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
- Polytope 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; Ex. 9.6.1].
Cousins
- 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.
- Simplex spherical code — Vertices of a dodecahedron can be split up into vertices of five tetrahedra, which are simplex spherical codes for \(n=3\) [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]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
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