Dodecahedron code 

Description

Spherical \((3,20,2-2\sqrt{5}/3)\) code whose codewords are the vertices of the dodecahedron (alternatively, the centers of the faces of a icosahedron, the dodecahedron's dual polytope).

Parents

Cousins

  • Dual polytope code — The icosahedron and dodecahedron are dual to each other.
  • Icosahedron code — The icosahedron and dodecahedron are dual to each other.
  • Golay code — The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see post by J. Baez for more details. To construct the Golay code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.
  • Petersen cycle code — The Petersen graph can be thought of as a dodecahedron with antipodes identified [2; Appx. A.2.1].
  • Pentakis dodecahedron code — The pentakis dodecahedron is the convex hull of the icosahedron and dodecahedron.
  • Simplex spherical code — Vertices of a dodecahedron can be split up into vertices of five tetrahedra, which are simplex spherical codes for \(n=3\) [3].
  • \([[16,4,3]]\) dodecahedral code — The encoder-respecting form of the \([[16,4,3]]\) dodecahedral code is the graph of vertices of a dodecahedron [4].
  • \([[30,8,3]]\) Bring code — The qubits and stabilizer generators of the \([[30,8,3]]\) Bring code lie on the vertices of the small stellated dodecahedron.
  • \([[14,3,3]]\) Rhombic dodecahedron surface code — The qubits of the \([[14,3,3]]\) rhombic dodecahedron surface code lie on the vertices of the small stellated dodecahedron.

References

[1]
S. P. Jain, J. T. Iosue, A. Barg, and V. V. Albert, “Quantum spherical codes”, Nature Physics (2024) arXiv:2302.11593 DOI
[2]
J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[3]
H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
[4]
J. Z. Lu, A. B. Khesin, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2024) arXiv:2411.14448
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: dodecahedron

Cite as:
“Dodecahedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dodecahedron
BibTeX:
@incollection{eczoo_dodecahedron, title={Dodecahedron code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dodecahedron} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/dodecahedron

Cite as:

“Dodecahedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dodecahedron

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/3d/dodecahedron.yml.