Description
Lattice in dimension \(7\).
A generating matrix for the lattice embedded in eight dimensions is [1] \begin{align} \begin{bmatrix} -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}
Cousins
- \([7,4,3]\) Hamming code— The \([7,4,3]\) Hamming code yields the \(E_7^{\perp}\) root lattice via Construction A [7].
- \([7,3,4]\) simplex code— The \([7,3,4]\) simplex code yields the \(E_7\) root lattice via Construction A [7][8; Exam. 10.5.3][1; pg. 138].
- \(E_7\) lattice-shell code
Member of code lists
Primary Hierarchy
Parents
The \([7,3,4]\) simplex code yields the \(E_7\) root lattice via Construction A [7][8; Exam. 10.5.3].
\(E_7\) 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 and highest known kissing number in seven dimensions."
- [7]
- J. H. Conway and N. J. A. Sloane, “On the Voronoi Regions of Certain Lattices”, SIAM Journal on Algebraic Discrete Methods 5, 294 (1984) DOI
- [8]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
Page edit log
- Victor V. Albert (2022-12-12) — most recent
Cite as:
“\(E_7\) root lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eseven