Dodecahedron code

Cousins

• Dual polytope code— The icosahedron and dodecahedron are dual to each other.
• Icosahedron code— The icosahedron and dodecahedron are dual to each other.
• \([24, 12, 8]\) Extended Golay code— The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see [2] for more details. To construct the 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.
• \([15,6,5]\) Petersen cycle code— The Petersen graph can be thought of as a dodecahedron with antipodes identified [3; 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 \(n=3\) [4].
• \([[16,4,3]]\) dodecahedral code— The encoder-respecting form of the \([[16,4,3]]\) dodecahedral code is the graph of vertices of a dodecahedron [5].
• \([[30,8,3]]\) Bring code— The qubits and stabilizer generators of the \([[30,8,3]]\) Bring code lie on the vertices of the small stellated dodecahedron.