Leech lattice

The Leech lattice is the unique even (http://planetmath.org/EvenLattice) unimodular lattice of dimensionPlanetmathPlanetmath (http://planetmath.org/Dimension2) 24 having no elements of norm 2. Its automorphism groupMathworldPlanetmathPlanetmath (http://planetmath.org/EquivalentCode) is the largest Conway group Co0 (sometimes denoted by 0). The quotientPlanetmathPlanetmath of Co0 by its center is called Co1, a sporadic simple group.

The construction of the Leech lattice below depends on the existence of the extended binary Golay code 𝒢24 (for a construction of the latter, see miracle octad generator).

1 Construction of the Leech lattice

Let Ω={1,2,,24} and assume we have constructed the Golay 𝒢24 on Ω. The Leech lattice Λ is the set of all points


in 24 where each xi is an integer, such that

  • For some integer m, we have xixjm(mod2) for all i,jΩ;

  • For any integer n, the set of coordinatesMathworldPlanetmathPlanetmath {iΩ:xin(mod4)} is in 𝒢24;

  • iΩxi4m(mod8).

2 Properties of the Leech lattice

1. The Leech lattice Λ is an unimodular lattice; in other words:

2. Let Λ(n)={xΛ:xx=2n}. Then |Λ(0)|=1, |Λ(1)|=0, |Λ(2)|=196560, |Λ(3)|=16773120, |Λ(4)|=398034000.

3. The automorphism group (http://planetmath.org/EquivalentCode) Aut(Λ) is the largest Conway group Co0 with order 8 315 553 613 086 720 000=222395472111323.

4. The group Co0 acts transitively (http://planetmath.org/LeftAction) on the sets Λ(2), Λ(3), Λ(4). For n=2,3, the imprimitivity blocks of the action of Co0 on Λ(n) are the sets {x,-x} where xΛ(n). The imprimitivity blocks of the action of Co0 on Λ(4) are sets of 48 vectors called . Any two distinct vectors in a are either or orthogonalMathworldPlanetmathPlanetmath, and are congruentMathworldPlanetmathPlanetmath (http://planetmath.org/QuotientGroup) modulo 2Λ.

5. Any vector in Λ is modulo 2Λ to a vector in Λ(n) for one of n=0,2,3,4. The imprimitivity blocks of the action of Co0 on these sets account for all classes (http://planetmath.org/EquivalenceClass) of Λ/2Λ:



  • 1 J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999.
Title Leech lattice
Canonical name LeechLattice
Date of creation 2013-03-22 18:43:23
Last modified on 2013-03-22 18:43:23
Owner monster (22721)
Last modified by monster (22721)
Numerical id 7
Author monster (22721)
Entry type Definition
Classification msc 20D08
Classification msc 20B25
Classification msc 11H56
Classification msc 11H06
Classification msc 51E10
Related topic BinaryGolayCode
Related topic MiracleOctadGenerator