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Polytope code

Description

Spherical code whose codewords are the vertices of a polytope, i.e., a geometrical figure bounded by lines, planes, and hyperplanes in either real [1] or complex [2] space. A polytope in two (three, four) dimensions is called a polygon (polyhedron, polychoron).

Notes

See Polytope Wiki and webpage by R. Klitzing for lists of polytopes.

Cousin

  • Quantum spherical code (QSC)— QSCs can be constructed by using vertices of polytopes for logical constellations. The logical constellations form the vertices of the code constellation, a polytope compound.

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Constant-energy spherical codeECC
Spherical code
Slepian group-orbit codeECC
Parents
Slepian group-orbit code
Vertices of polytope codes typically form an orbit of the polytope’s symmetry group.
Polytope code
Children
Polyhedron code
Three-dimensional polytope codes are polyhedron codes.
120-cell code
600-cell code
Rectified Hessian polyhedron code
\(2_{31}\) polytope code
\(3_{21}\) polytope code
\(2_{41}\) polytope code
Dual polytope code
Biorthogonal spherical code
Biorthogonal spherical codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a square (octahedron, 16-cell, \(n\)-orthoplex).
Hypercube code
Hypercube codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a square (cube, tesseract, \(n\)-cube).

References

[1]
H. S. M. Coxeter, Regular Polytopes (Courier Corporation, 1973)
[2]
H. S. M. Coxeter, Regular Complex Polytopes (Cambridge University Press, 1991)
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Zoo Code ID: polytope

Cite as:
“Polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/polytope, arXiv:2606.11484
BibTeX:
@incollection{eczoo_polytope,
title={Polytope code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/polytope}
}
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Permanent link:
https://errorcorrectionzoo.org/c/polytope

Cite as:

“Polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/polytope, arXiv:2606.11484

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