## Duke Mathematical Journal

### Sphere packing bounds via spherical codes

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 8 July 2014

https://projecteuclid.org/euclid.dmj/1404824306

Digital Object Identifier
doi:10.1215/00127094-2738857

Mathematical Reviews number (MathSciNet)
MR3229046

Zentralblatt MATH identifier
1296.05046

#### Citation

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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