# 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

## Parent

## Children

- Real-Clifford subgroup-orbit code — The orbit of any point under the real Clifford subgroup is a spherical 7-design [7], and some are 11-designs [8].
- 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 [9].
- 600-cell code — The 600-cell code forms a spherical 11-design that is unique up to equivalence [10].
- Disphenoidal 288-cell code — The disphenoidal 288-cell code forms a spherical 7-design [11].
- 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].
- 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 [5; Ch. 14].
- Cameron-Goethals-Seidel (CGS) isotropic subspace code
- Spherical sharp configuration — Spherical sharp configurations are spherical designs of strength \(2m-1\) for some \(m\).

## Cousins

- Combinatorial design code
- Slepian group-orbit code — Slepian group-orbit codes can form spherical designs [12,13]. Polynomial invariants of a discrete subgroup \(G\) of the orthogonal group can be used to determine the 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.
- \(D_4\) lattice-shell code — \(D_4\) \(2m\)-shell codes can form spherical designs [15].
- 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 [16,17].
- \(\Lambda_{24}\) Leech lattice-shell code — Smallest-shell \((24,196560,1)\) code is a tight and unique spherical 11-design [5; Ch. 3].

## 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]
- V. M. Sidelnikov, Journal of Algebraic Combinatorics 10, 279 (1999) DOI
- [8]
- 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
- [9]
- H. Cohn et al., “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
- [10]
- P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
- [11]
- 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
- [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]
- E. BANNAI, “Spherical t-designs which are orbits of finite groups”, Journal of the Mathematical Society of Japan 36, (1984) DOI
- [14]
- S. L. Sobolev, “Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations”, Selected Works of S.L. Sobolev 461 DOI
- [15]
- M. Hirao, H. Nozaki, and K. Tasaka, “Spherical designs and modular forms of the \(D_4\) lattice”, (2023) arXiv:2303.09000
- [16]
- B. B. Venkov. Even unimodular extremal lattices, algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165:43–48, 1984.
- [17]
- 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.

## Page edit log

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

## Cite as:

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