Binary group-orbit code[1,2] 

Description

Bianry legnth-\(n\) whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), which is a subgroup of the group of bit-string permutations and translations, i.e., the automorphism group of binary codes under the Hamming distance.

Parents

Child

  • Linear binary code — The set of codewords of a binary linear code can be thought of as an orbit of a particular codeword under the translation group formed by the code [3; Thm. 8.4.2]. However, binary group-orbit codes do not have to be linear; see [3; Remark 8.4.3].

Cousin

References

[1]
D. Slepian, “A Class of Binary Signaling Alphabets”, Bell System Technical Journal 35, 203 (1956) DOI
[2]
D. Slepian, “Some Further Theory of Group Codes”, Bell System Technical Journal 39, 1219 (1960) DOI
[3]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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Zoo Code ID: binary_group_orbit

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

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/binary_group_orbit.yml.