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\(2_{41}\) polytope code

Description

An antipodal spherical \((8,2160,1/2)\) code whose codewords are the vertices of the second-smallest shell of the \(E_8\) lattice [1][2; Table 10.3].

Notes

See the corresponding Bendwavy database entry [3].

Cousin

  • Witting polytope code— Vertices of the \(2_{41}\) and \(4_{21}\) polytopes minimize each other’s potential functions [1].

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Constant-energy spherical codeECC
Spherical code
Slepian group-orbit codeECC
Polytope code
Parents
Polytope code
\(E_8\) Gosset lattice-shell codeLattice-shell ECC
Codewords of the \(2_{41}\) polytope form the second-smallest shell of the \(E_8\) lattice [1].
Spherical designECC \(t\)-design
The \(2_{41}\) polytope code forms a spherical 7-design [1].
\(2_{41}\) polytope code

References

[1]
S. Borodachov, “Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials”, Aequationes mathematicae 98, 509 (2024) DOI
[2]
T. Ericson and V. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
[3]
R. Klitzing. “Bay.” Polytopes & their Incidence Matrices. bendwavy.org/klitzing/incmats/bay.htm
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Zoo Code ID: 241_polytope

Cite as:
“\(2_{41}\) polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/241_polytope
BibTeX:
@incollection{eczoo_241_polytope, title={\(2_{41}\) polytope code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/241_polytope} }
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Permanent link:
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Cite as:

“\(2_{41}\) polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/241_polytope

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/8d/241_polytope.yml.