Dual linear programming bounds for sphere packing via discrete reductions

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, 4, and 5 dimensional sphere packing problems. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar