Abstract: A new mathematical model, the linear homogeneous equations with polynomial coefficients for describing the synthesis problem, is presented in this paper. It gives a nature approach ro generalize the linear synthesis to nonlinear case. This method is used ro obtain a new solution for the multisequence synthesis. The Grbner bases theory in polynomial ring is used to present an efficient algorithm for the mathematical model. This turns out to be a generalization of Euclid algorithm. However, the new one has much brilliant prospects. As one of the important results, it is discovered that the new algorithm can be used to deduce an efficient decoding algorithm for a class of algebraic geometry codes constructed by Justesen, so the important open problem is solved.