# Algebraic-geometry codes, one-point codes, and evaluation codes

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DOI: 10.1007/s10623-007-9074-5

- Cite this article as:
- Bras-Amorós, M. Des Codes Crypt (2007) 43: 137. doi:10.1007/s10623-007-9074-5

## Abstract

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an \({\mathbb{F}}_q\)-algebra *A*, an order function \(\rho\) on *A* and given a surjective \({\mathbb{F}}_q\)-morphism of algebras \(\varphi: A\rightarrow {\mathbb{F}}_q^{n}\), the *i*th *evaluation code* with respect to \(A,\rho,\varphi\) is defined as the code \(C_i=\varphi(\{f\in A: \rho(f)\leq i\})\) . In this work it is shown that under a certain hypothesis on the \(\mathbb{F}_q\)-algebra *A*, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on *A* is that its field of fractions is a function field over \(\mathbb{F}_q\) and that *A* is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes *G*_{i} with associated divisors \(\Gamma_i\) is the sequence of evaluation codes associated to some \({\mathbb{F}}_q\)-algebra *A*, some order function \(\rho\) and some surjective morphism \(\varphi\) with \(\{f\in A: \rho(f)\leq i\}={\mathcal{L}}(\Gamma_i)\) if and only if it is a sequence of one-point codes.