Here is a list of spherical designs.

[Jump to code graph excerpt]

Code Relation
120-cell code The code forms a spherical 11-design because its vertices can be divided into five 600-cells, each of which forms said design.
24-cell code The 24-cell code is a spherical 5-design [1].
600-cell code The 600-cell code forms a spherical 11-design that is unique up to equivalence [2].
Binary PSK (BPSK) modulation format
Biorthogonal spherical code Biorthogonal spherical codes are the only tight spherical 3-designs [3; Tab. 9.3]. A suitable weighted union of the vertices of a hypercube and an orthoplex forms a weighted spherical 5-design in dimensions \(\geq 3\) [4; Sec. 8.6, Ex. 5-2][5; Exam. 2.6].
Cameron-Goethals-Seidel (CGS) isotropic subspace code
Disphenoidal 288-cell code The disphenoidal 288-cell code forms a spherical 7-design [6].
Dodecahedron code The dodecahedron code forms a spherical 5-design [7].
Hessian polyhedron code The Hessian polytope code forms a tight spherical 4-design [8; Exam. 7.3]. The double Hessian polytope code forms a spherical 5-design [9].
Hypercube code Hypercube codes form spherical 3-designs. The weighted union of the vertices of a hypercube and an orthoplex form a weighted spherical 5-design in dimensions \(\geq 3\) [4; Sec. 8.6, Ex. 5-2][5; Exam. 2.6].
Icosahedron code The icosahedron code forms a unique tight spherical 5-design [10][3; Exam. 9.6.1].
Kerdock spherical code Kerdock spherical codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) orthoplexes [11].
McLaughlin spherical code Both McLaughlin spherical codes are sharp configurations [12,13]. The \((22,275,1/6)\) code is a unique and tight spherical 4-design, while the \((23,552,1/5)\) code is a unique and tight spherical 5-design; see Ref. [12; Appx. A].
Pentakis dodecahedron code Vertices of the pentakis dodecahedron form a weighted spherical 9-design [14,15][5; Exam. 2.5].
Petersen spherical code The Petersen spherical code forms a spherical 2-design [16].
Phase-shift keying (PSK) modulation format
Polygon code A \(q\)-gon is a tight spherical \(q-1\) design.
Quadrature PSK (QPSK) modulation format
Real-Clifford subgroup-orbit code The orbit of any point under the real Clifford subgroup is a spherical 7-design [17], and some are 11-designs [18].
Rectified Hessian polyhedron code The rectified Hessian polyhedron code forms a spherical 5-design [19].
Simplex spherical code Simplex spherical codes are the only tight spherical 2-designs [3; Tab. 9.3]. The bi-simplex is a spherical 3-design since antipodal codes have zero averages over odd-degree polynomials.
Spherical design
Spherical sharp configuration Spherical sharp configurations are spherical designs of strength \(2m-1\) for some \(m\).
Witting polytope code The Witting polytope code forms a tight spherical 7-design [20][21; Ch. 14].
\(2_{31}\) polytope code The 126 vertices of the \(2_{31}\) polytope form a spherical 5-design [9].
\(2_{41}\) polytope code The \(2_{41}\) polytope code forms a spherical 7-design [22].
\(3_{21}\) polytope code The \(3_{21}\) polytope code forms a tight spherical 5-design [10,20][21; Ch. 14][12; Table 1].
\(BW_{32}\) lattice-shell code
\(\Lambda_{16}\) lattice-shell code
List (descendants): All descendants of Spherical design.

References

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