Wissenschaften 290, New York: Springer-Verlag, 1988.
  • [Delsarte et al. 77] P. Delsarte, J. M. Goethals, and J. J. Seidel. “Spherical Codes and Designs.” Geometriae Dedicata 6 (1977), 363–388.
  • [Dostert et al. 15] M. Dostert, C. Guzmán, F. M. de Oliveira Filho, and F. Vallentin. “New Upper Bounds for the Density of Translative Packings of Three-dimensional Convex Bodies with Tetrahedral Symmetry.” arXiv:1501.00168, 2015, 29pp.
  • [Fulton and Harris 04] W. Fulton and J. Harris. Representation Theory: A First Course, Graduate Texts in Mathematics 129. New YorkSpringer-Verlag, 2004.
  • [Gatermann and Parrilo 04] K. Gatermann and P. A. Parrilo. “Symmetry Groups, Semidefinite Programs, and Sums of Squares.” J. Pure Appl. Algebra 192 (2004), 95–128.
  • [Levenshtein 79] V. I. Levenshtein. “On Bounds for Packings in n-dimensional Euclidean Space.” Doklady Akademii Nauk SSSR 245 (1979), 1299–1303.
  • [Mittelmann and Vallentin 10] H. D. Mittelmann and F. Vallentin. “High-accuracy Semidefinite Programming Bounds for Kissing Numbers.” Exp. Math. 19 (2010), 175–179.
  • [Musin 08] O. R. Musin. “The Kissing Number in Four Dimensions.” Ann. Math. 168 (2008), 1–32.
  • [Nakata 10] M. Nakata. “A Numerical Evaluation of Highly Accurate Multiple-Precision Arithmetic Version of Semidefinite Programming Solver: SDPA-GMP,-QD and-DD.” In: 2010 IEEE International Symposium on Computer-Aided Control System Design, Tokyo, pp. 29–34, 2010.
  • [Odlyzko and Sloane 79] 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.
  • [Revol and Rouillier 05] N. Revol and F. Rouillier. “Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library.” Reliab. Comput. 11 (2005), 275–290.
  • [Schütte and van der Waerden 53] K. Schütte and B. L. van der Waerden. “Das Problem der dreizehn Kugeln.” Math. Ann. 125 (1953), 325–334.
  • [Serre 77] J.-P. Serre. Linear Representations of Finite Groups (Translated from the second French edition by Leonard L. Scott), Graduate Texts in Mathematics 42. New York: Springer-Verlag, 1977.
  • [Sturmfels 08] B. Sturmfels. Algorithms in Invariant Theory. Vienna: Springer-Verlag, 2008.
  • [Szegö 75] G. Szegö. Orthogonal Polynomials, Fourth edition. Providence: American Mathematical Society Colloquium Publications Volume XXIII, American Mathematical Society, 1975.
  • [Zinoviev and Ericson 99] V. A. Zinoviev and T. Ericson. “New Lower Bounds for Contact Numbers in Small Dimensions.” Prob. Info. Trans. 35 (1999), 287–294.

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