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Dual Linear Programming Bounds for Sphere Packing via Discrete Reductions

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arxiv 2206.09876 v3 pith:32FLJAAB submitted 2022-06-20 math.MG cs.ITmath.COmath.IT

Dual Linear Programming Bounds for Sphere Packing via Discrete Reductions

Rupert Li This is my paper
classification math.MG cs.ITmath.COmath.IT
keywords cohn-elkieslinearpackingboundsdualprogramspherebound
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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 $3 \leq d \leq 13$ 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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Linear Programming Bounds for Fibered Sphere Packings

    math.MG 2026-07 conditional novelty 8.0

    Linear programming bounds are sharp for some fibered sphere packings, collapse to the planar bound in a key six-dimensional case, and fail to prove the Cohn–Rajagopal conjecture for D5/A3.

  2. A dual linear programming bound for sphere packing in dimension 36

    math.MG 2026-07 conditional novelty 6.0

    An exact dual Cohn–Elkies certificate in dimension 36 proves the LP bound exceeds the Kschischang–Pasupathy packing density by ≥32.91, so that packing cannot be certified optimal by any Cohn–Elkies auxiliary function.

  3. A dual linear programming bound for sphere packing in dimension 36

    math.MG 2026-07 accept novelty 6.0

    In dimension 36, a modular-form certificate proves the Cohn–Elkies LP bound exceeds the density of the best known packing by at least a factor of 32.9.