Leech lattice

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Definition via a unique characterization

The Leech lattice is the unique lattice in 24-dimensional Euclidean space satisfying all of the following conditions:

Condition name What it means
unimodular lattice The lattice can be generated by a set of basis vectors such that the determinant of the 24 \times 24 matrix formed by these vectors is 1.
even lattice The square of the length of any vector in the lattice is even.
unit balls do not intersect The length of any non-zero vector in the lattice is greater than 2. Equivalently, unit balls centered at the points of the lattice have empty pairwise intersections.

Definition using the binary Golay code

The Leech lattice can be defined in terms of the binary Golay code.