Reed-Muller codes in the sum-rank metric

TL;DR

This paper introduces linearized Reed--Muller codes in the sum-rank metric, analyzing their parameters, minimum distance, and embedding properties, which could enhance decoding strategies in coding theory.

Contribution

It defines the sum-rank metric analogue of Reed--Muller codes using multivariate Ore polynomials and explores their parameters and embedding into algebraic geometry codes.

Findings

Codes have good parameters similar to classical Reed--Muller codes.

Lower bounds for minimum distance are established.

Many codes can be embedded into linearized algebraic geometry codes.

Abstract

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 arXiv:2303.08903, a property which could turn out to be useful in light of decoding.