A fast interpolation method for list decoding of RS and algebraic-geometric codes

Abstract

In this paper, we present an efficient algorithm to find a Grobner basis of the ideal set of polynomials I(V) based on the Berlekamp-Massey-Sakata (BMS) algorithm, which gives another efficient method of giving the solution at the first stage of the Sudan algorithm (1997). We also show that the interpolation problem can be generalized to find a Grobner basis of the ideal I(V;M) which consists of polynomials having zeros (/spl alpha//sub i/,/spl beta//sub i/)/spl isin/V with some multiplicity condition specified by a set M (/spl isin/Z/sub 0//sup 2/) of integer vectors. This Hermitian type of interpolation problem takes a role in the improved version of the Sudan algorithm. A modification of BMS the algorithm can be applied to solve this problem. On the other hand, for list decoding of one-point AG codes our method can be adapted to find a Grobner basis of a relevant ideal.