Description
Spherical \((3,12,2-2/\sqrt{5})\) code whose codewords are the vertices of the icosahedron (alternatively, the centers of the faces of a dodecahedron, the icosahedron's dual polytope).
Protection
Optimal configuration of 12 points in 3D space [1; pg. 76]. Saturates the absolute bound for antipodal codes [1; pg. 314].
Notes
See post by J. Baez for more details.
Parents
- Polyhedron code
- Spherical sharp configuration — The icosahedron is a sharp configuration [2,3].
- Spherical design — The icosahedron code forms a unique tight spherical 5-design [4][1; Exam. 9.6.1].
Cousins
- Dual polytope 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.
- Dodecahedron code — The icosahedron and dodecahedron are dual to each other.
- Pentakis dodecahedron code — The pentakis dodecahedron is the convex hull of the icosahedron and dodecahedron.
- \([[54,6,5]]\) five-covered icosahedral code — The encoder-respecting form of the \([[54,6,5]]\) five-covered icosahedral code is the graph of a five-cover of the icosahedron [5].
References
- [1]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [2]
- Andreev, Nikolay N. "An extremal property of the icosahedron." East J. Approx 2.4 (1996): 459-462.
- [3]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
- [4]
- P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs”, Geometriae Dedicata 6, 363 (1977) DOI
- [5]
- J. Z. Lu, A. B. Khesin, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2024) arXiv:2411.14448
Page edit log
- Victor V. Albert (2022-11-16) — most recent
Cite as:
“Icosahedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/icosahedron