In 2021, Augot, Couvreur, Lavauzelle and Neri proposed a construction of rank
metric analogues of Reed–Muller codes from twisted group algebras associated with an abelian
Galois extension. These codes are multivariate version of Gabidulin codes which are defined for
cyclic Galois extensions.
In this talk, I will present a probabilistic decoding for binary rank metric Reed-Muller codes that
corrects errors of rank up to half minimum distance. It rests on a recursive structure of the binary
Reed–Muller codes that can be considered as a rank analogue of the recursive Plotkin “(u — u+v)”
structure for Hamming metric binary Reed–Muller codes. Furthermore, I will discuss how we can
adapt this recursive decoder to construct new efficiently decodable matrix rank metric codes over
finite field extensions. This is based on a work with Alain Couvreur.

References:

[1] D. Augot, A. Couvreur, J. Lavauzelle, and A. Neri, Rank metric codes over ar-
bitrary Galois extensions and rank analogues of Reed-Muller codes, SIAM journal of applied algebra
and geometry, vol. 5 (2), 2021.
[2] A. Couvreur, R. Pratihar, Recursive decoding of binary rank Reed-Muller codes and Plotkin
construction for matrix codes, ISIT 2025, https://inria.hal.science/hal-04915230.