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| In [[computer science]], '''group codes''' are a type of [[coding theory|code]]. Group codes consist of
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| <math>n</math> [[linear block codes]] which are subgroups of <math>G^n</math>, where <math>G</math> is a finite [[Abelian group]].
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| A systematic group code <math>C</math> is a code over <math>G^n</math> of order <math>\left| G \right|^k</math> defined by <math>n-k</math> homomorphisms which determine the parity check bits. The remaining <math>k</math> bits are the information bits themselves.
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| == Construction ==
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| Group codes can be constructed by special [[generator matrix|generator matrices]] which resemble generator matrices of linear block codes except that the elements of those matrices are [[endomorphism]]s of the group instead of symbols from the code's alphabet. For example, consider the generator matrix
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| :<math>
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| 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} \\
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| \begin{pmatrix} 0 0 \\ 1 1 \end{pmatrix} \begin{pmatrix} 11 \\ 1 1 \end{pmatrix} \begin{pmatrix} 0 0 \\ 0 0 \end{pmatrix}
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| \end{pmatrix}
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| </math>
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| The elements of this matrix are <math>2\times 2</math> matrices which are endomorphisms. In this scenario, each codeword can be represented as
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| <math>g_1^{m_1} g_2^{m_2} ... g_r^{m_r}</math>
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| where <math>g_1,... g_r</math> are the [[Generating set of a group|generator]]s of <math>G</math>.
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| == References ==
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| * {{cite doi|10.1109/ISIT.1993.748676}}
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| * G. D. Forney, M. Trott, {{doi-inline|10.1109/18.259635|The dynamics of group codes : State spaces, trellis diagrams and canonical encoders}}, ''IEEE Trans. Inform. theory'', Vol '''39''' (1993), pages 1491-1593.
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| * V. V. Vazirani, Huzur Saran and B. S. Rajan, {{doi-inline|10.1109/18.556679|An efficient algorithm for constructing minimal trellises for codes over finite Abelian groups}}, ''IEEE Trans. Inform. Theory'' '''42''', No.6, (1996), 1839-1854.
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| * A. A. Zain, B. Sundar Rajan, "Dual codes of Systematic Group Codes over Abelian Groups", ''Appl. Algebra Eng. Commun. Comput.'' '''8'''(1): 71-83 (1996).
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| [[Category:Coding theory]]
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