Group-orbit code


Code whose set of codewords forms an orbit of some reference codeword under a subgroup of the automorphism group, i.e., the group of distance-preserving transformations on the metric space defined with the code's alphabet.


  • Error-correcting code (ECC) — Not all codes are group-orbit codes, and more generally one can classify codewords into orbits of the automorphism group [1].


  • Linear code over \(G\) — Since codewords of a linear code over \(G\) form a group, any codeword \(c\) can be obtained from any other codeword via action of a codeword. This means that the set of codewords can be thought of as an orbit of a particular codeword under the group. For example, see [2; Thm. 8.4.2] for the binary case. However, group orbit codes do not have to be linear; see [2; Remark 8.4.3].
  • Slepian group-orbit code — Slepian group-orbit codes are group-orbit codes on spheres. Binary group-orbit codes can be mapped into Slepian group-orbit codes via various mappings [2; Ch. 8].


J. H. Conway and N. J. A. Sloane, “Orbit and coset analysis of the Golay and related codes”, IEEE Transactions on Information Theory 36, 1038 (1990) DOI
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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Zoo Code ID: group_orbit

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

“Group-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.