In recent years, the notion of rank metric in the context of coding theory has known many interesting developments in terms of applications such as space time coding, network coding or public key cryptography. These applications raised the interest of the community for theoretical properties of this type of codes, such as the hardness of decoding in rank metric or better decoding algorithms. Among classical problems associated to codes for a given metric, the notion of code equivalence has always been of the greatest interest. In this article, we discuss the hardness of the code equivalence problem in rank metric for $\mathbb{F}_{q^m}$--linear and general rank metric codes. In the $\mathbb{F}_{q^m}$--linear case, we reduce the underlying problem to another one called Matrix Codes Right Equivalence Problem (MCREP). We prove the latter problem to be either in $\mathcal{P}$ or in $\mathcal{ZPP}$ depending of the ground field size. This is obtained by designing an algorithm whose principal routines are linear algebra and factoring polynomials over finite fields. It turns out that the most difficult instances involve codes with non trivial stabilizer algebras. The resolution of the latter case will involve tools related to finite dimensional algebras and the so--called Wedderburn--Artin theory. It is interesting to note that 30 years ago, an important trend in theoretical computer science consisted to design algorithms making effective major results of this theory. These algorithmic results turn out to be particularly useful in the present article. Finally, for general matrix codes, we prove that the equivalence problem (both left and right) is at least as hard as the well--studied {\em Monomial Equivalence Problem} for codes endowed with the Hamming metric.

中文翻译：

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等级度量中代码等价问题的难度

近年来，编码理论背景下的秩度量概念在空时编码、网络编码或公钥密码学等应用方面取得了许多有趣的发展。这些应用引起了社区对此类代码的理论特性的兴趣，例如等级度量解码的难度或更好的解码算法。在与给定度量的代码相关的经典问题中，代码等效性的概念一直是最受关注的。在本文中，我们讨论了 $\mathbb{F}_{q^m}$--linear 和一般秩度量代码在秩度量中代码等价问题的难度。在 $\mathbb{F}_{q^m}$--linear 情况下，我们将潜在问题简化为另一个称为矩阵码右等价问题 (MCREP) 的问题。我们证明后一个问题要么在 $\mathcal{P}$ 中，要么在 $\mathcal{ZPP}$ 中，这取决于地面场的大小。这是通过设计一种算法来获得的，该算法的主要例程是线性代数和有限域上的因式分解多项式。事实证明，最困难的实例涉及具有非平凡稳定器代数的代码。后一种情况的解决将涉及与有限维代数和所谓的 Wedderburn--Artin 理论相关的工具。有趣的是，30 年前，理论计算机科学的一个重要趋势是设计算法，使该理论产生有效的主要结果。这些算法结果在本文中特别有用。最后，对于一般矩阵码，