## Description

Encodes \(K\) states (codewords) in \(n\) coordinates over a group \(G\) such that the codewords form a subgroup of \(G^n\). In other words, the set of codewords is closed under the group operation.

The automorphism group of such codes is the group \(G^n\) formed by the action of \(G\) on each coordinate as well as the coordinate permutation group \(S_n\).

## Parents

- Group-alphabet code
- Group-orbit code — 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 [4; Remark 8.4.3].
- Block code

## Children

- Lattice-based code — Lattice-based codes are linear codes over \(G=\mathbb{R}^n\). Because any orthogonal matrix leaving the lattice invariant has a corresponding integer matrix (see lattice code description), integer representations of groups can be used to obtain lattices [5; Ch. 3, Sec. 4.2].
- Additive \(q\)-ary code — Additive \(q\)-ary codes are linear over \(G=GF(q)\) since Galois fields are Abelian groups under addition.
- \(R\)-linear code — \(R\)-linear codes are linear over \(G=R\) since rings are Abelian groups under addition.

## Cousins

- Group GKP code
- Slepian group-orbit code — Any finite-group code can be mapped to a Slepian group-orbit code by representing the group using orthogonal matrices [3].

## References

- [1]
- F. R. Kschischang, P. G. de Buda, and S. Pasupathy, “Block coset codes for M-ary phase shift keying”, IEEE Journal on Selected Areas in Communications 7, 900 (1989) DOI
- [2]
- G. D. Forney, “Geometrically uniform codes”, IEEE Transactions on Information Theory 37, 1241 (1991) DOI
- [3]
- H.-A. Loeliger, “Signal sets matched to groups”, IEEE Transactions on Information Theory 37, 1675 (1991) DOI
- [4]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [5]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI

## Page edit log

- Victor V. Albert (2022-11-07) — most recent
- Victor V. Albert (2022-03-24)

## Cite as:

“Linear code over \(G\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group_linear

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/groups/group_linear.yml.