Here is a list of polytope codes.

Code | Description |
---|---|

120-cell code | Spherical \((4,600,(7-3\sqrt{5})/4)\) code whose codewords are the vertices of the 120-cell. See [2][1; Table 1][3; Table 3] for realizations of the 600 codewords. |

24-cell code | Spherical \((4,24,1)\) code whose codewords are the vertices of the 24-cell. Codewords form the minimal lattice-shell code of the \(D_4\) lattice. |

600-cell code | Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. See [4; Table 1][3; Table 3] for realizations of the 120 codewords. A realization in terms of quaternions yields the 120 elements of the binary icosahedral group \(2I\) [5]. |

Binary PSK (BPSK) code | Encodes one bit of information into a constellation of antipodal points \(\pm\alpha\) for complex \(\alpha\). These points are typically associated with two phases of an electromagnetic signal. |

Biorthogonal spherical code | Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization. The code makes up the vertices of an \(n\)-orthoplex (a.k.a. hyperoctahedron or cross polytope). |

Cubeoctahedron code | Spherical \((3,12,1)\) code whose codewords are the vertices of the cubeoctahedron. Codewords form the minimal lattice-shell code of the \(D_3\) face-centered cubic (fcc) lattice. |

Hessian polyhedron code | Also known as the Schlafli configuration. Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) real polytope. Two copies of the code yield the \((6,54,1)\) double Hessian polyhedron code. |

Hypercube code | Spherical \((n,2^n,4/n)\) code whose codewords are vertices of an \(n\)-cube, i.e., all permutations and negations of the vector \((1,1,\cdots,1)\), up to normalization. |

Icosahedron code | 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). |

Phase-shift keying (PSK) code | A \(q\)-ary phase-shift keying (\(q\)-PSK) encodes one \(q\)-ary digit of information into a constellation of \(q\) points distributed equidistantly on a circle in \(\mathbb{C}\) or, equivalently, \(\mathbb{R}^2\). |

Polygon code | Spherical \((1,q,4\sin^2 \frac{\pi}{q})\) code for any \(q\geq1\) whose codewords are the vertices of a \(q\)-gon. Special cases include the line segment (\(q=2\)), triangle (\(q=3\)), square (\(q=4\)), pentagon (\(q=5\)), and hexagon (\(q=6\)). |

Polytope code | Spherical code whose codewords are the vertices of a polytope, i.e., a geometrical figure bounded by lines, planes, and hyperplanes [2]. Polytopes in two (three) real or complex dimensions are called polygons (polyhedra). |

Quadrature PSK (QPSK) code | Also known as quadriphase PSK, 4-PSK, or 4-QAM. Quaternary encoding into a constellation of four points distributed equidistantly on a circle. For the case of \(\pi/4\)-QPSK, the constellation is \(\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}\). |

Rectified Hessian polyhedron code | Spherical \((6,72,1)\) code whose codewords are the vertices of the rectified Hessian complex polyhedron and the \(1_{22}\) real polytope. Codewords form the minimal lattice-shell code of the \(E_6\) lattice. See [6; pg. 127][7; pg. 126] for realizations of the 72 codewords. |

Simplex spherical code | Spherical \((n,n+1,2+2/n)\) code whose codewords are all permutations of the \(n+1\)-dimensional vector \((1,1,\cdots,1,-n)\), up to normalization, forming an \(n\)-simplex. Codewords are all equidistant and their components add up to zero. For example, the spherical simplex code in \(n=3\) makes up the vertices of a tetrahedron. In general, the code makes up the vertices of an \(n\)-simplex. See [8; Sec. 7.7] for a parameterization. |

Snub-cube code | Spherical \((3,24,0.55384)\) code whose codewords are the vertices of the snub cube. |

Square-antiprism code | Spherical \((3,8,4(4-\sqrt{2})/7)\) code whose codewords are the vertices of the square antiprism. |

Witting polytope code | Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope, the \(4_{21}\) real polytope, and the minimal lattice-shell code of the \(E_8\) lattice. Taking its kissing configuration yields the \((7,56,1/3)\) spherical code. Both codes are optimal and unique up to equivalence [7,9,10]. |

## References

- [1]
- M. Waegell and P. K. Aravind, “Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell”, Foundations of Physics 44, 1085 (2014) arXiv:1309.7530 DOI
- [2]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [3]
- S. Mamone, G. Pileio, and M. H. Levitt, “Orientational Sampling Schemes Based on Four Dimensional Polytopes”, Symmetry 2, 1423 (2010) DOI
- [4]
- M. Waegell and P. K. Aravind, “Critical noncolorings of the 600-cell proving the Bell–Kochen–Specker theorem”, Journal of Physics A: Mathematical and Theoretical 43, 105304 (2010) arXiv:0911.2289 DOI
- [5]
- L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
- [6]
- H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
- [7]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [8]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [9]
- E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
- [10]
- H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448