Linear diophantine equations over polynomials and soft decoding of Reed-Solomon codes

This paper generalizes the classical Knuth-Schoumlnhage algorithm computing the greatest common divisor (gcd) of two polynomials for solving arbitrary linear Diophantine systems over polynomials in time, quasi-linear in the maximal degree. As an application, the following weighted curve fitting problem is considered: given a set of points in the plane, find an algebraic curve (satisfying certain degree conditions) that goes through each point the prescribed number of times. The main motivation for this problem comes from coding theory, namely, it is ultimately related to the list decoding of Reed-Solomon codes. This paper presents a new fast algorithm for the weighted curve fitting problem, based on the explicit construction of a Groebner basis. This gives another fast algorithm for the soft decoding of Reed-Solomon codes different from the procedure proposed by Feng, which works in time (w/r) ^O(1)nlog²n, where r is the rate of the code, and w is the maximal weight assigned to a vertical line