Biorthogonal spherical code 

Description

Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization.

For \(n=3\), biorthogonal spherical codewords make up the vertices of an octahedron. For \(n=4\), codewords make up the vertices of a 16-cell, or, equivalently, the Möbius-Kantor complex polygon. A quaternion realization of the vertices yields the quaternion group \(Q\). In general, the code makes up the vertices of an \(n\)-orthoplex.

The set of permutations of \((0,0,\cdots,0,1)\) forms an orthogonal set and yields the biorthogonal code when combined with the set of permutations of \((0,0,\cdots,0,-1)\).

Protection

Biorthogonal spherical codes saturate the absolute bound for antipodal codes and the third Rankin bound [1]. Biorthogonal codes are unique up to equivalence [1; pg. 19], which follows from saturating the Boroczky bound [2].

Parents

Child

Cousins

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]
G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels”, IEEE Transactions on Information Theory 44, 2384 (1998) DOI
[4]
Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
[5]
H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
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Zoo Code ID: biorthogonal

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

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/spherical/polytope/infinite/biorthogonal.yml.