Niemeier lattice[1]
Description
One of the 24 positive-definite even unimodular lattices of rank 24. The 24 lattices are \(D_{24}\), \(D_{16}E_8\), \(E_8^3\), \(A_{24}\), \(D_{12}^2\), \(A_{17}E_7\), \(D_{10}E_7^2\), \(A_{15}D_9\), \(D_8^3\), \(A_{12}^2\), \(A_{11}D_7E_6\), \(E_6^4\), \(A_9^2D_6\), \(D_6^4\), \(A_8^3\), \(A_7^2D_5^2\), \(A_6^4\), \(A_5^4D_4\), \(D_4^6\), \(A_4^6\), \(A_3^8\), \(A_2^{12}\), \(A_1^{24}\), and \(\Lambda_{24}\) (the Leech lattice) [2; Table 16.1].Cousins
- Self-dual linear code— The nine inequivalent \([24,12]\) doubly even self-dual codes [3] yield certain Niemeier lattices via Construction A [4]. Niemeier lattices can be constructed from ternary self-dual codes of length 24 [5].
- Self-dual code over \(\mathbb{Z}_q\)— Extremal Type II self-dual codes of length 24 over \(\mathbb{Z}_6\) yield Niemeier lattices [6].
- Octacode— The octacode is the glue code for the Niemeier lattice \(A_4^6\) [2; 3rd Ed., pg. liv].
- \([4,2,3]_3\) Tetracode— The tetracode is the glue code for the Niemeier lattice \(E_6^4\) [2; Ch. 16, pg. 408].
- \([6,3,4]_4\) Hexacode— The hexacode is the glue code for the Niemeier lattice \(D_4^6\) [2; Ch. 16, pg. 408].
- \([11,6,5]_3\) Ternary Golay code— The extended ternary Golay code is the glue code for the Niemeier lattice \(A^{12}_2\) [2; Ch. 16, pg. 408].
- \([24, 12, 8]\) Extended Golay code— The extended Golay code is the glue code for the Niemeier lattice \(A_1^{24}\) [2; Ch. 16, pg. 408].
- Harada-Kitazume code— Niemeier lattices can be constructed from quaternary codes over \(\mathbb{Z}_4\) via Construction \(A_4\) [7]. These codes are the Harada-Kitazume codes [4].
Member of code lists
Primary Hierarchy
Parents
Niemeier lattices are even and unimodular.
Niemeier lattices can be constructed from quaternary codes over \(\mathbb{Z}_4\) via Construction \(A_4\) [7]. These codes are the Harada-Kitazume codes [4].
Niemeier lattice
Children
The Leech lattice is the Niemeier lattice with minimal norm 4 [4]. Every Niemeier lattice is a sublattice of the Leech lattice [4,8]. 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 [2; Ch. 24, pg. 510]. The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [9,10]. The Leech lattice can be constructed from the extended quaternary Golay code via Construction \(A_4\) [2; 3rd Ed., pg. xxxiii] (see also [10–12]).
References
- [1]
- H.-V. Niemeier, “Definite quadratische formen der dimension 24 und diskriminante 1”, Journal of Number Theory 5, 142 (1973) DOI
- [2]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [3]
- V. Pless and N. J. A. Sloane, “On the classification and enumeration of self-dual codes”, Journal of Combinatorial Theory, Series A 18, 313 (1975) DOI
- [4]
- 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
- [5]
- P. S. Montague, “A new construction of lattices from codes over GF(3)”, Discrete Mathematics 135, 193 (1994) DOI
- [6]
- M. Harada and M. Kitazume, “Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices”, European Journal of Combinatorics 23, 573 (2002) DOI
- [7]
- A. Bonnecaze, P. Gaborit, M. Harada, M. Kitazume, and P. Solé, “Niemeier lattices and Type II codes over Z4”, Discrete Mathematics 205, 1 (1999) DOI
- [8]
- C. Dong, H. Li, G. Mason, and S. P. Norton, “Associative subalgebras of the Griess algebra and related topics”, (1996) arXiv:q-alg/9607013
- [9]
- E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
- [10]
- 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
- [11]
- J. H. Conway and N. J. A. Sloane, “Twenty-three constructions for the Leech lattice”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 381, 275 (1982) DOI
- [12]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
Page edit log
- Victor V. Albert (2022-11-08) — most recent
Cite as:
“Niemeier lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/niemeier