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.

Spherical designs with asymptotically optimal cardinality 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— The smallest-shell $(24,196560,1)$ code is a tight and unique spherical 11-design [21][7; Ch. 3]. The $(23,4600,1/3)$ [21–24][25; Table 1] and $(22,891,1/4)$ [22–24][25; Table 1] spherical codes are also sharp configurations.
Polygon code— A $q$-gon is a tight spherical $q-1$ design.
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].
Icosahedron code— The icosahedron code forms a unique tight spherical 5-design [1][6; Exam. 9.6.1].
24-cell code— The 24-cell code is a spherical 5-design [28].
Hessian polyhedron code— The Hessian polytope code forms a tight spherical 4-design [2; Exam. 7.3]. The double Hessian polytope code forms a spherical 5-design [20].
$3_{21}$ polytope code— The $3_{21}$ polytope code forms a tight spherical 5-design [1,21][7; Ch. 14][24; Table 1].
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]. A suitable weighted union of the vertices of a hypercube and an orthoplex forms a weighted spherical 5-design in dimensions $\geq 3$ [29; Sec. 8.6, Ex. 5-2][30; 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.
McLaughlin spherical code— Both McLaughlin spherical codes are sharp configurations [24,31]. 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 [32], and some are 11-designs [33].

Dodecahedron code

The dodecahedron code forms a spherical 5-design [34].

Pentakis dodecahedron code

Vertices of the pentakis dodecahedron form a weighted spherical 9-design [35,36][30; 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.

600-cell code

The 600-cell code forms a spherical 11-design that is unique up to equivalence [37].

Disphenoidal 288-cell code

The disphenoidal 288-cell code forms a spherical 7-design [38].

Rectified Hessian polyhedron code

The rectified Hessian polyhedron code forms a spherical 5-design [12].

$2_{31}$ polytope code

The 126 vertices of the $2_{31}$ polytope form a spherical 5-design [20].

$2_{41}$ polytope code

The $2_{41}$ polytope code forms a spherical 7-design [27].

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 $\geq 3$ [29; Sec. 8.6, Ex. 5-2][30; Exam. 2.6].

Kerdock spherical code

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

Spherical sharp configuration

Spherical sharp configurations are spherical designs of strength $2m-1$ for some $m$.

Petersen spherical code

The Petersen spherical code forms a spherical 2-design [40].

References

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Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2023-05-12)

Cite as:

“Spherical design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/spherical_design, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/spherical_design.yml.