Icosahedron code

Cousins

• Spherical design— The icosahedron code forms a unique tight spherical 5-design [3][1; Exam. 9.6.1].
• Dual polytope 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 [4] 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.
• 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. The pentakis dodecahedron and truncated icosahedron are dual to each other [5; pg. 55].
• 600-cell code— A realization of the 600-cell can be done in terms of icosians, which are quaternion coordinates of the 120 elements of the binary icosahedral group \(2I \cong 2.A_5\) (a.k.a. the icosian group) [7][6; Ch. 8, pg. 207].
• \([[54,6,5]]\) 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 [8].
• \(((7,2,3))\) Pollatsek-Ruskai code— Binary icosahedral group \(2I\) gates can be realized transversally in the Pollatsek-Ruskai code [9].