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.

翻译：我们利用多元Ore多项式引入和秩度量下Reed-Muller码的类比，称之为线性化Reed-Muller码。我们研究这些码的参数，计算其维数并给出最小距离的下界。我们的码具有相当好的参数，遵循与汉明度量下Reed-Muller码类似的界。最后，我们还证明许多新引入的线性化Reed-Muller码可嵌入某些线性化代数几何码中，这一性质在解码方面可能具有实用价值。