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Optimality and uniqueness of the Leech lattice among lattices. (English) Zbl 1213.11144

An even unimodular lattice \(\Lambda\in\mathbb R^n\) is a lattice such that \(|\Lambda|=1\), \(\langle x,y\rangle\in\mathbb Z\) for all \(x,y\in\Lambda\) and \(<x,x>\;\in 2\mathbb Z\) for all \(x\in \Lambda\). Such lattices exist only if \(n\) is a multiple of \(8\). Up to isometries of \(\mathbb R^n\), the unique exemple when \(n=8\) is the \(E_8\) lattice. There are \(24\) examples in \(\mathbb R_{24}\) among which the Leech lattice is the unique one containing no vector of length \(\sqrt{2}\), see [H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. 3rd ed. Grundlehren Math. Wiss. 290. New York, NY: Springer. (1999; Zbl 0915.52003)].
The authors prove: The Leech lattice is the unique densest lattice in \(\mathbb R^{24}\), up to scaling and isometries of \(\mathbb R^{24}\).
They also give a new proof of Blichfeldt-Vetcinkin’s [H. F. Blichfeldt, Math. Z. 39, 1–15 (1934; Zbl 0009.24403)] theorem: The \(E_8\) root lattice is the unique densest lattice in \(\mathbb R^8\), up to scaling and isometries of \(\mathbb R^8\).
The proof combines human reasoning with computer verification of the properties of certain explicit polynomials (less than one hour on a personal computer): the authors prove also that no sphere-packing in \(\mathbb R^{24}\) can exceed the Leech’s lattice density by a factor of more than \(1+1.65\cdot10^{-30}\).
{The method:} Section \(8\) of [H. Cohn and N. Elkies, Ann. Math. (2) 157, No. 2, 689–714 (2003; Zbl 1041.52011)] shows how to prove that the Leech lattice is the unique densest periodic packing (i.e., union of finitely many translates of a lattice) if a function from \(\mathbb R^{24}\) to \(\mathbb R\) with certain properties exists. To try to prove that the Leech lattice is the unique densest lattice in \(\mathbb R_{24}\) starting from this result would lead to computations far beyond what is possible on a computer. Having recalled the definitions of a spherical design in the sphere \(S^{n-1}\) (see also [P. Delsarte, J. M. Goethals and J. J. Seidel, Geom. Dedicata 6, 363–388 (1977; Zbl 0376.05015)]), and a \(k\)-association scheme, the authors use these objects to salvage the approach by using also complementary sophisticated arguments that take advantage of some special properties of Leech’s lattice: its automorphism group acts transitively on pair of minimal vectors with the same inner product, its minimal vector form an association scheme when pairs are grouped according to their inner products, and its minimal vectors form a spherical \(11\)-design.

MSC:

11H31 Lattice packing and covering (number-theoretic aspects)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
05B40 Combinatorial aspects of packing and covering
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References:

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