Recursive Decoding of Binary Rank Reed-Muller Codes and Plotkin Construction for Matrix Codes

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., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{G}= (\mathbb{Z} / 2 \mathbb{Z})^{m}$</tex>. 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.