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Niemeier lattice[1]

Description

One of the 24 positive-definite even unimodular lattices of rank 24.

Cousins

  • Self-dual linear code— The nine inequivalent \([24,12]\) doubly even self-dual codes [2] yield certain Niemeier lattices via Construction A [3]. Niemeier lattices can be constructed from ternary self-dual codes of length 24 [4].
  • Self-dual code over \(R\)— Extremal Type II self-dual codes of length 24 over \(\mathbb{Z}_6\) yield Niemeier lattices [5].
  • Octacode— The octacode can be used to construct a Niemeier lattice via Construction \(A_4\) [6].
  • Harada-Kitazume code— Niemeier lattices can be constructed from quaternary codes over \(\mathbb{Z}_4\) via Construction \(A_4\) [6]. These codes are the Harada-Kitazume codes [3].

Member of code lists

Primary Hierarchy

Classical Domain
Analog Kingdom
Analog codeECC
Sphere packing
Lattice-based codeECC
Dual lattice
Unimodular lattice
Parents
Unimodular lattice
Niemeier lattices are even and unimodular.
Niemeier lattice
Children
\(\Lambda_{24}\) Leech lattice
The \(\Lambda_{24}\) Leech lattice is the Niemeier lattice with minimal norm 4 [3]. Every Niemeier lattice is a sublattice of the Leech lattice [3,7].

References

[1]
H.-V. Niemeier, “Definite quadratische formen der dimension 24 und diskriminante 1”, Journal of Number Theory 5, 142 (1973) DOI
[2]
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
[3]
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
[4]
P. S. Montague, “A new construction of lattices from codes over GF(3)”, Discrete Mathematics 135, 193 (1994) DOI
[5]
M. Harada and M. Kitazume, “Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices”, European Journal of Combinatorics 23, 573 (2002) DOI
[6]
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
[7]
C. Dong, H. Li, G. Mason, and S. P. Norton, “Associative subalgebras of the Griess algebra and related topics”, (1996) arXiv:q-alg/9607013
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Zoo Code ID: niemeier

Cite as:
“Niemeier lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/niemeier
BibTeX:
@incollection{eczoo_niemeier, title={Niemeier lattice}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/niemeier} }
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Cite as:

“Niemeier lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/niemeier

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/lattice/niemeier.yml.