Spherical design code[1] 


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].


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.


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




  • 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].


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H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
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Zoo Code ID: 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
@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.