论文标题
在任意Galois扩展名和Reed-Muller代码的等级类似物上的等级代码
Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes
论文作者
论文摘要
本文在扩展字段中扩展了对配备任意Galois组$ G = \ Mathrm {gal}（\ MathBb {l}/\ MathBb {K k}）$的级别代码的研究。我们提出了一个将这些代码作为组代数$ \ mathbb {l} [g] $的子空间的框架，我们将这种观点与通常在$ \ mathbb {l}^n $或$ \ mathbb {k}^k}^n \ times n \ times n} $ n $ n $ n $ n $ n $ n $ n y $ n $ n $ n y $ n y $ n y $ n $ n = n $ n = n y $ n y $ n = n y n $ n = n y n $ n = n = n = n = n = n = n = n = n = \ mathbb {k}] $。然后，我们将错误校正对的概念调整到此上下文中，以便为这些代码提供一种非平凡的解码算法。然后，我们专注于$ g $是Abelian的情况，这使我们将CodeWord视为多元偏度多项式环的元素。我们证明，我们可以根据其程度限制这些多项式零的矢量空间的维度。该结果可以看作是Alon-füredi定理的类似物 - 以及施瓦茨 - 齐佩尔引理中的类似物。最后，我们在等级度量中构建了芦苇毛刺代码的对应物，并给出了它们的参数。在$ \ mathbb {l} $是kummer扩展的情况下，我们还显示了这些代码与经典芦苇毛刺代码之间的连接。
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.
