Reed-Muller codes in the sum-rank metric
Résumé
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, recently defined in [Algebraic geometry codes in the sum-rank metric, IEEE Trans. Inf. Theory70 (2024) 3345–3356], a property which could turn out to be useful in light of decoding.
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