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 real Clifford group [3–5].
Protection
BW lattices in dimension \(2^{m+1}\) have a nominal coding gain of \(2^{m/2}\). Their kissing number is \(K_{\text{min}} = \prod_{i=1}^{m+1} (2^i + 2)\).Cousins
- Reed-Muller (RM) code— BW lattices are lattice analogues of RM codes in that both can be constructed recursively via a \(|u|u+v|\) construction [6,7].
- Qubit stabilizer code— Stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the Clifford group is related to the symmetry group of the lattice [5].
- Modular-qudit stabilizer code— Modular-qudit stabilizer states can be mapped into the first lattice shell of a BW lattice over a cyclotomic field, while the modular-qudit Clifford group is related to the symmetry group of the lattice [5].
- \(\mathbb{Z}^n\) hypercubic lattice— The hypercubic lattice for \(n=2\) is the \(m=0\) BW lattice.
- \(C_{m,r}\) code— \(C_{m,r=1}\) codes give rise to certain BW lattices [8,9].
- Real-Clifford subgroup-orbit code— The automorphism group of BW lattices is a subgroup of index 2 of a real Clifford group [10,11] (see [3,5] for an explanation).
Member of code lists
Primary Hierarchy
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]
- C. Bachoc, “Designs, groups and lattices”, (2007) arXiv:0712.1939
- [5]
- V. Kliuchnikov and S. Schönnenbeck, “Stabilizer operators and Barnes-Wall lattices”, (2024) arXiv:2404.17677
- [6]
- E. L. Cusack, “Error control codes for QAM signalling”, Electronics Letters 20, 62 (1984) DOI
- [7]
- G. D. Forney, “Coset codes. I. Introduction and geometrical classification”, IEEE Transactions on Information Theory 34, 1123 (1988) DOI
- [8]
- P. Sole, "Generalized theta functions for lattice vector quantization", in Coding and Quantization, DIMACS Series in Dr,crete Mathenulies and Theoretical Computer Science, vol. 14. Providence, RH: American Math. Soc., 1993, pp. 27-32.
- [9]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- [10]
- 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
- [11]
- V. M. Sidelnikov, “On a finite group of matrices generating orbit codes on Euclidean sphere”, Proceedings of IEEE International Symposium on Information Theory 436 DOI
Page edit log
- Victor V. Albert (2022-11-08) — most recent
Cite as:
“Barnes-Wall (BW) lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/barnes_wall