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. A weighed spherical design is a generalization in which the average over codewords is non-uniform.

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].
Combinatorial design— Spherical designs can be thought of as Euclidean analogues of combinatorial designs [16].
\([23, 12, 7]\) Golay code— The dual of the Golay code forms a spherical 3-design under the antipodal mapping [1; Exam. 9.3].
\([6,3,4]_4\) Hexacode— The hexacode is a complex spherical 3-design when embedded into the complex sphere via the polyphase mapping [17].
\(D_4\) lattice-shell code— \(D_4\) \(2m\)-shell codes can form spherical designs [18].
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 [19,20]. 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 [21–24][25; Table 1].
Antiprism code— For the case when the two \(q\)-gons are such that the \(q=2,3\) cases reduce to the tetrahedron and octahedron, respectively, the antiprism is a spherical 3-design for \(q \geq 3\), and a \(2\)-design for \(q=2\) [26]. This can be seen as a consequence of [27; Lemma 6.11].
\(3_{21}\) polytope code— The \(3_{21}\) polytope code forms a tight spherical 5-design [1,21][7; Ch. 14] that is associated with.
McLaughlin spherical code— Both McLaughlin spherical codes are sharp configurations [24,28]. The \((22,275,1/6)\) code is a unique and tight spherical 4-design, while the \((23,552,1/5)\) code is a unique and tight spherical 5-design; see Ref. [24; Appx. A].

Member of code lists

Primary Hierarchy

Classical Domain

Spherical Kingdom

Constant-energy spherical codeECC

Spherical code

Parents

Spherical code

\(t\)-design

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 [29], and some are 11-designs [30].

Polygon code

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

Dodecahedron code

The dodecahedron code forms a spherical 5-design [31].

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 [32,33][34; 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 [35].

600-cell code

The 600-cell code forms a spherical 11-design that is unique up to equivalence [36].

Disphenoidal 288-cell code

The disphenoidal 288-cell code forms a spherical 7-design [37].

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].

\(2_{41}\) real polytope code

The \(2_{41}\) real polytope code forms a spherical 7-design [27].

Witting polytope code

The Witting polytope code forms a tight spherical 7-design [21][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 an orthoplex form a weighted spherical 5-design in dimensions \(\geq 3\) [38; Sec. 8.6, Ex. 5-2][34; Exam. 2.6][34; Exam. 2.6].

Hypercube code

Hypercube codes form spherical 3-designs. The weighted union of the vertices of a hypercube and an orthoplex form a weighted spherical 5-design in dimensions \(\geq 3\) [38; Sec. 8.6, Ex. 5-2][34; Exam. 2.6].

Simplex spherical code

Simplex spherical codes are the only tight spherical 2-designs [6; Tab. 9.3]. The bi-simplex is a spherical 3-design since antipodal codes have zero averages over odd-degree polynomials.

Kerdock spherical code

Kerdock spherical codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) orthoplexes [39].

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 [40].

References

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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.