Group code: Difference between revisions

Revision as of 05:22, 21 September 2005

Revision as of 05:28, 21 September 2005

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== 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...

<math>

\PsiG = \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}

</math>

The elements of this matrix are <math>2 \into 2</math> matrices which are endomorphisms. In this scenario, each codeword can be represented as

<math>c = (c_1,\hdots,c_n) = g_1^m_1 g_2^m2 \hdots g_r^ m_r</math>

where <math>g_1,\hdots g_r</math> are the generators of <math>G</math>.

[[Category:Error detection and correction]]