David González González (Department of Mathematics, Universidad de Salamanca)
Algebraic-geometric codes extend the classical construction of evaluation codes from the projective line to arbitrary algebraic curves over finite fields. Their parameters are closely related to geometric properties of the underlying curve and to the theory of divisors and Riemann-Roch spaces.
We begin by reviewing the classical construction in detail. Let $X$ be a smooth projective curve over a finite field $\mathbb{F}_{q}$, let
$$D = P_{1} + \dots + P_{n}$$
be a divisor supported on pairwise distinct rational points, and let $G$ be a divisor whose support is disjoint from that of $D$. The associated algebraic-geometric code is obtained from the evaluation map
$$\operatorname{ev}_{D}: L(G) \longrightarrow \mathbb{F}_{q}^{n}, \quad f \mapsto (f(P_{1}), \dots, f(P_{n})),$$
where $L(G)$ denotes the corresponding Riemann-Roch space. We explain how the length, dimension, and minimum distance of the resulting code are determined or bounded in terms of the divisors $D$ and $G$. In particular, the Riemann-Roch theorem provides the main tool for determining the dimension, while the degree of $G$ yields the classical lower bound for the minimum distance. This construction provides the geometric background required for the higher-order codes considered in the second part of the talk.
We then replace ordinary evaluation at the points $P_{i}$ by higher-order local evaluation. Instead of retaining only the value $f(P_{i})$, we record several coefficients of the local expansion of $f$ at each point. These coefficients can be described using Hasse-Schmidt derivations, which provide a natural framework for higher-order differentiation in positive characteristic. By assigning a multiplicity to each evaluation point, one obtains a family of differential Goppa codes in which the classical algebraic-geometric construction appears as the multiplicity-one case.
We study how these multiplicities affect the parameters and structure of the resulting codes. Higher-order evaluation allows several symbols to be obtained from a single rational point and therefore provides a way of constructing longer codes without increasing the number of evaluation points. We illustrate this construction through explicit examples and compare it with the classical case.
Finally, we consider the metric structure induced by the higher-order coordinates. At each evaluation point, the local coefficients form an ordered block, and the vanishing of its initial coefficients reflects the order of vanishing of the corresponding function. This makes the Niederreiter-Rosenbloom-Tsfasman (NRT) metric a natural metric for the higher-order construction. We discuss how the NRT metric captures this local ordered structure and how it behaves under changes of local parameters, in comparison with the usual Hamming metric.
Poster