Abelian Codes in Principal Ideal Group Algebras
Abelian Codes in Principal Ideal Group Algebras
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DOI：
10.1109/tit.2012.2236383
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发表时间：
2013-05
影响因子：
2.5
通讯作者：
Somphong Jitman;S. Ling;Hongwei Liu;Xiaoli Xie
中科院分区：
文献类型：
--
作者：
Somphong Jitman;S. Ling;Hongwei Liu;Xiaoli Xie
研究了主理想群代数中的交换码。本文首先给出了任意群代数中阿贝尔码的代数刻画，并给出了一些一般性结果。对于可看作半单群代数上的循环码的PIGA中的阿贝尔码，证明了PIGA中的每一个阿贝尔码都有生成元和校验元。这些类似于循环码的生成器和奇偶校验多项式。给出了PIGA中欧氏自对偶和欧氏自正交阿贝尔码的一个刻划和一个计数，推广了最近关于自对偶循环码的类似结果。此外，建立了PIGA中可逆互补对偶阿贝尔码的结构，进一步推广了可逆互补对偶循环码的结果。最后，研究了交换码在PIGA中的渐近性质。利用半单群代数中交换码的最小距离，给出了非半单群代数中交换码的最小距离的一个上界。阿贝尔码在一个非半简单的PIGA，然后被证明是渐近坏的，类似的情况下，重复根循环码。
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.