\(E_8\) Gosset lattice[1]
Description
Unimodular even BW lattice in dimension \(8\), consisting of the Cayley integral octonions rescaled by \(\sqrt{2}\). The lattice corresponds to the \([8,4,4]\) Hamming code via Construction A.
A generator matrix is \begin{align} M = \begin{bmatrix} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix}. \tag*{(1)}\end{align}
Protection
The \(E_8\) lattice has a nominal coding gain of \(2\). It exhibits the densest lattice packing [2–5], densest packing [6], and highest kissing number of 240 [7] in eight dimensions.
Notes
Popular summary of solution to the sphere-packing problem in Quanta Magazine.
Parents
- Root lattice
- Barnes-Wall (BW) lattice
- Construction-\(A\) code — The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice code via Construction A [8; Exam. 10.5.2].
- Unimodular lattice — The \(E_8\) Gosset lattice code is even and unimodular.
- Universally optimal sphere packing — The \(E_8\) Gosset lattice code is universally optimal [9].
Cousins
- \([8,4,4]\) extended Hamming code — The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice code via Construction A [8; Exam. 10.5.2].
- Octacode — The octacode yields the \(E_8\) Gosset lattice code via Construction \(A_4\) [10,11].
- \(q\)-ary Hamming code — The \([4,2,3]_3\) ternary Hamming code can be used to obtain the \(E_8\) Gosset lattice code [8; Exam. 10.5.5].
- Repetition code — The \([8,1,8]\) repetition code can be used to obtain the \(E_8\) Gosset lattice code [8; Exam. 10.7.1].
- Spherical sharp configuration — Several spherical sharp configrations are derived from the \(E_8\) Gosset lattice code [12].
- \(E_8\) Gosset lattice-shell code
References
- [1]
- Gosset, Thorold. "On the regular and semi-regular figures in space of n dimensions." Messenger of Mathematics 29 (1900): 43-48.
- [2]
- Blichfeldt, H. F. "On the minimum value of positive real quadratic forms in 6 variables." Bulletin of American Math. Soc 31 (1925): 386.
- [3]
- H. F. Blichfeldt, “The minimum value of quadratic forms, and the closest packing of spheres”, Mathematische Annalen 101, 605 (1929) DOI
- [4]
- H. F. Blichfeldt, “The minimum values of positive quadratic forms in six, seven and eight variables”, Mathematische Zeitschrift 39, 1 (1935) DOI
- [5]
- G. L. Watson, “The Class-Number of a Positive Quadratic Form”, Proceedings of the London Mathematical Society s3-13, 549 (1963) DOI
- [6]
- H. Cohn, A. Kumar, S. Miller, D. Radchenko, and M. Viazovska, “The sphere packing problem in dimension \(24\)”, Annals of Mathematics 185, (2017) arXiv:1603.06518 DOI
- [7]
- Vetchinkin, N. M. "Uniqueness of classes of positive quadratic and highest kissing number of 240 in eight dimensions"
- [8]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [9]
- H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, “Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas”, (2022) arXiv:1902.05438
- [10]
- A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Lecture Notes in Computer Science 194 (1994) DOI
- [11]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- [12]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
Page edit log
- Victor V. Albert (2022-11-08) — most recent
Cite as:
“\(E_8\) Gosset lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eeight