Year: 2014
Type: article
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.
Source: Duke Mathematical Journal
Authors: Henry Cohn, Yufei Zhao
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2138683031