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\(E_8\) Gosset lattice-shell code

Description

Spherical code whose codewords are points on the \(E_8\) Gosset lattice normalized to lie on the unit sphere.

The minimal shell of the lattice yields the \((8,240,1)\) code, whose codewords form the vertices of the \(4_{21}\) real polytope and the Witting complex polytope.

Protection

Smallest-shell code yields an optimal solution to the kissing problem in 8D [1,2].

Cousin

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Constant-energy spherical codeECC
Spherical code
Lattice-shell code
Parents
Lattice-shell code
\(E_8\) Gosset lattice-shell code
Children
\(2_{41}\) real polytope code
Codewords of the \(2_{41}\) real polytope form the second-smallest shell of the \(E_8\) lattice [3].
Witting polytope code
Witting polytope codewords form the minimal shell of the \(E_8\) lattice.

References

[1]
A. Korkine and G. Zolotareff, “Sur les formes quadratiques”, Mathematische Annalen 6, 366 (1873) DOI
[2]
E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[3]
S. Borodachov, “Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials”, Aequationes mathematicae 98, 509 (2024) DOI
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Zoo Code ID: eeight_shell

Cite as:
“\(E_8\) Gosset lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eeight_shell
BibTeX:
@incollection{eczoo_eeight_shell, title={\(E_8\) Gosset lattice-shell code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/eeight_shell} }
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Permanent link:
https://errorcorrectionzoo.org/c/eeight_shell

Cite as:

“\(E_8\) Gosset lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eeight_shell

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/lattice_shell/eeight_shell.yml.