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Snub-cube code

Alternative Names: Snub cuboctahedron code.

Description

Spherical \((3,24,\frac{2(t^2+1)}{3t^2+2t+2})\) code whose codewords are the vertices of a snub cube, normalized to lie on the unit sphere. Here, \(t \approx 1.839\) is the tribonacci constant, the real root of \(t^3-t^2-t-1=0\), and the minimum distance squared is approximately \(0.55384\).

Protection

Optimal configuration of 24 points on \(S^2\) [1; pg. 78]; see also the recent discussion in [2; Sec. 1.3, Conj. 1.4], which cites Robinson’s geometric proof of optimality and asks for a semidefinite-programming proof.

Notes

See the corresponding Bendwavy database entry [3].

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Constant-energy spherical codeECC
Spherical code
Slepian group-orbit codeECC
Polytope code
Polyhedron code
Parents
Polyhedron code
Snub-cube code

References

[1]
T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
[2]
H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
[3]
R. Klitzing, “Snic”, Polytopes & their Incidence Matrices URL
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Zoo Code ID: snub_cube

Cite as:
“Snub-cube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/snub_cube, arXiv:2606.11484
BibTeX:
@incollection{eczoo_snub_cube,
title={Snub-cube code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/snub_cube}
}
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Permanent link:
https://errorcorrectionzoo.org/c/snub_cube

Cite as:

“Snub-cube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/snub_cube, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/3d/snub_cube.yml.