Linear codes in the Hamming metric have played a central role in error correction since the 1950s and have been extensively studied. In contrast, the theory of codes in the sum-rank metric is still in its early stages, with only a few known constructions.
A cornerstone of coding theory in the Hamming metric is the family of Reed–Solomon (RS) codes, which are constructed by evaluating univariate polynomials at distinct elements of a finite field $F_{q}$ . RS codes have optimal parameters, however, their length is by definition limited by the size of $ F_{q}$. Two classical approaches to overcome this limitation, while maintaining control on the parameters, are considering multivariate polynomials, giving rise to Reed–Muller (RM) codes, and evaluating rational function at points on algebraic curves, leading to Algebraic Geometry (AG) codes.
The sum-rank analogue of RS codes is the family of linearized Reed–Solomon (LRS) codes (see U. Martínez-Peñas 2018), which also achieve optimal parameters but face a similar length restriction as RS codes. In this talk, inspired by the similarities between RS and LRS codes,we will introduce analogues of RM and AG codes in the sum-rank metric, known as linearized Reed–Muller (LRM) codes (see E. Berardini and X. Caruso 2025) and linearized Algebraic Geometry (LAG) codes (see E. Berardini and X. Caruso 2024).
We will begin by reviewing key background on sum-rank metric codes and univariate Ore polynomials. Afterwards, we will introduce the theory of multivariate Ore polynomials and their evaluation, leading to the construction of linearized Reed–Muller codes and an analysis of their parameters. Then, we will develop the theory of Riemann–Roch spaces over Ore polynomial rings with coefficients in the function field of a curve, leveraging the classical framework of divisors and Riemann–Roch spaces on curves. Using this foundation, we will construct linearized AG codes, providing lower bounds on their dimension and minimum distance. We will conclude the talk by sketching some related works in progress.
