Patent ID: 7502988

Claim:
A method of decoding a one-point algebraic geometric code defined on an algebraic curve of the kind C(a,b) represented by an equation of degree b in X and of degree a in Y, said method comprising, for any received word r, a step of identifying each point (x,y) of a locating set of a code such that a component of said received word r labeled by that point is erroneous, the method comprising the following steps: counting the number l of different values of x appearing in the points associated with erroneous components of r, each of the values of x defining an erroneous aggregate of which the elements are all the λ(x) points (x,y p (x)), where p=0, . . . , λ(x)−1, of the locating set corresponding to that value of x, calculating, for i=0, . . . , 2l−1 and j=0, . . . , λ max −1, where λ max is the maximum value of the cardinals λ(x) of the erroneous aggregates, the extended error syndromes σ j (i)=<Y j X i |e>, where e is a transmission error affecting r and Y j X i represents the word of which the components are equal to the value taken by the monomial Y j X i at the points of the locating set, implementing, for j=0, . . . , λ max −1, by the error syndromes polynomial S j ⁡ ( Z ) = ∑ i = 0 2 ⁢ l - 1 ⁢ σ j ⁡ ( i ) ⁢ Z i , an error correction algorithm adapted to Reed-Solomon codes, so as to calculate the errors E j (x) in the components of a Reed-Solomon codeword defined over the same Galois field as the algebraic geometric code which are labeled by the l values of x associated with an erroneous aggregate, and calculating, for each value of x such that there is at least one value j for which E j (x) is non-zero, the estimations ê(x,y p (x)) of the respective errors in the components r(x,y p (x)), where p=0, . . . , λ(x)−1, of r by solving the system of equations E j ⁡ ( x ) = ∑ p = 0 λ ⁡ ( x ) - 1 ⁢ ⁢ [ y p ⁡ ( x ) ] j ⁢ e ^ ⁡ ( x , y p ⁡ ( x ) ) , where ⁢ ⁢ j = 0 , … ⁢ , λ ⁡ ( x ) - 1.