Biorthogonal spherical code


Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization. Biorthogonal spherical codes are the only tight spherical 3-designs [1; Tab. 9.3].

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


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.
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.
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.
@incollection{eczoo_biorthogonal, 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.