Duke Mathematical Journal

Sphere packing bounds via spherical codes

Henry Cohn and Yufei Zhao

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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.

Article information

Duke Math. J. Volume 163, Number 10 (2014), 1965-2002.

First available in Project Euclid: 8 July 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05B40: Packing and covering [See also 11H31, 52C15, 52C17]
Secondary: 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31] 11H31: Lattice packing and covering [See also 05B40, 52C15, 52C17]


Cohn, Henry; Zhao, Yufei. Sphere packing bounds via spherical codes. Duke Math. J. 163 (2014), no. 10, 1965--2002. doi:10.1215/00127094-2738857. https://projecteuclid.org/euclid.dmj/1404824306.

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