Binary group-orbit code

Binary group-orbit code[1,2]

Description

Binary length-\(n\) code whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), where \(G\) is a subgroup of the Hamming-space isometry group generated by coordinate permutations and bitwise translations.

Cousin

Slepian group-orbit code— Binary group-orbit codes can be mapped into Slepian group-orbit codes via various mappings [3; Ch. 8].

Member of code lists

Primary Hierarchy

Parents

Binary group-orbit codes are group-orbit codes in Hamming space.

Binary group-orbit code

Children

Linear binary codeGray

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].

Two-in-five code

The two-in-five code is a binary group-orbit code with group \(S_5\).

One-hot code

The one-hot code is a binary group-orbit code with the cyclic permutation group \(\mathbb{Z}_n\).

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.

Page edit log

Victor V. Albert (2022-11-18) — most recent

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.