Dual Linear Programming Bounds for Sphere Packing via Discrete Reductions

Abstract

The Cohn-Elkies linear program for sphere packing, which was used to solve the 8 and 24 dimensional cases, is conjectured to not be sharp in any other dimension d > 2. By mapping feasible points of this infinite-dimensional linear program into a finite-dimensional problem via discrete reduction, we provide a general method to obtain dual bounds on the Cohn-Elkies linear program. This reduces the number of variables to be finite, enabling computer optimization techniques to be applied. Using this method, we prove that the Cohn-Elkies bound cannot come close to the best packing densities known in dimensions 3 ≤ d ≤ 13 except for the solved case d = 8. In particular, our dual bounds show the Cohn-Elkies bound is unable to solve the 3 and 4 dimensional sphere packing problems.

From sphere packing to Fourier interpolation

“…Cohn and Elkies conjectured that they were equal in those cases, and the solutions of the sphere packing problem in these dimensions come from proving this conjecture. 2 For comparison, it is known that the linear programming bound cannot be sharp in dimensions three through five [33], six [18], twelve, or sixteen [15], and it is likely that the only sharp cases are dimensions one, two, eight, and twenty-four.…”

Six-dimensional sphere packing and linear programming

“…It is a pleasure to thank both Henry Cohn and Henri Cohen for their helpful advice. Thanks also to Rupert Li, Henry Cohn, David de Laat, and Andrew Salmon for sharing results from earlier versions of their articles [24] and [8]. Thanks again to David de Laat for advice on computing the Cohn-Elkies function with [23].…”

“…This proves that the Cohn-Elkies bound is not sharp in dimensions 3, 4, and 5, and gives bounds in all dimensions 3 ≤ D ≤ 13 (ignoring the sharp case D = 8). In dimension 6, the bound from [24] is 0.07632412 in terms of center density, whereas one would need roughly 0.0794 to conclude from Theorem 1.1 that the LP bound is not sharp. However, we imagine the only obstacle is computational power (and the current value would already suffice if one knew that E 6 is a densest packing, or if the SDP bound could be improved enough).…”

“…There are other approaches to rigorously estimating the LP bound. Li [24] discretizes Theorem 1.4, replacing R D by a grid, and using duality for the resulting finite-dimensional linear programs rather than Theorem 1.5. This proves that the Cohn-Elkies bound is not sharp in dimensions 3, 4, and 5, and gives bounds in all dimensions 3 ≤ D ≤ 13 (ignoring the sharp case D = 8).…”

“…Their result, in terms of density rather than center density, shows that the LP bound is at least 2 −τ D+o(D) where 1.1], as are the highest known densities and the numerical LP bounds computed in [1]. Li's dual bound on the LP bound from [24] for D ≤ 13, and the values for D = 12, 16 from Cohn-Triantafillou [9, Table 6.1], have been converted from center density to density. The value from Theorem 1.2 is shown in bold.…”