Work

Reed–Muller codes in the sum-rank metric

Year: 2025

Type: article

Abstract: 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.

Source: Journal of Algebra and Its Applications

DOI: 10.1142/s0219498825410191

Authors: Elena Berardini, Xavier Caruso

Institutions: Centre National de la Recherche Scientifique, Université de Bordeaux, Institut de Mathématiques de Bordeaux

Language: English

Topic: Coding theory and cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funder: Agence Nationale de la Recherche

Award: algeBrA, pRoofs, Algorithms, Curves and sUrfaces for coDes and their Applications