Description
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.
Protection
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].
Parents
- Polytope code — Simplex spherical codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a triangle (tetrahedron, 5-cell, \(n\)-simplex).
- Spherical sharp configuration
- Spherical design — Simplex spherical codes are the only tight spherical 2-designs [1; Tab. 9.3].
- Permutation spherical code
Cousins
- Icosahedron code — Vertices of a dodecahedron can be split up into vertices of five tetrahedra, which are simplex spherical codes for \(n=3\) [3].
- Binary antipodal code — Binary simplex codes map to \((2^m,2^m+1)\) simplex spherical codes under the antipodal mapping [4; Sec. 6.5.2][1; pg. 18]. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.
- \([2^m-1,m,2^{m-1}]\) simplex code — Binary simplex codes map to \((2^m,2^m+1)\) simplex spherical codes under the antipodal mapping [4; Sec. 6.5.2][1; pg. 18]. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.
- Polyphase code — Simplex spherical codes for dimension \(n=(p-1)/2\) with \(p\) an odd prime admit a polyphase realization [1; Sec. 7.7].
- Petersen spherical code — Codewords of the Petersen spherical code correspond to midpoints of the \(5\)-cell [5].
- \([[2^r-1,1,3]]\) simplex code — Each \([[2^r-1,1,3]]\) simplex code is a color code whose qubits are placed on the vertices, edges, and faces of an \((r-1)\)-simplex [6,7].
References
- [1]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [2]
- K. Boroczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hung. 32 (1978), 243–261.
- [3]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [4]
- Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
- [5]
- 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
- [6]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [7]
- 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
Page edit log
- Victor V. Albert (2022-11-15) — most recent
Cite as:
“Simplex spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/simplex_spherical