Biorthogonal spherical code 

Also known as Cross polytope code, Hyperoctahedron code, Orthoplex code, Co-cube code.


Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization. The code makes up the vertices of an \(n\)-orthoplex (a.k.a. hyperoctahedron or cross polytope).

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 \(Q\).

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 \((0,0,\cdots,0,-1)\).


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].





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.
G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels”, IEEE Transactions on Information Theory 44, 2384 (1998) DOI
Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
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Zoo Code ID: biorthogonal_spherical

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