\(E_7\) root lattice

Description

Lattice in dimension \(7\).

The \(E_7\) root lattice exhibits the densest lattice packing [1–5].

Root lattice
Construction-\(A\) code — The \([7,4,3]\) Hamming code yields the \(E_7^{\perp}\) root lattice code via Construction A [6]. The \([7,3,4]\) little Hamming code yields the \(E_7\) root lattice code via the same construction [6][7; Exam. 10.5.3].

\([7,4,3]\) Hamming code — The \([7,4,3]\) Hamming code yields the \(E_7^{\perp}\) root lattice code via Construction A [6]. The \([7,3,4]\) little Hamming code yields the \(E_7\) root lattice code via the same construction [6][7; Exam. 10.5.3].
\(E_7\) lattice-shell code

[1]: Blichfeldt, H. F. "On the minimum value of positive real quadratic forms in 6 variables." Bulletin of American Math. Soc 31 (1925): 386.
[2]: H. F. Blichfeldt, “The minimum value of quadratic forms, and the closest packing of spheres”, Mathematische Annalen 101, 605 (1929) DOI
[3]: H. F. Blichfeldt, “The minimum values of positive quadratic forms in six, seven and eight variables”, Mathematische Zeitschrift 39, 1 (1935) DOI
[4]: G. L. Watson, “The Class-Number of a Positive Quadratic Form”, Proceedings of the London Mathematical Society s3-13, 549 (1963) DOI
[5]: Vetchinkin, N. M. "Uniqueness of classes of positive quadratic and highest known kissing number in seven dimensions."
[6]: 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
[7]: T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.

Page edit log

“\(E_7\) root lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eseven