On the hardness of code equivalence problems in rank metric

In the 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. Among classical problems associated to codes for a given metric, the notion of code equivalence (to decide if two codes are isometric) has always been of the greatest interest, for its cryptographic applications or its deep connexions to the graph isomorphism problem. 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 {\em Matrix Codes Right Equivalence Problem}. 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 {\em stabilizer algebras}. The resolution of the latter case will involve tools related to finite dimensional algebras and 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.