Spherical design

Spherical design[1]

Description

Spherical code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the real sphere. A spherical code is a spherical design of strength \(t\), i.e., a \(t\)-design, if the average of any polynomial of degree up to \(t\) over its codewords is equal to the average over the entire sphere. A weighed spherical design is a generalization in which the average over codewords is non-uniform.

Spherical designs can also be defined for complex spheres, and there are ways to convert between the two [2; Lemma 3.6].

Protection

The number of points \(|X|\) of an \(n\)-dimensional spherical design \(X\) is bounded by [1] \begin{align} |X|\geq\begin{cases} {n+s-1 \choose n-1}+{n+s-2 \choose n-1} & t=2s\\ 2{n+s-1 \choose n-1} & t=2s+1 \end{cases}~, \tag*{(1)}\end{align} and designs saturating the above inequality are called tight.

Spherical designs with asymptotically optimal cardinality exist for all \(n\) [3], proving the Korevaar-Meyers conjecture [4].

Notes

See Refs. [5,6,8–10][7; pg. 89] for reviews and examples on spherical designs.

Cousins

Slepian group-orbit code— Slepian group-orbit codes can form spherical designs for real [11,12] or complex spheres [13]. Polynomial invariants of a discrete subgroup \(G\) of the orthogonal group can be used to determine the real design strength of orbits of \(G\) [14]. Let \(t+1\) be the degree of the lowest-degree \(G\)-invariant polynomial that is not a polynomial in the norm \(\left\Vert x\right\Vert^2\). Then, any orbit under \(G\) forms a Slepian group-orbit code that is also a spherical \(t\)-design.
Unimodular lattice— A union of \(t\) shells of self-dual lattices and their shadows form spherical \(t\)-designs [15].
Combinatorial design— Spherical designs can be thought of as Euclidean analogues of combinatorial designs [16].
\([23, 12, 7]\) Golay code— The dual of the Golay code forms a spherical 3-design under the antipodal mapping [1; Exam. 9.3].
\([6,3,4]_4\) Hexacode— The hexacode is a complex spherical 3-design when embedded into the complex sphere via the polyphase mapping [17].
\(D_4\) lattice-shell code— \(D_4\) \(2m\)-shell codes can form spherical designs [18].
Lattice-shell code— Nonempty \(2m\)-shell codes of extremal even unimodular lattices in \(n\) dimensions form spherical \(t\)-designs with \(t=11\) (\(t=7\), \(t=3\)) if \(n \equiv 0\) (\(n \equiv 8\), \(n\equiv 16\)) modulo 24 [19,20]. Shells of \(A_n\) and \(D_n\) lattices form infinite families of spherical 3-designs [5; Exam. 2.9].
\(\Lambda_{24}\) Leech lattice-shell code— The smallest-shell \((24,196560,1)\) code is a tight and unique spherical 11-design [21][7; Ch. 3]. The \((23,4600,1/3)\) [21–24][25; Table 1] and \((22,891,1/4)\) [22–24][25; Table 1] spherical codes are also sharp configurations.
Polygon code— A \(q\)-gon is a tight spherical \(q-1\) design.
Antiprism code— For the case when the two \(q\)-gons are such that the \(q=2,3\) cases reduce to the tetrahedron and octahedron, respectively, the antiprism is a spherical 3-design for \(q \geq 3\), and a \(2\)-design for \(q=2\) [26]. This can be seen as a consequence of [27; Lemma 6.11].
Icosahedron code— The icosahedron code forms a unique tight spherical 5-design [1][6; Exam. 9.6.1].
24-cell code— The 24-cell code is a spherical 5-design [28].
Hessian polyhedron code— The Hessian polytope code forms a tight spherical 4-design [2; Exam. 7.3]. The double Hessian polytope code forms a spherical 5-design [20].
\(3_{21}\) polytope code— The \(3_{21}\) polytope code forms a tight spherical 5-design [1,21][7; Ch. 14][24; Table 1].
Witting polytope code— The Witting polytope code forms a tight spherical 7-design [21][7; Ch. 14].
Biorthogonal spherical code— Biorthogonal spherical codes are the only tight spherical 3-designs [6; Tab. 9.3]. A suitable weighted union of the vertices of a hypercube and an orthoplex forms a weighted spherical 5-design in dimensions \(\geq 3\) [29; Sec. 8.6, Ex. 5-2][30; Exam. 2.6].
Simplex spherical code— Simplex spherical codes are the only tight spherical 2-designs [6; Tab. 9.3]. The bi-simplex is a spherical 3-design since antipodal codes have zero averages over odd-degree polynomials.
McLaughlin spherical code— Both McLaughlin spherical codes are sharp configurations [24,31]. 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. [24; Appx. A].

Member of code lists

Primary Hierarchy

Classical Domain

Spherical Kingdom

Constant-energy spherical codeECC

Spherical code

Parents

Spherical code

\(t\)-design

Spherical designs are designs on real or complex spheres.

Spherical design

Children

Real-Clifford subgroup-orbit code

The orbit of any point under the real Clifford subgroup is a spherical 7-design [32], and some are 11-designs [33].

Dodecahedron code

The dodecahedron code forms a spherical 5-design [34].

Pentakis dodecahedron code

Vertices of the pentakis dodecahedron form a weighted spherical 9-design [35,36][30; Exam. 2.5].

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.

600-cell code

The 600-cell code forms a spherical 11-design that is unique up to equivalence [37].

Disphenoidal 288-cell code

The disphenoidal 288-cell code forms a spherical 7-design [38].

Rectified Hessian polyhedron code

The rectified Hessian polyhedron code forms a spherical 5-design [12].

\(2_{31}\) polytope code

The 126 vertices of the \(2_{31}\) polytope form a spherical 5-design [20].

\(2_{41}\) polytope code

The \(2_{41}\) polytope code forms a spherical 7-design [27].

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 \(\geq 3\) [29; Sec. 8.6, Ex. 5-2][30; Exam. 2.6].

Kerdock spherical code

Kerdock spherical codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) orthoplexes [39].

Spherical sharp configuration

Spherical sharp configurations are spherical designs of strength \(2m-1\) for some \(m\).

Petersen spherical code

The Petersen spherical code forms a spherical 2-design [40].

References

[1]: P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs”, Geometriae Dedicata 6, 363 (1977) DOI
[2]: A. Roy and S. Suda, “Complex spherical designs and codes”, (2011) arXiv:1104.4692
[3]: A. Bondarenko, D. Radchenko, and M. Viazovska, “Optimal asymptotic bounds for spherical designs”, (2011) arXiv:1009.4407
[4]: J. Korevaar and J. L. H. Meyers, “Spherical faraday cage for the case of equal point charges and chebyshev-type quadrature on the sphere”, Integral Transforms and Special Functions 1, 105 (1993) DOI
[5]: E. Bannai and E. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres”, European Journal of Combinatorics 30, 1392 (2009) DOI
[6]: T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
[7]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[8]: Z. Xiang, “Explicit spherical designs”, Algebraic Combinatorics 5, 347 (2022) DOI
[9]: C. J. Colbourn and J. H. Dinitz, editors , Handbook of Combinatorial Designs (Chapman and Hall/CRC, 2006) DOI
[10]: J. D. McEwen and Y. Wiaux, “A Novel Sampling Theorem on the Sphere”, IEEE Transactions on Signal Processing 59, 5876 (2011) arXiv:1110.6298 DOI
[11]: E. BANNAI, “Spherical t-designs which are orbits of finite groups”, Journal of the Mathematical Society of Japan 36, (1984) DOI
[12]: 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
[13]: M. Mohammadpour and S. Waldron, “Complex spherical designs from group orbits”, (2024) arXiv:2308.02499
[14]: S. L. Sobolev, “Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations”, Selected Works of S.L. Sobolev 461 DOI
[15]: C. Pache, “Shells of selfdual lattices viewed as spherical designs”, (2005) arXiv:math/0502313
[16]: E. Bannai, E. Bannai, and Y. Zhu, “A survey on tight Euclidean t-designs and tight relative t-designs in certain association schemes”, Proceedings of the Steklov Institute of Mathematics 288, 189 (2015) DOI
[17]: V. V. Albert, private communication, 2024
[18]: M. Hirao, H. Nozaki, and K. Tasaka, “Spherical designs and modular forms of the \(D_4\) lattice”, (2023) arXiv:2303.09000
[19]: B. B. Venkov, “Even unimodular extremal lattices, algebraic geometry and its applications”, Trudy Matematicheskogo Instituta imeni V. A. Steklova 165, 43–48 (1984)
[20]: B. Venkov, “Réseaux et designs sphériques”, in Réseaux euclidiens, designs sphériques et formes modulaires, volume 37 of Monographies de l’Enseignement Mathématique, pages 10–86. Enseignement Mathématique, Geneva, 2001
[21]: E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[22]: R. A. Wilson, “Vector stabilizers and subgroups of Leech lattice groups”, Journal of Algebra 127, 387 (1989) DOI
[23]: H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
[24]: 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
[25]: H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202
[26]: V. V. Albert, private communication, 2025
[27]: S. Borodachov, “Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials”, Aequationes mathematicae 98, 509 (2024) DOI
[28]: 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
[29]: A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice Hall, 1971)
[30]: 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
[31]: P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Universal minima of discrete potentials for sharp spherical codes”, (2023) arXiv:2211.00092
[32]: V. M. Sidelnikov, “Spherical 7-Designs in 2 n -Dimensional Euclidean Space”, Journal of Algebraic Combinatorics 10, 279 (1999) DOI
[33]: V. M. Sidelnikov, “Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial”, Algebra i Analiz 11(4), 183–203 (1999); St. Petersburg Mathematical Journal 11(4), 673–686 (2000)
[34]: S. P. Jain, J. T. Iosue, A. Barg, and V. V. Albert, “Quantum spherical codes”, Nature Physics 20, 1300 (2024) arXiv:2302.11593 DOI
[35]: J. M. Goethals and J. J. Seidel, “Cubature Formulae, Polytopes, and Spherical Designs”, The Geometric Vein 203 (1981) DOI
[36]: D. Hughes and S. Waldron, “Spherical (t,t)-designs with a small number of vectors”, Linear Algebra and its Applications 608, 84 (2021) DOI
[37]: P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
[38]: 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
[39]: H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
[40]: 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

Page edit log

Victor V. Albert (2023-05-12) — most recent

Cite as:

“Spherical design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/spherical_design

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/spherical_design.yml.