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Dual linear programming bounds for sphere packing via modular forms

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arxiv 1909.04772 v2 pith:6Y54UTQR submitted 2019-09-10 math.MG math.NT

Dual linear programming bounds for sphere packing via modular forms

Henry Cohn , Nicholas Triantafillou This is my paper
classification math.MG math.NT
keywords linearpackingprogrammingbounddimensionsdualformsmodular
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.

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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. Cusp Form Dimensions, Lattice Uniqueness, and LP Sharpness for Sphere Packing in Dimensions 8 and 24

    math.CO 2026-04 unverdicted novelty 7.0

    Three independent conditions on cusp forms, dual LP obstructions, and extremal CFTs are conjectured to be equivalent for d ≡ 0 mod 8 and together explain LP sharpness exclusively in dimensions 8 and 24.

  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.