Also known as Quasi-\(G\) code.
Description
A \(q\)-ary linear code based on a finite group \( G \) of order \(n/\ell\) for some index \(\ell\). The code is a right submodule of the direct sum of \(\ell\) copies of the group algebra \(\mathbb{F}_q G\). A quasi group-algebra code for an Abelian group is called an Abelian quasi group-algebra code.
Parent
- Linear \(q\)-ary code — A linear code is a quasi group-algebra code for a group \(G\) and index \(\ell\) if and only if \(G\) is isomorphic to a regular subgroup of the code's permutation automorphism group which acts freely of index \(\ell\) on the coordinates [1; Thm. 3.5].
Child
- Group-algebra code — A quasi group-algebra code of index \(\ell=1\) is a group-algebra code.
Cousin
- Quasi-cyclic code — A quasi group-algebra code for \(G\) being the cyclic group is a quasi-cyclic \(q\)-ary linear code.
References
- [1]
- M. Borello and W. Willems, “On the algebraic structure of quasi group codes”, (2021) arXiv:1912.09167
Page edit log
- Victor V. Albert (2023-11-01) — most recent
Cite as:
“Quasi group-algebra code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quasi_group