24-cell code[1]
Alternative names: Icositetrachoron code, Octaplex code, Hyperdiamond code.
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].
Cousins
- Group-alphabet code— The 24-cell code has a quaternion-coordinate realization as the 24 elements of the binary tetrahedral group \(2T\), one of the three exceptional finite subgroups of \(SU(2)\) [3].
- 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 [11], but comes quite close [12; Exam. 12.4.29].
- Sharp configuration— The 12 antipodal pairs of the 24-cell code form a sharp configuration and a 2-design in \(\mathbb{R}P^3\) [13].
- \(t\)-design— The 12 antipodal pairs of the 24-cell code form a sharp configuration and a 2-design in \(\mathbb{R}P^3\) [13].
- Real projective space code— The 12 antipodal pairs of the 24-cell code form a sharp configuration and a 2-design in \(\mathbb{R}P^3\) [13]. This is a special case of a family of real projective plane codes, constructed using Kerdock codes [14] (cf. [15]).
- Kerdock code— The 12 antipodal pairs of the 24-cell code form a sharp configuration and a 2-design in \(\mathbb{R}P^3\) [13]. This is a special case of a family of real projective plane codes, constructed using Kerdock codes [14] (cf. [15]).
- \(D_4\) hyper-diamond lattice— The Voronoi cell of the \(D_4\) lattice is a 24-cell [16; Ch. 21, pg. 464].
- 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]. The vertices of a 24-cell are a union of the vertices of a tesseract and a 16-cell [17; Exam. 2.6].
- Hypercube code— The vertices of a 24-cell are a union of the vertices of a tesseract and a 16-cell [17; Exam. 2.6].
- 2T-qutrit code— The \(2T\)-qutrit code is constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group \(2T\), a.k.a. the 24-cell.
- 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.
Member of code lists
Primary Hierarchy
Parents
The 24-cell is self-dual.
The 24-cell code is the minimal shell of the \(D_4\) lattice.
The 24-cell code is equivalent to the real Clifford subgroup-orbit code for \(n=4\).
The 24-cell code is a spherical 5-design [11].
24-cell code
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]
- L. Schläfli, Theorie der vielfachen Kontinuität, vol. 38 (Zürcher & Furrer, 1901)
- [5]
- O. R. Musin, “The kissing number in four dimensions”, (2006) arXiv:math/0309430
- [6]
- R. Klitzing. “Ico.” Polytopes & their Incidence Matrices. bendwavy.org/klitzing/incmats/ico.htm
- [7]
- P. H. Schoute (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]
- H. Cohn, J. H. Conway, N. D. Elkies, and A. Kumar, “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
- [12]
- P. Boyvalenkov, D. Danev, “Linear programming bounds.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [13]
- 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
- [14]
- V. I. Levenshtein, “Bounds on the maximal cardinality of a code with bounded modulus of the inner product.” Soviet Mathematics Doklady 25(2) (1982): 526-531
- [15]
- H. Cohn, A. Kumar, and G. Minton, “Optimal simplices and codes in projective spaces”, Geometry & Topology 20, 1289 (2016) arXiv:1308.3188 DOI
- [16]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [17]
- S. Borodachov, P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Energy bounds for weighted spherical codes and designs via linear programming”, (2024) arXiv:2403.07457
Page edit log
- Shubham P. Jain (2026-05-05) — most recent
- Victor V. Albert (2022-11-23)
Cite as:
“24-cell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/24cell