Dual Linear Programming Bounds for Sphere Packing via Discrete Reductions
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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.
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