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The first soft-decision decoding algorithm for algebraic-geometric codes is presented. This algorithm is developed based on the Koetter-Vardy algorithm, which was proposed for the decoding of Reed-Solomon codes. The interpolation process is modified by redefining the zero condition of a trivariate polynomial and introducing complexity reducing methods. Simulation results show that 0.7 and 1.7-dB performance improvements over the hard-decision decoding bound can be achieved in AWGN and quasi-static Rayleigh fading channels, respectively.