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Group code
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<h2 id="construction">Construction</h2> <p>Group codes can be constructed by special <a href="/facts/Generator_matrix/V76ugWLN">generator matrices</a> which resemble generator matrices of linear block codes except that the elements of those matrices are <a href="/facts/Endomorphism/s7NNlSea">endomorphisms</a> of the group instead of symbols from the code's alphabet. For example, considering the generator matrix </p> G = ( ( 00 11 ) ( 01 01 ) ( 11 01 ) ( 00 11 ) ( 11 11 ) ( 00 00 ) ) {\displaystyle 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}}} <p>the elements of this matrix are 2 × 2 {\displaystyle 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 {\displaystyle g_{1}^{m_{1}}g_{2}^{m_{2}}...g_{r}^{m_{r}}} where g 1 , . . . g r {\displaystyle g_{1},...g_{r}} are the <a href="/facts/Generating_set_of_a_group/B9q5lnFY">generators</a> of G {\displaystyle G} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Group_coded_recording/rz5tW8Jq">Group coded recording</a> (GCR)</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Watkinson, John (1990). "3.4. Group codes". <i>Coding for Digital Recording</i>. Stoneham, MA, USA: <a href="/facts/Focal_Press/pQEAhNVv">Focal Press</a>. pp. 51–61. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-240-51293-8.</li> <li>Biglieri, Ezio; Elia, Michele (1993-01-17). "Construction of Linear Block Codes Over Groups". <a href="/facts/IEEE_International_Symposium_on_Information_Theory/HKqq9e1G"><i>Proceedings. IEEE International Symposium on Information Theory (ISIT)</i></a>. p. 360. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FISIT.1993.748676">10.1109/ISIT.1993.748676</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7803-0878-7. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123694385">123694385</a>.</li> <li><a href="/facts/George_David_Forney/dKvnKPT2">Forney, George David</a>; Trott, Mitch D. (1993). "The dynamics of group codes: State spaces, trellis diagrams and canonical encoders". <i><a href="/facts/IEEE_Transactions_on_Information_Theory/NbsWT0E4">IEEE Transactions on Information Theory</a></i>. 39 (5): 1491–1593. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F18.259635">10.1109/18.259635</a>.</li> <li><a href="/facts/Vijay_Virkumar_Vazirani/QzVR2f9n">Vazirani, Vijay Virkumar</a>; Saran, Huzur; Rajan, B. Sundar (1996). "An efficient algorithm for constructing minimal trellises for codes over finite Abelian groups". <i><a href="/facts/IEEE_Transactions_on_Information_Theory/NbsWT0E4">IEEE Transactions on Information Theory</a></i>. 42 (6): 1839–1854. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.7058">10.1.1.13.7058</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F18.556679">10.1109/18.556679</a>.</li> <li>Zain, Adnan Abdulla; Rajan, B. Sundar (1996). "Dual codes of Systematic Group Codes over Abelian Groups". <i>Applicable Algebra in Engineering, Communication and Computing (AAECC)</i>. 8 (1): 71–83.</li></ul>