\(\Lambda_{24}\) 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\(_0\).

A generator matrix for the symplectic version [2; Appx. 2] is \begin{align} \left(\begin{smallmatrix}24 & -10 & -10 & -10 & -10 & -10 & -10 & -10 & -10 & -10 & -10 & -10 & 3 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6\\ -10 & 6 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & -1 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3\\ -10 & 4 & 6 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & -1 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2\\ -10 & 4 & 4 & 6 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & -1 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2\\ -10 & 4 & 4 & 4 & 6 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & -1 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3\\ -10 & 4 & 4 & 4 & 4 & 6 & 4 & 4 & 4 & 4 & 4 & 4 & -1 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2\\ -10 & 4 & 4 & 4 & 4 & 4 & 6 & 4 & 4 & 4 & 4 & 4 & -1 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2\\ -10 & 4 & 4 & 4 & 4 & 4 & 4 & 6 & 4 & 4 & 4 & 4 & -1 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2\\ -10 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 6 & 4 & 4 & 4 & -1 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3\\ -10 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 6 & 4 & 4 & -1 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3\\ -10 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 6 & 4 & -1 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3\\ -10 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 6 & -1 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2\\ 3 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 16 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4\\ 6 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ 6 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & 4 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ 6 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & 4 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ 6 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & 4 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ 6 & -2 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & 4 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2\\ 6 & -2 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2\\ 6 & -3 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2\\ 6 & -3 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2\\ 6 & -3 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2\\ 6 & -2 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2\\ 6 & -3 & -2 & -2 & -3 & -2 & -2 & -2 & -3 & -3 & -3 & -2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 \end{smallmatrix}\right). \tag*{(1)}\end{align}

Protection

The Leech lattice has a nominal coding gain of \(4\). It exhibits the densest packing [3] and highest kissing number of 196560 in 24 dimensions.

Notes

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

Cousins

\(A_2\) triangular lattice— The \(A_2\) hexagonal lattice can be specified as a section of the Leech lattice [4; Fig. 6.2].
\(D_3\) face-centered cubic (fcc) lattice— The \(D_3\) lattice can be specified as a section of the Leech lattice [4; Fig. 6.2]. The Leech lattice can also 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 [5,6].
\(D_4\) hyper-diamond lattice— The \(D_4\) lattice can be specified as a section of the Leech lattice [4; Fig. 6.2].
\(E_6\) root lattice— The \(E_6\) lattice can be specified as a section of the Leech lattice [4; Fig. 6.2].
\(E_7\) root lattice— The \(E_7\) lattice can be specified as a section of the Leech lattice [4; Fig. 6.2].
\(E_8\) Gosset lattice— The \(E_8\) lattice can be specified as a section of the Leech lattice [4; Fig. 6.2]. The Leech lattice admits a Turyn-type construction from two copies of the \(E_8\) lattice in the same space [4; Ch. 8, pg. 211].
Combinatorial design— The Leech lattice is completely determined by the Steiner system \(S(5,8,24)\) formed by the octads of the extended Golay code [4; Ch. 12, pg. 335, Thm. 6].
\([24, 12, 8]\) Extended Golay code— Two copies of the 24-dimensional packing obtained from the extended Golay code can be fitted together without overlap to form the Leech lattice [4; Ch. 5, pg. 145]. Half of the lattice can be obtained using Construction \(B^{\star}\) [7; Exam. 10.7.3].
\([11,6,5]_3\) Ternary Golay code— A 12-dimensional complex version of the Leech lattice can be obtained from the ternary Golay code [8,9][4; pg. 200].
Extended quaternary Golay code— The Leech lattice can be constructed from the extended quaternary Golay code via Construction \(A_4\) [4; 3rd Ed., pg. xxxiii] (see also [9–11]).
Spherical sharp configuration— Several spherical sharp configurations are derived from the Leech lattice [12].
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 [5,6].
Modulation scheme— Codewords of the Leech lattice have been proposed to be used for a modulation scheme [13].
Pseudo Golay code— The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [11,14].
Self-dual code over \(\mathbb{Z}_4\)— Each 4-frame of the Leech lattice corresponds to an extremal Type II self-dual code over \(\mathbb{Z}_4\) [15].
\(\Lambda_{16}\) Barnes-Wall lattice— The \(\Lambda_{16}\) Barnes-Wall lattice can be obtained from the Leech lattice by restricting to a fixed 16-dimensional subspace of an involution [4; Ch. 4, pg. 131].
Coxeter-Todd \(K_{12}\) lattice— The Coxeter-Todd lattice can be realized as a subset of the Leech lattice [4; Ch. 4, pg. 128].
Higman-Sims graph-adjacency code— The Higman-Sims graph occurs in the Leech lattice [4].
\(\Lambda_{24}\) Leech lattice-shell code

Member of code lists

Primary Hierarchy

Niemeier latticeLattice 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]. In the holy construction for the Niemeier lattice \(A_2^{12}\), the combinations for which the sum of all coefficients is zero form a copy of the Leech lattice [4; Ch. 24, pg. 510].

Construction \(A_4\) codeLattice ECC

The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [11,14]. The Leech lattice can be constructed from the extended quaternary Golay code via Construction \(A_4\) [4; 3rd Ed., pg. xxxiii] (see also [9–11]).

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]: P. Buser and P. Sarnak, “On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)”, Inventiones Mathematicae 117, 27 (1994) DOI
[3]: 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
[4]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[5]: 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
[6]: A. R. Hammons, 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
[7]: T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[8]: R. A. Wilson, “The complex leech lattice and maximal subgroups of the Suzuki group”, Journal of Algebra 84, 151 (1983) DOI
[9]: “Twenty-three constructions for the Leech lattice”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 381, 275 (1982) DOI
[10]: A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
[11]: G. W. Moore and R. K. Singh, “Beauty and the Beast Part 2: Apprehending the Missing Supercurrent”, Communications in Mathematical Physics 406, (2025) arXiv:2309.02382 DOI
[12]: 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
[13]: G. R. Lang and F. M. Longstaff, “A Leech lattice modem”, IEEE Journal on Selected Areas in Communications 7, 968 (1989) DOI
[14]: E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
[15]: 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
[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/dual/leech.yml.