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Further lattice packings in high dimensions
Author(s) -
Bos A.,
Conway J. H.,
Sloane N. J. A.
Publication year - 1982
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300012262
Subject(s) - mathematics , lattice (music) , euclidean geometry , iterated function , euclidean space , logarithm , spheres , dimension (graph theory) , combinatorics , high dimensional , space (punctuation) , pure mathematics , mathematical analysis , geometry , linguistics , philosophy , physics , astronomy , artificial intelligence , computer science , acoustics
Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝ m with density Δ satisfyinglog 2 Δ ∼ − m log 2 * m , as m → ∞ wherelog 2 * m is the smallest value of k for which the k ‐th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high‐dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.