  We introduce the sum-rank metric analogue of Reed-Muller codes, which we
called linearized Reed-Muller codes, using multivariate Ore polynomials. We
study the parameters of these codes, compute their dimension and give a lower
bound for their minimum distance. Our codes exhibit quite good parameters,
respecting a similar bound to Reed-Muller codes in the Hamming metric. Finally,
we also show that many of the newly introduced linearized Reed--Muller codes
can be embedded in some linearized Algebraic Geometry codes, a property which
could turn out to be useful in light of decoding.
