Optimality and uniqueness of the Leech lattice among lattices


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 $\mathbb{R}^8$.


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