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# Effective Construction of Algebraic Geometry Codes

Goppa's construction of linear codes using algebraic geometry, the so-called geometric Goppa or AG codes, was a major breakthrough in the history of coding theory. In particular, it was the first (and only) construction leading to a family of codes with parameters above the Gilbert-Varshamov bound [41].

There exist several (essentially equivalent) ways to construct AG codes starting from a smooth projective curve defined over a finite field . Mainly, we should like to mention the L-, resp. the -construction.

Given rational points and a rational divisor on having disjoint support with the divisor , the AG code , resp. , is the image of the -linear map

 ev    resp. resresres

In practice, there are two main difficulties when looking for an effective method to compute the generator matrices of such codes: Given a plane (singular) model of , how to compute the places of and how to compute a basis for the linear system (cf. below), resp. for the vector space of rational one-forms

One possible solution, making use of blowing-up theory (to compute the places of ) and of the (classical) Brill-Noether algorithm (for the computation of a basis of ) is presented in [28,19]. In the following, we should like to point to the modified approach of Campillo and Farrán [3], using Hamburger-Noether expansions instead of blowing-up theory, and to present in some detail the resulting algorithm as implemented in the computer algebra system SINGULAR [17].

Subsections

Next: Preliminaries, Notations Up: Three Algorithms in Algebraic Previous: Integral closure of an
Christoph Lossen
2001-03-21