Description
Lattice in dimension \(6\).
A generating matrix for the lattice embedded in eight dimensions is [1] \begin{align} \begin{bmatrix} 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 root \(E_6\) lattice exhibits the densest lattice packing [2–6] and highest known kissing number in six dimensions.Cousins
- \(q\)-ary repetition code— The \([3,1,3]_3\) ternary repetition code can be used to obtain the \(E_6\) root lattice [7; Exam. 10.5.4].
- \(E_6\) lattice-shell code
- Hessian polyhedron code— The 27 Hessian polyhedron codewords are intimately related to the \(E_6\) Lie group [8].
Member of code lists
Primary Hierarchy
Parents
\(E_6\) root lattice
References
- [1]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [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]
- Vetchinkin, N. M. "Uniqueness of classes of positive quadratic forms, on which values of Hermite constants are reached for 6≤n≤8." Trudy Matematicheskogo Instituta imeni VA Steklova 152 (1980): 34-86.
- [7]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [8]
- L. Manivel, “Configurations of lines and models of Lie algebras”, (2005) arXiv:math/0507118
Page edit log
- Victor V. Albert (2022-11-29) — most recent
Cite as:
“\(E_6\) root lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/esix