# \(\Lambda_{24}\) Leech lattice code[1]

## Description

Even unimodular lattice in 24 dimensions that exhibits optimal packing. Its automorphism group is the Conway group \(.0\) a.k.a. Co\(_0\).

## Protection

The \(\Lambda_{24}\) Leech lattice has a nominal coding gain of \(4\). It exhibits the densest packing [2] and highest kissing number of 196560 in 24 dimensions.

## Notes

Popular summary of solution to the sphere-packing problem in Quanta Magazine.

## Parents

- Niemeier lattice code
- Universally optimal sphere packing — The \(\Lambda_{24}\) Leech lattice code is universally optimal [3].

## Cousins

- Golay code — The \(\Lambda_{24}\) Leech lattice can be obtained by lifting the Golay code to \(\mathbb{Z}_4\) [4], appending a parity check, and applying construction \(A_4\) [5] (see also [6]). Half of the lattice can be obtained in a different construction [7; Ex. 10.7.3].
- Spherical sharp configuration — Several spherical sharp configrations are derived from the \(\Lambda_{24}\) Leech lattice [8].
- Octacode — The Leech lattice can be constructed via the Turyn construction and the holy construction using the octacode as the glue code [9].
- \(\Lambda_{24}\) Leech lattice-shell code

## References

- [1]
- J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
- [2]
- H. Cohn et al., “The sphere packing problem in dimension \(24\)”, Annals of Mathematics 185, (2017) arXiv:1603.06518 DOI
- [3]
- H. Cohn et al., “Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas”, (2022) arXiv:1902.05438
- [4]
- A. R. Hammons et al., “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
- [5]
- A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Algebraic Coding 194 (1994) DOI
- [6]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [7]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [8]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
- [9]
- Feng-Wen Sun and H. C. A. van Tilborg, “The Leech lattice, the octacode, and decoding algorithms”, IEEE Transactions on Information Theory 41, 1097 (1995) DOI

## Page edit log

- Victor V. Albert (2022-11-08) — most recent

## Cite as:

“\(\Lambda_{24}\) Leech lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/leech