arXiv 2006.144892020-06-25cs.ITmath.ITmath.NTmath.RA

任意Galois扩张上的秩度量码和Reed-Muller码的秩模拟

Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes

Daniel Augot, Alain Couvreur, Julien Lavauzelle, Alessandro Neri

备注：: 26 pages, 1 figure

摘要：

本文推广了带有任意伽罗瓦群$G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$的扩域$\mathbb{L}$上的秩度量码的研究。我们提出了一个框架，研究这些代码作为子空间的群代数$\mathbb{L}[G]$，我们认为这一点与通常的概念秩度量码在$\mathbb{L}^N$或$\mathbb{K}^{N\times N}$，其中$N = [\mathbb{L} : \mathbb{K}]$。然后，我们适应这个上下文中的纠错对的概念，以提供一个非平凡的解码算法，这些代码。然后，我们专注于$G$是阿贝尔的情况下，这使我们看到码字的多元斜多项式环的元素。我们证明，我们可以绑定这些多项式的零向量空间的维数，取决于他们的程度。这个结果可以被看作是阿隆-富雷迪定理的一个类似物--并通过施瓦茨-齐佩尔引理--在秩度量中。最后，我们构造了Reed-Muller码在秩度量下的对应码，并给出了它们的参数。我们还显示了这些代码和经典的Reed-Muller码的情况下，$\mathbb{L}$是一个Kummer扩展之间的连接。

英文摘要：

This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group algebra $\mathbb{L}[G]$, and we relate this point of view with usual notions of rank-metric codes in $\mathbb{L}^N$ or in $\mathbb{K}^{N\times N}$, where $N = [\mathbb{L} : \mathbb{K}]$. We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes. We then focus on the case where $G$ is abelian, which leads us to see codewords as elements of a multivariate skew polynomial ring. We prove that we can bound the dimension of the vector space of zeroes of these polynomials, depending of their degree. This result can be seen as an analogue of Alon-Füredi theorem -- and by means, of Schwartz-Zippel lemma -- in the rank metric. Finally, we construct the counterparts of Reed-Muller codes in the rank metric, and we give their parameters. We also show the connection between these codes and classical Reed-Muller codes in the case where $\mathbb{L}$ is a Kummer extension.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

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