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) code
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) code
Polygon code A \(q\)-gon is a tight spherical \(q-1\) design.
Quadrature PSK (QPSK) code
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

References

[1]
H. Cohn, J. H. Conway, N. D. Elkies, and A. Kumar, “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
[2]
P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
[3]
T. Ericson and V. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
[4]
A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice Hall, 1971)
[5]
S. Borodachov, P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Energy bounds for weighted spherical codes and designs via linear programming”, (2024) arXiv:2403.07457
[6]
N. J. A. Sloane, R. H. Hardin, and P. Cara, “Spherical designs in four dimensions”, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674) 253 DOI
[7]
S. P. Jain, J. T. Iosue, A. Barg, and V. V. Albert, “Quantum spherical codes”, Nature Physics 20, 1300 (2024) arXiv:2302.11593 DOI
[8]
A. Roy and S. Suda, “Complex spherical designs and codes”, (2011) arXiv:1104.4692
[9]
B. Venkov. Réseaux et designs sphériques. In Réseaux euclidiens, designs sphériques et formes modulaires, volume 37 of Monogr. Enseign. Math., pages 10–86. Enseignement Math., Geneva, 2001
[10]
P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs”, Geometriae Dedicata 6, 363 (1977) DOI
[11]
H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
[12]
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
[13]
P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Universal minima of discrete potentials for sharp spherical codes”, (2023) arXiv:2211.00092
[14]
J. M. Goethals and J. J. Seidel, “Cubature Formulae, Polytopes, and Spherical Designs”, The Geometric Vein 203 (1981) DOI
[15]
D. Hughes and S. Waldron, “Spherical (t,t)-designs with a small number of vectors”, Linear Algebra and its Applications 608, 84 (2021) DOI
[16]
C. Bachoc and F. Vallentin, “Optimality and uniqueness of the (4,10,1/6) spherical code”, Journal of Combinatorial Theory, Series A 116, 195 (2009) arXiv:0708.3947 DOI
[17]
V. M. Sidelnikov, “Spherical 7-Designs in 2 n -Dimensional Euclidean Space”, Journal of Algebraic Combinatorics 10, 279 (1999) DOI
[18]
V. M. Sidelnikov, “Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial”, Algebra i Analiz, 11:4 (1999), 183–203; St. Petersburg Mathematical Journal, 11:4 (2000), 673–686
[19]
P. de la Harpe and C. Pache, “Spherical designs and finite group representations (some results of E. Bannai)”, European Journal of Combinatorics 25, 213 (2004) DOI
[20]
E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[21]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[22]
S. Borodachov, “Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials”, Aequationes mathematicae 98, 509 (2024) DOI