 
Summary: Linear Diophantine equations over polynomials and soft decoding of
ReedSolomon codes
Michael Alekhnovich #
October 20, 2004
Abstract
We generalize the classical KnuthSch˜onhage algorithm computing GCD of two polynomials
for solving arbitrary linear Diophantine systems over polynomials in time, quasilinear in the
maximal degree. As an application, we consider the following weighted curve fitting problem:
given a set of points in the plain, 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 the coding theory, namely it is ultimately related to the list decoding of
ReedSolomon codes.
We present a new fast algorithm for the weighted curve fitting problem, based on the ex
plicit construction of Groebner basis. This gives another fast algorithm for the soft decoding
of ReedSolomon codes di#erent from the procedure proposed by Feng, which works in time
( w
r
) O(1) n log 2 n, where r is the rate of the code, and w is the maximal weight assigned to a
vertical line.
1. Preliminaries
