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.Member of code lists
Primary Hierarchy
Parents
Not all codes are group-orbit codes, and more generally one can classify codewords into orbits of the automorphism group [1].
Group-orbit code
Children
Binary group-orbit codes are group-orbit codes in Hamming space.
Linear code over \(G\)Lattice-based Linear \(q\)-ary Evaluation MDS GRS GRM QR Projective geometry Tanner \(q\)-ary LDPC Divisible Self-dual additive Self-dual linear Linear code over \(\mathbb{Z}_q\) Gray
The set of codewords of a linear code over \(G\) can be thought of as an orbit of a particular codeword under the group formed by the code. However, group orbit codes do not have to be linear [2; Remark 8.4.3].
All codewords of a cyclic code can be obtained from any codeword via cyclic shifts, meaning that the code consists of only one orbit.
A \(q\)-ary group-orbit code hosts a transitive group action. If the action is also free, then the code is a group-algebra code.
Slepian group-orbit codes are group-orbit codes on spheres.
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.
Page edit log
- Victor V. Albert (2022-07-18) — most recent
Cite as:
“Group-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group_orbit