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

Description

Even unimodular lattice in 24 dimensions that exhibits optimal packing.

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

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
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Zoo Code ID: leech

Cite as:
\(\Lambda_{24}\) Leech lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/leech
BibTeX:
@incollection{eczoo_leech, title={\(\Lambda_{24}\) Leech lattice code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/leech} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/lattice/bw/leech.yml.