Group code

In Error correction and detection, Group codes are $n$ length 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.

Construction

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, consider the generator matrix below... $G={\begin{pmatrix}{\begin{pmatrix}00\\11\end{pmatrix}}{\begin{pmatrix}01\\01\end{pmatrix}}{\begin{pmatrix}11\\01\end{pmatrix}}\\{\begin{pmatrix}00\\11\end{pmatrix}}{\begin{pmatrix}11\\11\end{pmatrix}}{\begin{pmatrix}00\\00\end{pmatrix}}\end{pmatrix}}$ The elements of this matrix are Failed to parse (unknown function "\into"): {\displaystyle 2 \into 2} matrices which are endomorphisms. In this scenario, each codeword can be represented as Failed to parse (unknown function "\hdots"): {\displaystyle c = (c_1,\hdots,c_n) = g_1^m_1 g_2^m2 \hdots g_r^ m_r} where Failed to parse (unknown function "\hdots"): {\displaystyle g_1,\hdots g_r} are the generators of $G$.