Solving a congruence on a graded algebra by a subresultant sequence and its application

Let G be a submonoid of nonnegative integers. Let g be an element of G-graded algebra with a G-basis, and f be an element of a nonzero ideal of the algebra. In this paper, we consider solving for p and r the congruence pf ¿ r (mod g) such that deg(r) - deg(p) is less than a given integer and p has the minimal degree among all solutions. We show how to solve the congruence by using a so-called subresultant sequence of f and g. We also give an algorithm to find a solution explicitly. As an application, we use this algorithm to decode geometric Goppa codes