24-cell code 


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 [1; Table 3] for another realization. A realization in terms of quaternions yields the 24 elements of the binary tetrahedral group \(2T\) [2].

Figure I: Projection of the coordinates of the 24-cell.


Code yields an optimal solution to the kissing problem in 4D [3,4].


See post by G. Egan for more details.



  • Universally optimal spherical code — The 24-cell code is not universally optimal [8], but comes quite close [9; 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 [10] (cf. [11]).
  • Sharp configuration — The 12 sets of antipodal pairs of the 24-cell code form a sharp configuration in the projective space \(\mathbb{R}P^3\) [12].
  • Biorthogonal spherical code — Vertices of a 24-cell can be split up into vertices of three 16-cells, which are biorthogonal spherical codes for \(n=4\) [6].
  • 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.


S. Mamone, G. Pileio, and M. H. Levitt, “Orientational Sampling Schemes Based on Four Dimensional Polytopes”, Symmetry 2, 1423 (2010) DOI
L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
Schläfli, L. (1901). Theorie der vielfachen Kontinuität (Vol. 38). Zürcher & Furrer.
O. R. Musin, “The kissing number in four dimensions”, (2006) arXiv:math/0309430
Schoute, P. H. (1903). Mehrdimensionale Geometrie, Vol. 2 (Die Polytope).
H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
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
H. Cohn et al., “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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).
H. Cohn, A. Kumar, and G. Minton, “Optimal simplices and codes in projective spaces”, Geometry & Topology 20, 1289 (2016) arXiv:1308.3188 DOI
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
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Zoo Code ID: 24cell

Cite as:
“24-cell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/24cell
@incollection{eczoo_24cell, title={24-cell code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/24cell} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/24cell/24cell.yml.