In this article, we investigate the decoding of the rank metric Reed–Muller codes introduced by Augot, Couvreur, Lavauzelle and Neri in 2021. These codes are defined from Abelian Galois extensions extending the construction of Gabidulin codes over arbitrary cyclic Galois extensions. We propose a polynomial time algorithm that rests on the structure of Dickson matrices, works on any such code and corrects any error of rank up to half the minimum distance.

We give a recursive decoding algorithm of the rank metric Reed-Muller codes introduced by Augot, Couvreur, Lavauzelle and Neri in 2021 for the binary case, i.e.,$\mathbf{G}= (\mathbb{Z} / 2 \mathbb{Z})^{m}$. In a broad range of parameters, this recursive decoding algorithm has better complexity compared to a recently proposed decoding algorithm based on Dickson matrices. Furthermore, imitating the recursive structure, we introduce a Plotkin-like construction of matrix rank metric codes over finite fields answering a long-standing open question. We also provide a decoding algorithm associated to this construction.