Abstract
We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24.
Highlights
The sphere packing problem asks for the densest packing of spheres into Euclidean space
What fraction of Rn can be covered by congruent balls that do not intersect except along their boundaries? This problem fits into a broad framework of packing problems, including error-correcting codes
[KL] uses this technique to prove the best bounds known for sphere packing density in high dimensions
Summary
The sphere packing problem asks for the densest packing of spheres into Euclidean space. [KL] uses this technique to prove the best bounds known for sphere packing density in high dimensions. We develop linear programming bounds that apply directly to sphere packing, and study these bounds numerically to prove the best bounds known[1] for sphere packing in dimensions 4 through 36. If linear programming bounds can be used to prove the optimality of these lattices, it would not come as a complete surprise, because other packing problems in these dimensions can be solved . (The solutions in dimensions 8 and 24 are obtained from the minimal nonzero vectors in the E8 and Leech lattices.) Because we know a priori that the answer must be an integer, any upper bound within less than 1 of the truth would suffice. We could define admissibility more broadly, to include every function to which Poisson summation applies, but the restricted definition above appears to cover all the useful cases, and is more concrete
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