Barnes-Wall (BW) lattice code[1,2] 


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].


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





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
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
G. Nebe, E. M. Rains, and N. J. A. Sloane, “The invariants of the Clifford groups”, (2000) arXiv:math/0001038
V. Kliuchnikov and S. Schonnenbeck, “Stabilizer operators and Barnes-Wall lattices”, (2024) arXiv:2404.17677
E. L. Cusack, “Error control codes for QAM signalling”, Electronics Letters 20, 62 (1984) DOI
G. D. Forney, “Coset codes. I. Introduction and geometrical classification”, IEEE Transactions on Information Theory 34, 1123 (1988) DOI
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
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

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“Barnes-Wall (BW) lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@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={} }
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“Barnes-Wall (BW) lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.