Linear codes using skew polynomials with automorphisms and derivations
Linear codes using skew polynomials with automorphisms and derivations
复制标题
使用具有自同构和导数的倾斜多项式的线性代码
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
F. Ulmer
中科院分区：
文献类型：
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作者：
D. Boucher;F. Ulmer
在这项工作中，定义的代码作为模块的自同构类型的斜多项式环推广到斜多项式环，其乘法定义使用的自同构和导子。这产生了更一般的一类代码，在某些情况下，产生更好的距离界限比模块斜码只构造一个自同构。扩展Gabidulin码的方法，我们引入了新的概念，评估斜多项式与导子和相应的评估代码。我们提出了几种方法来推广Reed-Solomon和BCH码的模块偏斜码和两个类，我们表明，这样的Reed-Solomon型偏斜码的对偶是一个评估偏斜码。我们推广的解码算法，由于Gabidulin的秩度量，并推导出家庭的最大距离可分和最大秩距离码。
In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several approaches to generalize Reed-Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed-Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank metric and derive families of Maximum Distance Separable and Maximum Rank Distance codes.