# 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.

## Notes

## Parents

- Spherical code
- \(t\)-design — Spherical designs are designs on real or complex spheres.

## Children

- Hexacode — The hexacode is a complex spherical 3-design [9].
- Real-Clifford subgroup-orbit code — The orbit of any point under the real Clifford subgroup is a spherical 7-design [10], and some are 11-designs [11].
- 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 [12].
- 600-cell code — The 600-cell code forms a spherical 11-design that is unique up to equivalence [13].
- Disphenoidal 288-cell code — The disphenoidal 288-cell code forms a spherical 7-design [14].
- \(3_{21}\) polytope code — The \(3_{21}\) polytope code forms a 5-design [15][5; 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 [16].
- Icosahedron code — The icosahedron code forms a unique tight spherical 5-design [1][4; Ex. 9.6.1].
- Biorthogonal spherical code — Biorthogonal spherical codes are the only tight spherical 3-designs [4; Tab. 9.3].
- Simplex spherical code — Simplex spherical codes are the only tight spherical 2-designs [4; Tab. 9.3].
- Polygon code — A \(q\)-gon a tight spherical \(q-1\) design.
- Witting polytope code — The Witting polytope code forms a tight spherical 7-design [15][5; Ch. 14].
- Kerdock spherical code — Kerdock codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [17].
- 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 two-design [18].

## Cousins

- Slepian group-orbit code — Slepian group-orbit codes can form spherical designs [16,19]. Polynomial invariants of a discrete subgroup \(G\) of the orthogonal group can be used to determine the design strength of orbits of \(G\) [20]. 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 [21].
- 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 [22].
- 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 [23,24]. 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 [15,25–27][28; Table 1].

## References

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- H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202

## 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