Abelian codes in principal ideal group algebras
We study abelian codes in principal ideal group algebras (PIGAs). We first give an algebraic characterization of abelian codes in any group algebra and provide some general results. For abelian codes in a PIGA, which can be viewed as cyclic codes over a semisimple group algebra, it is shown that eve...
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sg-ntu-dr.10356-1072322019-12-06T22:27:08Z Abelian codes in principal ideal group algebras Jitman, Somphong Ling, San Liu, Hongwei Xie, Xiaoli School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Applied mathematics::Information theory We study abelian codes in principal ideal group algebras (PIGAs). We first give an algebraic characterization of abelian codes in any group algebra and provide some general results. For abelian codes in a PIGA, which can be viewed as cyclic codes over a semisimple group algebra, it is shown that every abelian code in a PIGA admits generator and check elements. These are analogous to the generator and parity-check polynomials of cyclic codes. A characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in a PIGA are given, which generalize recent analogous results for self-dual cyclic codes. In addition, the structures of reversible and complementary dual abelian codes in a PIGA are established, again extending results on reversible and complementary dual cyclic codes. Finally, asymptotic properties of abelian codes in a PIGA are studied. An upper bound for the minimum distance of abelian codes in a non-semisimple PIGA is given in terms of the minimum distance of abelian codes in semisimple group algebras. Abelian codes in a non-semisimple PIGA are then shown to be asymptotically bad, similar to the case of repeated-root cyclic codes. 2013-11-25T07:18:46Z 2019-12-06T22:27:08Z 2013-11-25T07:18:46Z 2019-12-06T22:27:08Z 2013 2013 Journal Article Jitman, S., Ling, S., Liu, H., & Xie, X. (2013). Abelian Codes in Principal Ideal Group Algebras. IEEE Transactions on Information Theory, 59(5), 3046-3058. 0018-9448 https://hdl.handle.net/10356/107232 http://hdl.handle.net/10220/17834 http://dx.doi.org/10.1109/TIT.2012.2236383 en IEEE transactions on information theory |
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DRNTU::Science::Mathematics::Applied mathematics::Information theory Jitman, Somphong Ling, San Liu, Hongwei Xie, Xiaoli Abelian codes in principal ideal group algebras |
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We study abelian codes in principal ideal group algebras (PIGAs). We first give an algebraic characterization of abelian codes in any group algebra and provide some general results. For abelian codes in a PIGA, which can be viewed as cyclic codes over a semisimple group algebra, it is shown that every abelian code in a PIGA admits generator and check elements. These are analogous to the generator and parity-check polynomials of cyclic codes. A characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in a PIGA are given, which generalize recent analogous results for self-dual cyclic codes. In addition, the structures of reversible and complementary dual abelian codes in a PIGA are established, again extending results on reversible and complementary dual cyclic codes. Finally, asymptotic properties of abelian codes in a PIGA are studied. An upper bound for the minimum distance of abelian codes in a non-semisimple PIGA is given in terms of the minimum distance of abelian codes in semisimple group algebras. Abelian codes in a non-semisimple PIGA are then shown to be asymptotically bad, similar to the case of repeated-root cyclic codes. |
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School of Physical and Mathematical Sciences |
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School of Physical and Mathematical Sciences Jitman, Somphong Ling, San Liu, Hongwei Xie, Xiaoli |
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Article |
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Jitman, Somphong Ling, San Liu, Hongwei Xie, Xiaoli |
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Jitman, Somphong |
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Abelian codes in principal ideal group algebras |
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Abelian codes in principal ideal group algebras |
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Abelian codes in principal ideal group algebras |
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Abelian codes in principal ideal group algebras |
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Abelian codes in principal ideal group algebras |
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abelian codes in principal ideal group algebras |
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2013 |
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https://hdl.handle.net/10356/107232 http://hdl.handle.net/10220/17834 http://dx.doi.org/10.1109/TIT.2012.2236383 |
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