Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes
Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes
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DOI：
10.1006/jsco.2001.0442
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发表时间：
2001-05
期刊：
影响因子：
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通讯作者：
E. Byrne;P. Fitzpatrick
中科院分区：
文献类型：
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作者：
E. Byrne;P. Fitzpatrick
根据有限域上格罗瓦环上格罗瓦基的一般公式，建立了格罗瓦环上格罗瓦基的理论。我们的处理包括除法算法，Grobner基的表征，以及Buchberger？年代的算法。其中一个应用是解决伽罗瓦环上交替码的解码问题。为此，我们考虑模块M= {(a, b):aS？b modxr}是所谓的交替码关键方程的所有解，其中S是一个综合征多项式。在解码中，一个特解(？, ?)？求满足一定条件的M，在M的Grobner基中可以找到这样的解。利用本文第一部分介绍的技术，我们给出了一个返回所需解的算法。
We develop a theory of Grobner bases over Galois rings, following the usual formulation for Grobner bases over finite fields. Our treatment includes a division algorithm, a characterization of Grobner bases, and an extension of Buchberger?s algorithm. One application is towards the problem of decoding alternant codes over Galois rings. To this end we consider the module M= {(a, b) :aS?b modxr} of all solutions to the so-called key equation for alternant codes, where S is a syndrome polynomial. In decoding, a particular solution (?, ?) ?M is sought satisfying certain conditions, and such a solution can be found in a Grobner basis of M. Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution.