Group code
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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.
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. 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 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 .
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