In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech (Canad. J. Math. 16 (1964), 657--682). It is the unique lattice with the following list of properties:
- It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1.
- It is even; i.e., the square of the length of any vector in Λ is an even integer.
- The shortest length of any non-zero vector in Λ is 2.
The Leech lattice can be explicitly constructed as the set of vectors of the form 2−3/2(a1, a2, ..., a24) where the ai are integers such that
and the set of coordinates i such that ai belongs to any fixed residue class (mod 4) is a word in the binary Golay code.
The Leech lattice is highly symmetrical. Its automorphism group is the double cover of the Conway group Co1; its order is approximately 8.3(10)18.
See:
- Conway, J. H.; Sloane, N. J. A. (1999). Sphere packings, lattices and groups. (3rd ed.) With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften, 290. New York: Springer-Verlag. ISBN 0-387-98585-9.