Barnes-Wall (BW) lattice code[1,2] 

Description

Member of a family of \(2^{m+1}\)-dimensional lattices, denoted as BW\(_{2^{m+1}}\), that are the densest lattices known. Members include the integer square lattice \(\mathbb{Z}^2\), \(D_4\), the Gosset \(E_8\) lattice, and the \(\Lambda_{16}\) lattice, corresponding to \(m\in\{0,1,2,3\}\), respectively.

Its automorphism group is the Clifford group [3,4].

Protection

BW lattices in dimension \(2^{m+1}\) have a nominal coding gain of \(2^{m/2}\). Their kissing number is \(K_{\text{min}} = \prod_{i=1}^{m+1} (2^i + 2)\).

Parent

Children

Cousins

References

[1]
E. S. Barnes and G. E. Wall, “Some extreme forms defined in terms of Abelian groups”, Journal of the Australian Mathematical Society 1, 47 (1959) DOI
[2]
M. Broué and M. Enguehard, “Une famille infinie de formes quadratiques entières; leurs groupes d’automorphismes”, Annales scientifiques de l’École normale supérieure 6, 17 (1973) DOI
[3]
G. Nebe, E. M. Rains, and N. J. A. Sloane, “The invariants of the Clifford groups”, (2000) arXiv:math/0001038
[4]
V. Kliuchnikov and S. Schonnenbeck, “Stabilizer operators and Barnes-Wall lattices”, (2024) arXiv:2404.17677
[5]
E. L. Cusack, “Error control codes for QAM signalling”, Electronics Letters 20, 62 (1984) DOI
[6]
G. D. Forney, “Coset codes. I. Introduction and geometrical classification”, IEEE Transactions on Information Theory 34, 1123 (1988) DOI
[7]
V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform. 33 (1997), 35–54 (1997); English translation in Problems Inform. Transmission 33 (1997), 29–44
[8]
V. M. Sidelnikov, “On a finite group of matrices generating orbit codes on Euclidean sphere”, Proceedings of IEEE International Symposium on Information Theory DOI
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Zoo Code ID: barnes_wall

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

“Barnes-Wall (BW) lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/barnes_wall

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