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.

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.

Optimal spherical designs 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].
\([23, 12, 7]\) Golay code— The dual of the Golay code forms a spherical 3-design under the antipodal mapping [1; Exam. 9.3].
Hexacode— The hexacode is a complex spherical 3-design when embedded into the complex sphere via the polyphase mapping [16].
\(D_4\) lattice-shell code— \(D_4\) \(2m\)-shell codes can form spherical designs [17].
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 [18,19]. 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— Smallest-shell \((24,196560,1)\) code is a tight and unique spherical 11-design [7; Ch. 3]. The \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes are spherical 7- and 5-designs, respectively [20–23][24; Table 1].

Member of code lists

Primary Hierarchy

Classical Domain

Spherical Kingdom

Constant-energy codeECC

Spherical code

Parents

Spherical code

\(t\)-designECC

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 [25], and some are 11-designs [26].

Polygon code

A \(q\)-gon is a tight spherical \(q-1\) design.

Dodecahedron code

The dodecahedron code forms a spherical 5-design [27].

Icosahedron code

The icosahedron code forms a unique tight spherical 5-design [1][6; Exam. 9.6.1].

Pentakis dodecahedron code

Vertices of the pentakis dodecahedron form a weighted spherical 9-design [28,29][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.

24-cell code

The 24-cell code is a spherical 5-design [31].

600-cell code

The 600-cell code forms a spherical 11-design that is unique up to equivalence [32].

Disphenoidal 288-cell code

The disphenoidal 288-cell code forms a spherical 7-design [33].

\(3_{21}\) polytope code

The \(3_{21}\) polytope code forms a 5-design [20][7; Ch. 14].

Hessian polyhedron code

The Hessian polytope code forms a tight spherical 4-design [2; Exam. 7.3].

Rectified Hessian polyhedron code

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

Witting polytope code

The Witting polytope code forms a tight spherical 7-design [20][7; Ch. 14].

Biorthogonal spherical code

Biorthogonal spherical codes are the only tight spherical 3-designs [6; Tab. 9.3]. The weighted union of the vertices of a hypercube and a cross polytope form a weighted spherical 5-design in dimensions \(\geq 3\) [30; Exam. 2.6].

Hypercube code

Hypercube codes form spherical 3-designs. The weighted union of the vertices of a hypercube and a cross polytope form a weighted spherical 5-design in dimensions \(\geq 3\) [30; Exam. 2.6].

Simplex spherical code

Simplex spherical codes are the only tight spherical 2-designs [6; Tab. 9.3].

Kerdock spherical code

Kerdock codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [34].

Cameron-Goethals-Seidel (CGS) isotropic subspace code

Spherical sharp configuration

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

Petersen spherical code

The Peterson spherical code forms a spherical 2-design [35].

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
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[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
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[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]: V. V. Albert, private communication, 2024.
[17]: M. Hirao, H. Nozaki, and K. Tasaka, “Spherical designs and modular forms of the \(D_4\) lattice”, (2023) arXiv:2303.09000
[18]: B. B. Venkov. Even unimodular extremal lattices, algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165:43–48, 1984.
[19]: 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.
[20]: E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[21]: R. A. Wilson, “Vector stabilizers and subgroups of Leech lattice groups”, Journal of Algebra 127, 387 (1989) DOI
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[24]: H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202
[25]: V. M. Sidelnikov, Journal of Algebraic Combinatorics 10, 279 (1999) DOI
[26]: 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 Math. J., 11:4 (2000), 673–686
[27]: S. P. Jain, J. T. Iosue, A. Barg, and V. V. Albert, “Quantum spherical codes”, Nature Physics (2024) arXiv:2302.11593 DOI
[28]: J. M. Goethals and J. J. Seidel, “Cubature Formulae, Polytopes, and Spherical Designs”, The Geometric Vein 203 (1981) DOI
[29]: D. Hughes and S. Waldron, “Spherical (t,t)-designs with a small number of vectors”, Linear Algebra and its Applications 608, 84 (2021) DOI
[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]: 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
[32]: P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
[33]: 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
[34]: H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
[35]: 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.