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

- Preliminaries, Notations
- Symbolic Hamburger-Noether expressions
- Adjoint curves and Brill-Noether residue theorem
- Computational Aspects
- Example