Encyclopedia

Group code

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated May 10, 2025

In coding theory, group codes are a type of code. Group codes consist of $n$ linear block codes which are subgroups of $G^{n}$ , where $G$ is a finite Abelian group.

A systematic group code $C$ is a code over $G^{n}$ of order $\left|G\right|^{k}$ defined by $n-k$ homomorphisms which determine the parity check bits. The remaining $k$ bits are the information bits themselves.

01Construction

Group codes can be constructed by special generator matrices which resemble generator matrices of linear block codes except that the elements of those matrices are endomorphisms of the group instead of symbols from the code's alphabet. For example, considering the generator matrix

$G={\begin{pmatrix}{\begin{pmatrix}00\\11\end{pmatrix}}{\begin{pmatrix}01\\01\end{pmatrix}}{\begin{pmatrix}11\\01\end{pmatrix}}\\{\begin{pmatrix}00\\11\end{pmatrix}}{\begin{pmatrix}11\\11\end{pmatrix}}{\begin{pmatrix}00\\00\end{pmatrix}}\end{pmatrix}}$

the elements of this matrix are $2\times 2$ matrices which are endomorphisms. In this scenario, each codeword can be represented as $g_{1}^{m_{1}}g_{2}^{m_{2}}...g_{r}^{m_{r}}$ where $g_{1},...g_{r}$ are the generators of $G$ .

Watch videos about Group codeExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Group code”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Group code

01Construction

Sources and credits

Related topics

Group coded recording