New upper bounds on sphere packings I

By Henry Cohn, Noam D. Elkies. 2001

400 citations recorded by the source corpus.

We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related results and conjectures. This article is in the arXiv as: arXiv:math.MG/0110010

Subfields: Computer Graphics and Computer-Aided Design, Electrical and Electronic Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/84e3d8ee19d69eb6dbad3932070740e3c44fccdd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Graphics and Computer-Aided Design

In Electrical and Electronic Engineering

In Numerical Analysis

Citations in the corpus

A conceptual breakthrough in sphere packing (2016, methodology): “The existence of such a function had been conjectured by Cohn and Elkies in 2003, but what sort of function it might be remained mysterious despite considerable effort.”
Kissing numbers, sphere packings, and some unexpected proofs (2004, methodology): “Independently, Delsarte’s by now classical approach has also recently been adapted by Henry Cohn and Noam Elkies [5] to deal with optimal sphere packings more directly and more effectively than had been possible before.”
The sphere packing problem in dimension 24 (2016, background): “, and as in Section 8 of [2] this condition implies that the Leech lattice is the unique densest periodic packing in R24.”
Predicting Lattice Reduction (2008, background): “For other values of n, the best numerical upper bounds known are given in [6].”
Fourier analysis, linear programming, and densities of distance avoiding sets in R^n (2008, methodology): “Recently, Bukh [5], using harmonic analysis and ideas resembling Szémeredi’s regularity lemma, showed that inf{md1,...,dN (Rn) : d1, . . . , dN > 0} drops to zero exponentially in N : he shows that there is a number r , strictly greater than 1, which depends only on N and n so that if d2/d1 > r,…”
The Crystallization Conjecture: A Review (2015, background): “Note however that, in dimensions d = 4, 8, 24, there are special lattices which are believed to solve the best packing problem [57].”
The sphere packing problem in dimension 8The sphere packing problem in dimension 8 (2016, background): “The following theorem is the key result of [4]: Theorem 2 (Cohn, Elkies [4]).”
The densest lattice in twenty-four dimensions (2004, methodology): “swer, with no need to reconstruct it from scratch. 6. Future prospects There is no likelihood that our techniques will work in any dimension except 1, 2, 8, or 24. They depend on using the methods of [CE] to prove nearly sharp upper bounds for the packing density, and that does not seem to happen in”
Jammed hard-particle packings: From Kepler to Bernal and beyond (2010, background): “Cohn and Elkies, 2003 obtained and computed linear programming upper bounds, which provided improvement over Rogers’ upper bounds for dimensions 4 through 36, but it is not yet known whether they improve upon the Kabatiaksky-Levenshtein upper bounds for large d. Cohn and Kumar, 2009 used these…”
Upper bounds for packings of spheres of several radii (2012, methodology): “When N = 1 and when the convex body K1 is centrally symmetric (an assumption which is in fact not needed) then this theorem reduces to the linear programming method of Cohn and Elkies [8].”