The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties: The last condition is equivalent to the condition that unit balls centered at the points of Λ24 do not overlap.
Conway (1983) showed that the Leech lattice is isometric to the set of simple roots (or the Dynkin diagram) of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II25,1.
More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting up to four.
It was used to communicate with the Voyager probes, as it is much more compact than the previously-used Hadamard code.
Quantizers, or analog-to-digital converters, can use lattices to minimise the average root-mean-square error.
The Leech lattice is a good solution to this problem, as the Voronoi cells have a low second moment.
The vertex algebra of the two-dimensional conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R24/Λ24 and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group.
This monster vertex algebra was also used to prove the monstrous moonshine conjectures.
A 24x24 generator (in row convention) for the Leech Lattice is given by the following matrix divided by
: [1] 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 for each fixed residue class modulo 4, the 24 bit word, whose 1s correspond to the coordinates i such that ai belongs to this residue class, is a word in the binary Golay code.
The Golay code, together with the related Witt design, features in a construction for the 196560 minimal vectors in the Leech lattice.
Leech lattice (L mod 8) can be directly constructed by combination of the 3 following sets,
The existence of such an integral vector of Lorentzian norm zero relies on the fact that 12 + 22 + ... + 242 is a perfect square (in fact 702); the number 24 is the only integer bigger than 1 with this property (see cannonball problem).
This was conjectured by Édouard Lucas, but the proof came much later, based on elliptic functions.
in this construction is really the Weyl vector of the even sublattice D24 of the odd unimodular lattice I25.
In the complex construction of the Leech lattice, the binary Golay code is replaced with the ternary Golay code, and the Mathieu group M24 is replaced with the Mathieu group M12.
The quotient by this kernel is a nonsingular bilinear form taking values in (1/2)Z.
Chapman (2001) described a construction using a skew Hadamard matrix of Paley type.
Raji (2005) constructed the Leech lattice using higher power residue codes over the ring
It also has far fewer symmetries than the 24-dimensional hypercube and simplex, or even the Cartesian product of three copies of the E8 lattice.
Each cross contains 24 mutually orthogonal vectors and their negatives, and thus describe the vertices of a 24-dimensional orthoplex.
Each of these crosses can be taken to be the coordinate system of the lattice, and has the same symmetry of the Golay code, namely 212 × |M24|.
Conway, Parker & Sloane (1982) showed that the covering radius of the Leech lattice is
; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space.
Cohn & Kumar (2009) showed that it gives the densest lattice packing of balls in 24-dimensional space.
Spherical codes are higher-dimensional analogues of Tammes problem, which arose as an attempt to explain the distribution of pores on pollen grains.
An example of a spherical code in three dimensions is the set of the 12 vertices of the regular icosahedron.
The theta function of an integral lattice is often written as a power series in
so that the coefficient of qn gives the number of lattice vectors of squared norm 2n.
(Ronan, p. 155) Bei dem Versuch, eine Form aus einer solchen Klasse wirklich anzugeben, fand ich mehr als 10 verschiedene Klassen in Γ24 Witt (1941, p. 324), has a single rather cryptic sentence mentioning that he found more than 10 even unimodular lattices in 24 dimensions without giving further details.