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15 July 2014 Sphere packing bounds via spherical codes
Henry Cohn, Yufei Zhao
Duke Math. J. 163(10): 1965-2002 (15 July 2014). DOI: 10.1215/00127094-2738857

Abstract

The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their argument and improve their bound by a constant factor using a simple geometric argument, and we extend the argument to packings in hyperbolic space, for which it gives an exponential improvement over the previously known bounds. Additionally, we show that the Cohn–Elkies linear programming bound is always at least as strong as the Kabatiansky–Levenshtein bound; this result is analogous to Rodemich’s theorem in coding theory. Finally, we develop hyperbolic linear programming bounds and prove the analogue of Rodemich’s theorem there as well.

Citation

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Henry Cohn. Yufei Zhao. "Sphere packing bounds via spherical codes." Duke Math. J. 163 (10) 1965 - 2002, 15 July 2014. https://doi.org/10.1215/00127094-2738857

Information

Published: 15 July 2014
First available in Project Euclid: 8 July 2014

zbMATH: 1296.05046
MathSciNet: MR3229046
Digital Object Identifier: 10.1215/00127094-2738857

Subjects:
Primary: 05B40
Secondary: 11H31 , 52C17

Rights: Copyright © 2014 Duke University Press

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Vol.163 • No. 10 • 15 July 2014
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