Optimality and uniqueness of the Leech lattice among lattices

2004/03/16 by Henry Cohn, Abhinav Kumar · 1 voice · 3 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Computability, Logic, AI Algorithms #Conjecture #Lattice (music) #Leech #Mathematical Approximation and Integration #Sphere packing #Uniqueness #math.MG

paper · pdf · doi:10.4007/annals.2009.170.1003

published as Annals of Mathematics 170 (2009), 1003-1050 · 39 pages

arxiv published 2004/03/16 · openalex publication_date 2009/11/01 · openalex created_date 2016/06/24 · arxiv created 2017/08/22 · arxiv updated 2017/08/22 · openalex updated_date 2026/08/05

Abstract

We prove that the Leech lattice is the unique densest lattice in R²4. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R²4 can exceed the Leech lattice's density by a factor of more than 1+1.65*10^(-30), and we give a new proof that E₈ is the unique densest lattice in R⁸.

Citations

• Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites.
• New upper bounds on sphere packings I
• Laminated Lattices

Cited by

• Densest local sphere-packing diversity. II. Application to three dimensions
• Improved sphere packing lower bounds from Hurwitz lattices
• Leech lattice [wikipedia]

Discussions

• There's a sense in which this isn't a surprise because of an amazing theorem by Cohn et al. For any "reasonable" utility function on lattice packings in 24D, the solution is the Leech lattice. https:/ [bsky, 0 points, 0 comments]