# 24-cell code[1]

## Description

Spherical \((4,24,1)\) code whose codewords are the vertices of the 24-cell. Codewords form the minimal lattice-shell code of the \(D_4\) lattice.

A realization of the codewords consists of the 24 permutations of the four vectors \((0,0,\pm 1,\pm 1)\); see [2; Table 3] for another realization. A realization in terms of quaternion coordinates yields the 24 elements of the binary tetrahedral group \(2T\) [3].

## Protection

## Notes

See post by G. Egan for more details.

## Parents

- Dual polytope code — The 24-cell is self-dual.
- Real-Clifford subgroup-orbit code — The 24-cell code is equivalent to the real Clifford subgroup-orbit code for \(n=4\).
- Spherical design — The 24-cell code is a spherical 5-design [6].

## Cousins

- 600-cell code — Vertices of a 600-cell can be split up into vertices of five 24-cells [7–9].
- 120-cell code — Vertices of a 120-cell can be split up into vertices of five 600-cells [8,10], and vertices of a 600-cell can be split up into vertices of five 24-cells [7–9]. Therefore, vertices of a 120-cell can be split up into vertices of 25 24-cells.
- Disphenoidal 288-cell code — Vertices of a disphenoidal 288-cell can be split up into vertices of a 24-cell and its dual 24-cell [3; Sec. 8.6].
- Universally optimal spherical code — The 24-cell code is not universally optimal [6], but comes quite close [11; Ex. 12.4.29].
- Kerdock code — The 24-cell is a special case of a family of codes for real projective planes, constructed using Kerdock codes [12] (cf. [13]).
- Sharp configuration — The 12 sets of antipodal pairs of the 24-cell code form a sharp configuration in the projective space \(\mathbb{R}P^3\) [14].
- Biorthogonal spherical code — Vertices of a 24-cell can be split up into vertices of three 16-cells, which are biorthogonal spherical codes for \(n=4\) [8].
- Clifford subgroup-orbit QSC — Logical constellations of the Clifford subgroup-orbit code for \(r=1\) form vertices of 24-cells when mapped into the real sphere, while code constellations form vertices of a disphenoidal 288-cell.

## References

- [1]
- L. Kollros, “An Attempt to determine the twenty-seven Lines upon a Surface of the third Order, and to divide such Surfaces into Species in Reference to the Reality of the Lines upon the Surface”, Gesammelte Mathematische Abhandlungen 198 (1953) DOI
- [2]
- S. Mamone, G. Pileio, and M. H. Levitt, “Orientational Sampling Schemes Based on Four Dimensional Polytopes”, Symmetry 2, 1423 (2010) DOI
- [3]
- L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
- [4]
- Schläfli, L. (1901). Theorie der vielfachen Kontinuität (Vol. 38). Zürcher & Furrer.
- [5]
- O. R. Musin, “The kissing number in four dimensions”, (2006) arXiv:math/0309430
- [6]
- H. Cohn et al., “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
- [7]
- Schoute, P. H. (1903). Mehrdimensionale Geometrie, Vol. 2 (Die Polytope).
- [8]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [9]
- M. Waegell and P. K. Aravind, “Critical noncolorings of the 600-cell proving the Bell–Kochen–Specker theorem”, Journal of Physics A: Mathematical and Theoretical 43, 105304 (2010) arXiv:0911.2289 DOI
- [10]
- M. Waegell and P. K. Aravind, “Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell”, Foundations of Physics 44, 1085 (2014) arXiv:1309.7530 DOI
- [11]
- P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [12]
- Levenshtein, V. I. (1982). Bounds on the maximal cardinality of a code with bounded modulus of the inner product. In Soviet Math. Dokl (Vol. 25, No. 2, pp. 526-531).
- [13]
- H. Cohn, A. Kumar, and G. Minton, “Optimal simplices and codes in projective spaces”, Geometry & Topology 20, 1289 (2016) arXiv:1308.3188 DOI
- [14]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI

## Page edit log

- Victor V. Albert (2022-11-23) — most recent

## Cite as:

“24-cell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/24cell