Divide-and-conquer interpolation for list decoding of reed-solomon codes

The most computationally demanding step in algebraic soft-decision decoding, as well as Sudan-type list-decoding, of Reed-Solomon codes is bivariate polynomial interpolation. The interpolation problem consists of computing a polynomial Q(X,Y) that passes through a given set of points P with prescribed multiplicities M. We propose a new divide-and-conquer method that can potentially reduce the complexity of interpolation. Specifically, we split the interpolation problem {P,M} into two problems {P₁,M₁} and {P₂,M₂}, and then show that the intersection of the corresponding polynomial ideals I(P₁,M₁)capI(P₂,M₂) is equal to their product. Our divide-and-conquer approach differs from the one suggested by Feng-Giraud [G.L.Feng, (2002)] in that the interpolation subproblems {P₂,M₂} and {P₂ ,M₂} are solved independently and only afterwards their solutions are merged. This makes it possible to solve these problems in parallel