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.
Parent
- Slepian group-orbit code — Vertices of polytope codes typically form an orbit of the polytope's symmetry group.
Children
- Polyhedron code — Three-dimensional polytope codes are polyhedron codes.
- 120-cell code
- 600-cell code
- \(3_{21}\) polytope code
- Hessian polyhedron code
- Rectified Hessian polyhedron 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 triangle (square, cube, \(n\)-cube).
Cousins
- 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.
- Ball color code — Polytopes dual to the hyperoctahedron, truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron are used to construct 3D ball color codes.
- \([[30,8,3]]\) Bring code — Bring code and related codes listed in [3; Table 1] arrange qubits and stabilizer generators on star polyhedra.
- \([[14,3,3]]\) Rhombic dodecahedron surface code — The rhombic dodecahedron surface code arranges qubits and stabilizer generators on polytopes.
References
- [1]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [2]
- H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
- [3]
- J. Conrad, C. Chamberland, N. P. Breuckmann, and B. M. Terhal, “The small stellated dodecahedron code and friends”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170323 (2018) arXiv:1712.07666 DOI
Page edit log
- Victor V. Albert (2022-11-16) — most recent
Cite as:
“Polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/polytope