\(\Lambda_{24}\) Leech lattice

\(\Lambda_{24}\) Leech lattice[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 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.

Cousins

\([23, 12, 7]\) Golay code— The Leech lattice can be obtained by lifting the Golay code to \(\mathbb{Z}_4\) [3], appending a parity check, and applying Construction \(A_4\) [4] (see also [5,6]). Half of the lattice can be obtained in a different construction [7; Exam. 10.7.3].
Ternary Golay code— The Leech lattice can be obtained by from the ternary Golay code [5].
Spherical sharp configuration— Several spherical sharp configrations are derived from the 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; one of these constructions uses eight copies of the \(D_3\) fcc lattice [9,10].
\(D_3\) face-centered cubic (fcc) lattice— The Leech lattice can be constructed via the Turyn construction and the holy construction using the octacode as the glue code; one of these constructions uses eight copies of the \(D_3\) fcc lattice [9,10].
Modulation scheme— Codewords of the Leech lattice have been proposed to be used for a modulation scheme [11].
Pseudo Golay code— The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [12,13].
Self-dual code over \(\mathbb{Z}_q\)— Each 4-frame of the Leech lattice corresponds to an extremal Type II self-dual code over \(\mathbb{Z}_4\) [14].
Higman-Sims graph-adjacency code— The Higman-Sims graph occurs in the Leech lattice [6].
Extended quaternary Golay code— The Leech lattice can be constructed from the extended quaternary Golay code [13,15].
\(\Lambda_{24}\) Leech lattice-shell code

Member of code lists

Primary Hierarchy

Lattice-based codeECC

Dual lattice

Unimodular lattice

Niemeier latticeLattice-based ECC

Parents

Niemeier lattice

The Leech lattice is the Niemeier lattice with minimal norm 4 [16]. Every Niemeier lattice is a sublattice of the Leech lattice [16,17].

Construction \(A_4\) codeLattice-based ECC

The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [12,13]. The Leech lattice can be obtained by lifting the Golay code to \(\mathbb{Z}_4\) [3], appending a parity check, and applying Construction \(A_4\) [4] (see also [5,6]). Half of the lattice can be obtained in a different construction [7; Exam. 10.7.3].

Universally optimal sphere packingUniversally optimal ECC

The Leech lattice is universally optimal [18].

\(\Lambda_{24}\) Leech lattice

References

[1]: J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
[2]: H. Cohn, A. Kumar, S. Miller, D. Radchenko, and M. Viazovska, “The sphere packing problem in dimension \(24\)”, Annals of Mathematics 185, (2017) arXiv:1603.06518 DOI
[3]: A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
[4]: A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Lecture Notes in Computer Science 194 (1994) DOI
[5]: “Twenty-three constructions for the Leech lattice”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 381, 275 (1982) 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
[10]: A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
[11]: G. R. Lang and F. M. Longstaff, “A Leech lattice modem”, IEEE Journal on Selected Areas in Communications 7, 968 (1989) DOI
[12]: E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
[13]: G. W. Moore and R. K. Singh, “Beauty And The Beast Part 2: Apprehending The Missing Supercurrent”, (2023) arXiv:2309.02382
[14]: A. Munemasa and R. A. L. Betty, “Classification of extremal type II \(\)\mathbb {Z}_4\(\)-codes of length 24”, Designs, Codes and Cryptography 92, 771 (2023) DOI
[15]: A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
[16]: M. Harada and M. Kitazume, “Z4-Code Constructions for the Niemeier Lattices and their Embeddings in the Leech Lattice”, European Journal of Combinatorics 21, 473 (2000) DOI
[17]: C. Dong, H. Li, G. Mason, and S. P. Norton, “Associative subalgebras of the Griess algebra and related topics”, (1996) arXiv:q-alg/9607013
[18]: H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, “Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas”, (2022) arXiv:1902.05438

Page edit log

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

Cite as:

“\(\Lambda_{24}\) Leech lattice”, 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/sphere_packing/lattice/bw/leech.yml.