\(\Lambda_{24}\) 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 [9].
- Modulation scheme— Codewords of the Leech lattice have been proposed to be used for a modulation scheme [10].
- Pseudo Golay code— The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [11,12].
- Self-dual code over \(\mathbb{Z}_q\)— Each 4-frame of the Leech lattice corresponds to an extremal Type II self-dual code over \(\mathbb{Z}_4\) [13].
- Higman-Sims graph-adjacency code— The Higman-Sims graph occurs in the Leech lattice [6].
- Quaternary Golay code— The Leech lattice can be constructed from the quaternary Golay code [12,14].
- \(\Lambda_{24}\) Leech lattice-shell code
Primary Hierarchy
Parents
The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [11,12]. 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].
The Leech lattice is universally optimal [17].
\(\Lambda_{24}\) 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]
- G. R. Lang and F. M. Longstaff, “A Leech lattice modem”, IEEE Journal on Selected Areas in Communications 7, 968 (1989) DOI
- [11]
- E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
- [12]
- G. W. Moore and R. K. Singh, “Beauty And The Beast Part 2: Apprehending The Missing Supercurrent”, (2023) arXiv:2309.02382
- [13]
- 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
- [14]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- [15]
- 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
- [16]
- C. Dong, H. Li, G. Mason, and S. P. Norton, “Associative subalgebras of the Griess algebra and related topics”, (1996) arXiv:q-alg/9607013
- [17]
- 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