Simplex spherical code 


Spherical \((n,n+1,2+2/n)\) code whose codewords are all permutations of the \(n+1\)-dimensional vector \((1,1,\cdots,1,-n)\), up to normalization, forming an \(n\)-simplex. Codewords are all equidistant and their components add up to zero. For example, the spherical simplex code in \(n=3\) makes up the vertices of a tetrahedron. In general, the code makes up the vertices of an \(n\)-simplex. See [1; Sec. 7.7] for a parameterization.


Simplex spherical codes saturate the absolute bound, the Levenshtein bound and, for \(2 < \rho \leq 4\), the first two Rankin bounds [1]. All simplex codes are unique up to equivalence [1; pg. 18], which follows from saturating the Boroczky bound [2].




T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
K. Boroczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hung. 32 (1978), 243–261.
H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
C. Bachoc and F. Vallentin, “Optimality and uniqueness of the (4,10,1/6) spherical code”, Journal of Combinatorial Theory, Series A 116, 195 (2009) arXiv:0708.3947 DOI
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
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Zoo Code ID: simplex_spherical

Cite as:
“Simplex spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_simplex_spherical, title={Simplex spherical code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Simplex spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.