Reed–Muller codes in the sum-rank metric

We introduce the sum-rank metric analog 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, recently defined in Berardini and Caruso [Algebraic geometry codes in the sum-rank metric, IEEE Trans. Inf. Theory 70 (2024) 3345–3356] a property which could turn out to be useful in light of decoding.