Abstract This paper provides a survey of the existing literature on the decoding of algebraic-geometric codes. Definitions, theorems and cross references will be given. We show what has been done, discuss what still has to be done and pose some open problems. The following subjects are examined in a more or less historical order.

Abstract The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition. 1

We prove a formula for the minimum distance of two-point codes on a Hermitian curve. 1

Constructing quantum error-correcting codes for p m-state systems from classical error-correcting codes