Group code

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In computer science, group codes are a type of code. Group codes consist of n linear block codes which are subgroups of Gn, where G is a finite Abelian group.

A systematic group code C is a code over Gn 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.

[edit] 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

G = \begin{pmatrix} \begin{pmatrix} 0 0 \\ 1 1 \end{pmatrix} \begin{pmatrix} 0 1 \\ 0 1 \end{pmatrix} \begin{pmatrix} 1 1 \\ 0 1 \end{pmatrix} \\ \begin{pmatrix} 0 0 \\ 1 1 \end{pmatrix} \begin{pmatrix} 11 \\ 1 1 \end{pmatrix} \begin{pmatrix} 0 0 \\ 0 0 \end{pmatrix} \end{pmatrix}

The elements of this matrix are 2x2 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 g1,...gr are the generators of G.

[edit] References

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