Spherical design code[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.

Notes

See Refs. [3,4,6,7][5; pg. 89] for reviews and examples on spherical designs.

Parent

Children

Cousins

  • Combinatorial design code
  • Slepian group-orbit code — Slepian group-orbit codes can form spherical designs [14,18]. Polynomial invariants of a discrete subgroup \(G\) of the orthogonal group can be used to determine the design strength of orbits of \(G\) [19]. 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 code — A union of \(t\) shells of self-dual lattices and their shadows form spherical \(t\)-designs [20].
  • Golay code — The dual of the Golay code forms a spherical three-design under the antipodal mapping [1; Exam. 9.3].
  • \(D_4\) lattice-shell code — \(D_4\) \(2m\)-shell codes can form spherical designs [21].
  • 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 [22,23]. Shells of \(A_n\) and \(D_n\) lattices form infinite families of spherical 3-designs [3; Exam. 2.9].
  • \(\Lambda_{24}\) Leech lattice-shell code — Smallest-shell \((24,196560,1)\) code is a tight and unique spherical 11-design [5; Ch. 3]. The \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes are spherical 7- and 5-designs, respectively [15,2426][27; Table 1].

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]
E. Bannai and E. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres”, European Journal of Combinatorics 30, 1392 (2009) DOI
[4]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[5]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[6]
Z. Xiang, “Explicit spherical designs”, Algebraic Combinatorics 5, 347 (2022) DOI
[7]
C. J. Colbourn and J. H. Dinitz, editors , Handbook of Combinatorial Designs (Chapman and Hall/CRC, 2006) DOI
[8]
V. V. Albert, private communication, 2024.
[9]
V. M. Sidelnikov, Journal of Algebraic Combinatorics 10, 279 (1999) DOI
[10]
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
[11]
H. Cohn et al., “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
[12]
P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
[13]
N. J. A. Sloane, R. H. Hardin, and P. Cara, “Spherical designs in four dimensions”, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674) DOI
[14]
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
[15]
E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[16]
H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
[17]
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
[18]
E. BANNAI, “Spherical t-designs which are orbits of finite groups”, Journal of the Mathematical Society of Japan 36, (1984) DOI
[19]
S. L. Sobolev, “Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations”, Selected Works of S.L. Sobolev 461 DOI
[20]
C. Pache, “Shells of selfdual lattices viewed as spherical designs”, (2005) arXiv:math/0502313
[21]
M. Hirao, H. Nozaki, and K. Tasaka, “Spherical designs and modular forms of the \(D_4\) lattice”, (2023) arXiv:2303.09000
[22]
B. B. Venkov. Even unimodular extremal lattices, algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165:43–48, 1984.
[23]
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.
[24]
R. A. Wilson, “Vector stabilizers and subgroups of Leech lattice groups”, Journal of Algebra 127, 387 (1989) DOI
[25]
H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
[26]
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
[27]
H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202
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Zoo Code ID: spherical_design

Cite as:
“Spherical design code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/spherical_design
BibTeX:
@incollection{eczoo_spherical_design, title={Spherical design code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/spherical_design} }
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“Spherical design code”, 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.