Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the codelength is smaller than the number of rational points on the curve, then this method can correct up to 1.2 (d*-L)/2-s errors, where d* is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all rational functions which have no poles except at P, and it is somehow similar to a polynomial ring. For a special kind of geometric Goppa code, namely C/sub Omega /(D,mP), the decoding algorithm is reduced to solving the key equation in the affine ring, which can be carried out by the subresultant sequence in the affine ring with complexity O(n/sup 3/), where n is the length of codewords.>

Traditional methods of representing rational functions on curves are unwieldy and unsuitable for solution of many problems. This paper describes a simple and elegant representation of elements of the affine coordinate ring of an algebraic curve and describes efficient, easy to implement algorithms to perform addition, subtraction, multiplication and polynomial evaluation. This data structure overcomes many of the disadvantages of more unwieldy traditional representations. Elements are represented as vectors of elements of the ground field in a manner similar to the representation of polynomials of one variable as an array of coefficients. This data structure is a fundamental ingredient in the author's decoding method for algebraic geometry codes. The rational function approximation techniques used for decoding could not have been described with multivariate polynomials or truncated infinite series.