The famous Berlekamp-Massey iteration algorithm is often used to give the solution of the synthesis problem of the sequence. In this paper

we present a more reasonable mathematical model

the linear homogeneous equations with polynomial coefficients

to describe the synthesis problem. The Grobner base theory in polynomial ring is used to present an efficient algorithm for the mathematical model. It turns out to be a generalization of Euclid’s algorithm. The new algorithm can be used to deduce an efficient decoding for a class of algebraic geometry codes constructed by Justesen.