Group-orbit code

Description

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.

Parent

  • 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].

Children

  • 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].

References

[1]
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
[2]
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. https://errorcorrectionzoo.org/c/group_orbit
BibTeX:
@incollection{eczoo_group_orbit, title={Group-orbit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_orbit} }
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https://errorcorrectionzoo.org/c/group_orbit

Cite as:

“Group-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group_orbit

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/properties/group_orbit.yml.