Here is a list of polytope codes.

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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 explicit realizations of its 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 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. icosian group) [6][5; Ch. 8, pg. 207].
Antiprism code Spherical \((3,2q)\) code for \(q \geq 2\) whose codewords are the vertices of a \(q\)-antiprism.
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).
Cuboctahedron code Spherical \((3,12,1)\) code whose codewords are the vertices of the cuboctahedron. Codewords form the minimal lattice-shell code of the \(D_3\) face-centered cubic (fcc) lattice.
Dodecahedron code Spherical \((3,20,2-2\sqrt{5}/3)\) code whose codewords are the vertices of the dodecahedron (alternatively, the centers of the faces of a icosahedron, the dodecahedron’s dual polytope).
Dual polytope code For a spherical code whose codewords are vertices of a convex polytope, the dual code consists of codewords corresponding to the facets of the polar dual polytope. For regular polytopes, these dual codewords can be represented by the normalized centers of the facets of the original polytope. The dual codewords make up the vertices of the polytope dual to the original polytope.
Hessian polyhedron code 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 (a.k.a. diplo-Schläfli) code. The code can be obtained from the Schläfli graph [7; Ch. 9]. The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in the complex projective plane [8–12].
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).
Pentakis dodecahedron code Spherical \((3,32,(9-\sqrt{5})/6)\) code whose codewords are the vertices of the pentakis dodecahedron, the convex hull of the icosahedron and dodecahedron.
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 (\(q=6\)).
Polyhedron code A polytope code in three dimensions, i.e., a spherical code whose codewords form vertices of a polyhedron.
Polytope code Spherical code whose codewords are the vertices of a polytope, i.e., a geometrical figure bounded by lines, planes, and hyperplanes in either real [2] or complex [13] space. A polytope in two (three, four) dimensions is called a polygon (polyhedron, polychoron).
Quadrature PSK (QPSK) code A 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, i.e., the 72 roots of \(E_6\) after normalization to the unit sphere. See [13; pg. 127][5; pg. 126] for realizations of the 72 codewords.
Rhombic dodecahedron code Spherical \((3,14,2-2/\sqrt{3})\) code whose codewords are the normalized vertices of the rhombic dodecahedron. Equivalently, the codewords are the union of the vertices of a cube and an octahedron on the unit sphere.
Rhombicuboctahedron code Spherical \((3,24,4/(5+2 \sqrt{2}))\) code whose codewords are the vertices of the rhombicuboctahedron.
Self-dual polytope code A spherical code whose codewords are the vertices of a self-dual polytope.
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. Simplex spherical codewords in 2 (3, 4) dimensions form the vertices of a triangle (tetrahedron, 5-cell) In general, the code makes up the vertices of an \(n\)-simplex. The union of a simplex and its antipodal simplex forms the vertices of a bi-simplex, which has \(2(n+1)\) vertices.
Snub-cube code Spherical \((3,24,0.55384)\) code whose codewords are the vertices of a snub cube, normalized to lie on the unit sphere.
Square-antiprism code Spherical \((3,8,4(4-\sqrt{2})/7)\) code whose codewords are the vertices of the square antiprism [7; pg. 72].
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. The code is optimal and unique up to equivalence [5,14,15]. Antipodal pairs of points of the \(4_{21}\) real polytope code correspond to the 120 tritangent planes of a canonical sextic curve in \(\mathbb{C}P^3\) [9–12].
\(3_{21}\) polytope code Spherical \((7,56,1/3)\) code whose codewords are the vertices of the \(3_{21}\) real polytope (a.k.a. the Hess polytope). The vertices form the kissing configuration of the Witting polytope code. The 1-skeleton of this polytope is the Gosset graph [2]. The code is optimal and unique up to equivalence [5,14,15]. Antipodal pairs of points of the \(3_{21}\) polytope code correspond to the 28 bitangent lines of a general quartic plane curve in the complex project plane [9–12].

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]
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[3]
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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]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
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[7]
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[8]
H. S. M. Coxeter, “The Polytope 2 21 Whose Twenty-Seven Vertices Correspond to the Lines to the General Cubic Surface”, American Journal of Mathematics 62, 457 (1940) DOI
[9]
P. du Val, “On the Directrices of a Set of Points in a Plane”, Proceedings of the London Mathematical Society s2-35, 23 (1933) DOI
[10]
Arnold, V. I. (1999). Symplectization, complexification and mathematical trinities. The Arnoldfest, 23-37.
[11]
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
[12]
Y.-H. He and J. McKay, “Sporadic and Exceptional”, (2015) arXiv:1505.06742
[13]
H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
[14]
E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[15]
H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448