Optimality and uniqueness of the Leech lattice among lattices

Abstract

We prove that the Leech lattice is the unique densest lattice in $\mathbb{R}^{24}$. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in $\mathbb{R}^{24}$ can exceed the Leech lattice’s density by a factor of more than $1+1.65\cdot 10^{-30}$, and we give a new proof that $E_8$ is the unique densest lattice in $\mathbb{R}^8$.

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  • [Vo] G. Voronoi, "Propriétés des formes quadratiques positives parfaites," J. Reine Angew. Math.\/, vol. 133, pp. 97-178, 1908.
    @article{ Vo,
      author={Voronoi, G.},
      TITLE={Propriétés des formes quadratiques positives parfaites},
      JOURNAL={J. Reine Angew. Math.\/},
      VOLUME={133},
      YEAR={1908},
      PAGES={97--178},
      }

Authors

Henry Cohn

Microsoft Research New England
One Memorial Drive
Cambridge, MA 02142
United States

Abhinav Kumar

Department of Mathematics
Room 2-169
Massachusetts Institute of Technology
Cambridge, MA 02139
United States