## Description

Spherical \((4,48,2-\sqrt{2})\) code [1; Ex. 1.2.6] whose codewords are the vertices of the disphenoidal 288-cell. Codewords are the union of two 24-point lattice shells of the \(D_4\) lattice. The first shell consists of the 24 permutations of the four vectors \((0,0,\pm 1,\pm 1)\), and the second of the 16 vectors \((\pm 1,\pm 1,\pm 1,\pm 1)\) and the 8 permutations of the vectors \((0,0,0,\pm 2)\). A realization in terms of quaternions yields the 48 elements of the binary octahedral group \(2O\) [2; Sec. 8.6].

## Parents

- \(D_4\) lattice-shell code — Disphenoidal 288-cell codewords are the union of two 24-point shells of the \(D_4\) lattice, with each shell making up the vertices of a 24-cell.
- Spherical design — The disphenoidal 288-cell code forms a spherical 7-design [3].

## Child

- 24-cell code — Vertices of a disphenoidal 288-cell can be split up into vertices of a 24-cell and its dual 24-cell [2; Sec. 8.6].

## Cousins

- Real-Clifford subgroup-orbit code — The disphenoidal 288-cell code is a group-orbit code with the group being the real Clifford group in \(4\) dimensions.
- 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]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [2]
- L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
- [3]
- N. J. A. Sloane, R. H. Hardin, and P. Cara, “Spherical designs in four dimensions”, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674) DOI

## Page edit log

- Victor V. Albert (2022-12-19) — most recent

## Cite as:

“Disphenoidal 288-cell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/disphenoidal288cell