Leech lattice

Builds on Lattice (group)Euclidean space Golay code

Description

The Leech lattice $\Lambda_{24}$ is a lattice in the Euclidean space $\R^{24}$ , discovered by John Leech. Up to isometry, it is the unique lattice $L \subset \R^{24}$ with these properties:

$L$ is unimodular: it has a basis whose Gram matrix has determinant $1$ ;
$L$ is even: $x \cdot x$ is an even integer for every $x \in L$ ;
$L$ has no vector $x$ with $x \cdot x = 2$ .

So its shortest nonzero vectors have norm $x \cdot x = 4$ . There are $196560$ of them.

Remarks

The Leech lattice can be constructed from the extended binary Golay code. Its automorphism group $\mathrm{Co}_0$ has order $8315553613086720000$ , and its quotient by $\set{\pm 1}$ is the Conway group $\mathrm{Co}_1$ . The $196560$ shortest vectors show that the kissing number in dimension $24$ is at least $196560$ , and in fact it equals $196560$ . The sphere packing obtained from the Leech lattice is the densest sphere packing in $\R^{24}$ ; this was proved by Cohn, Kumar, Miller, Radchenko and Viazovska. Even unimodular lattices exist only in dimensions divisible by $8$ . In dimension $24$ there are $24$ of them up to isometry, and the Leech lattice is the only one without vectors of norm $2$ .

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Conway groups

Tag	`XETV3657DG`
URI	`m2://XETV3657DG`
Kind	object
Writing	W0 stub
MSC	11H06, 11H31, 11H56
Uses	Lattice (group), Euclidean space, Golay code
Same as	Wikidata Q2510203 Wikipedia Leech lattice
Changed	2 November 2026

Leech lattice

Description

Remarks

Reading path

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Further reading

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