Rank-Metric Codes over Arbitrary Galois Extensions and Rank Analogues of Reed-Muller Codes

This paper extends the study of rank-metric codes in extension fields equipped with an arbitrary finite Galois group. We propose a framework for studying these codes as subspaces of the skew group algebra over the Galois group, and we relate this point of view with usual notions of vector rank-metric codes and matrix codes. We then adapt the notion of error-correcting pairs to this context, in order to provide a nontrivial decoding algorithm for these codes. We then focus on the case where the Galois group 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 zeros of these polynomials, depending of their degree. This result can be seen as an analogue of the Alon-F\" uredi theorem-and by means of the 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 of a Kummer extension.