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Maximal packing density of n-dimensional Euclidean space with equal balls

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Abstract

An estimate δ n ⩽ 2-n(0.5237+o(1)) is obtained for the maximal packing density of n-dimensional Euclidean space with equal balls for n→ ∞. This result is based on an improvement in a corresponding upper estimate for the maximal packing density of the unit (n-1)-dimensional sphere with spherical caps of fixed angular radius.

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Literature cited

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    C. Rogers, Packing and Covering, Cambridge Univ. Press (1970).

  2. 2.

    V. M. Sidel'nikov, “On the closest packing of balls on the surface of an n-dimensional Euclidean sphere and the number of binary code vectors with given code distance,” Dokl. Akad. Nauk SSSR,213, No. 5, 1029–1032 (1973).

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    V. I. Levenshtein, “On the minimal redundancy of binary codes which correct an error,” Probl. Pered. Inform.,9, No. 2, 26–42 (1974).

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    R. A. Rankin, “The closest packing of spherical caps in n-dimensions,” Proc. Glasgow Math. Assoc,2, 139–144 (1955).

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Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 301–311, August, 1975.

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Levenshtein, V.I. Maximal packing density of n-dimensional Euclidean space with equal balls. Mathematical Notes of the Academy of Sciences of the USSR 18, 765–771 (1975). https://doi.org/10.1007/BF01818046

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Keywords

  • Euclidean Space
  • Packing Density
  • Dimensional Sphere
  • Maximal Packing
  • Equal Ball