Decoding Algebraic-Geometric Codes by solving a key equation

The recent work about the problem of decoding Algebraic-Geometric Codes has led to an algorithm (e.g., see [2,3]). Another algorithm has been given by Porter, see [6,7,8,9,10], generalizing Berlekamp's decoding algorithm. The main step is to solve a so-called "key-equation". For this purpose, Porter gave a generalization of Euclid's algorithm for functions on curves. Unfortunately, therefore he had to impose some strong restrictions to the code and its underlying curve, such that the resulting algorithm works only for a very small class of Algebraic-Geometric Codes. Recently, the generalized Euclidian algorithm was investigated and corrected by Porter, Shen and Pellikaan ([11]) and Shen ([12]). Here, we will show how to generalize Porters ideas to all Algebraic-Geometric Codes and moreover, how to solve the key equation by simple linear algebra operations. Two observations on Porter's methods have motivated our work: