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


Even unimodular lattice in 24 dimensions that exhibits optimal packing.


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.


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



  • 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


J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
H. Cohn et al., “The sphere packing problem in dimension \(24\)”, Annals of Mathematics 185, (2017) arXiv:1603.06518 DOI
H. Cohn et al., “Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas”, (2022) arXiv:1902.05438
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
A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Algebraic Coding 194 (1994) DOI
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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
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

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