# Leech lattice

A particular lattice (see also Lattice of points; Geometry of numbers) in $\mathbf R^{24}$ defined by J. Leech in 1967 [a1] using the close relations between packing of balls and error-correcting binary codes (cf. Error-correcting code), and in particular a code defined by M.J.E. Golay. At present there are many alternative descriptions known.

The automorphism group of the Leech lattice (*i.e.* the group of orthogonal transformations mapping it onto itself) turned out to be of great importance in the search of the sporadic simple groups, see [a2] and Sporadic simple group.

The Newton or kissing number of a ball in $\mathbf R^n$ is the maximum number of non-overlapping balls of the same radius as the given ball that can touch the given ball. Its values are known only for $n=2,3,8,24$, and are 6, 12, 240, and 196560, respectively. For $n=24$ this was established by A.M. Odlyzko and N.J.A. Sloane and V.I. Levenshtein independently. The (unique) arrangement for which the Newton number in $\mathbf R^{24}$ is attained is given by a ball packing where the centres belong to the Leech lattice. It has been conjectured that the Leech lattice provides the densest lattice packing of balls in $\mathbf R^{24}$. As a first step towards a verification of this it has been shown that among all lattice packings of balls the Leech lattice ball packing locally has maximum density, cf. [a5]. For a comprehensive survey of the Leech lattice see [a3].

#### References

[a1] | J. Leech, "Note on sphere packings" Canad. J. Math. , 19 (1967) pp. 251–267 |

[a2] | R.L. Griess, "The friendly giant" Invent. Math. , 69 (1982) pp. 1–102 Zbl 0498.20013 |

[a3] | J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups" , Springer (1988) |

[a4] | P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint) |

[a5] | P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989) |

[a6] | T.M. Thompson, "From error-correcting codes through sphere packings to simple groups" , Math. Assoc. Amer. (1983) |

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Leech lattice.

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