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

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 script F sign q-algebra A, an order function ρ on A and given a surjective script F signq-morphism of algebras φ: A → script F signqn, the ith evaluation code with respect to A,ρ,φ is defined as the code Ci=φ({f ∈ A: ρ(f) ≤ i}). In this work it is shown that under a certain hypothesis on the script F signq-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 script F signq and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G i with associated divisors Γi is the sequence of evaluation codes associated to some script F signq-algebra A, some order function ρ and some surjective morphism φ with {f ∈ A: ρ(f) ≤ i}= L}}(Γi) if and only if it is a sequence of one-point codes. © Springer Science+Business Media, LLC 2007.