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.[-]

Projective Reed Muller Codes constitute an interesting class of linear codes, which was introduced by Gilles Lachaud in 1988. Questions about their minimum distance are intimately related to the question about the maximum possible number of F-rational points in the m-dimensional projective space on a hypersurface of degree d in m+1 variables with coefficients in a finite field F. Michael Tsfasman gave a conjectural formula for this maximum possible number of points on such hypersurfaces, and the conjecture was soon proved in the affirmative by Jean-Pierre Serre. In all these works, it is generally assumed that the degree d is at most q, where q is the number of elements in F. Anders Sørensen considered in 1991 more general projective Reed Muller codes where d can be larger than q. From a coding theoretical perspective, it is more natural to consider this larger class. Sørensen proposed a formula for the minimum distance in the general case, and also studied the duals of the projective Reed-Muller codes.
We shall revisit the work of Sorensen by pointing out some minor inaccuracies in his proof of the minimum distance. We then propose an alternative proof. Further, we address the question of obtaining a characterization of the minimum weight codewords of projective Reed Muller codes.
This is a joint work with Rati Ludhani. [-]

In general the computation of the weight enumerator of a code is hard and even harder so for the coset leader weight enumerator. Generalized Reed Solomon codes are MDS, so their weight enumerators are known and its formulas depend only on the length and the dimension of the code. The coset leader weight enumerator of an MDS code depends on the geometry of the associated projective system of points. We consider the coset leader weight enumerator of $F_{q}$-ary Generalized Reed Solomon codes of length q + 1 of small dimensions, so its associated projective system is a normal rational curve. For instance in case of the $\left [ q+1,3,q-1 \right ]_{q}$ code where the associated projective system of points consists of the q + 1 points of a plane conic, the answer depends whether the characteristic is odd or even. If the associated projective system of points of a $\left [ q+1,4,q-2 \right ]_{q}$ code consists of the q + 1 points of a twisted cubic, the answer depends on the value of the characteristic modulo 6.[-]

In the field of coding theory, Goppa's construction of error-correcting codes on algebraic curves has been widely studied and applied. As noticed by M. Tsfasman and S. Vlădut¸, this construction can be generalized to any algebraic variety. This talk aims to shed light on the case of surfaces and expand the understanding of Goppa's construction beyond curves. After discussing the motivations for considering codes from higher–dimensional varieties, we will compare and contrast codes from curves and codes from surfaces, notably regarding the computation of their parameters, their local properties, and asymptotic constructions.[-]