Fast Computation of Roots of Polynomials over Function Fields and Fast List Decoding of Algebraic Geometric Codes

Suppose C is a [n, k, d] code over Fq, t < n is a positive integer. For any received vector y = (y1, · · · , yn) ∈ Fq , we refer to any code word c in C satisfying d(c,y) ≤ t as a tconsistent code word. A decoding problem is in fact the problem of finding an effective (or efficient) algorithm which can find t-consistent code words, and we call such an algorithm a decoding algorithm that can correct t errors. The classical decodings (or call unique decodings) only consider the algorithms which can correct τ = d−1 2 or fewer errors. It is clear that in any Hamming sphere in Fq of radius ≤ τ , there exist at most one code word of a [n, k, d] code. We call τ the error correction bound of the code. On the other hand, if the number of errors t ≥ τ then there may exist several different consistent code words. A list decoding is a decoding algorithm which tries to construct a list of all consistent code words. Thus, a list decoding algorithm makes it is possible to recover the information from errors beyond the traditional error correction bound. The list decoding problem was first defined by Elias [2]. In [11], Sudan proposed a list decoding algorithm for generalized Reed-Solomon codes. Shokrollahi and Wasserman generalized Sudan’s algorithm and derived a list decoding scheme for algebraic geometric codes [10]. However, for codes of higher rates, these algorithms do not improve the classical decoding algorithms, i.e., these algorithms are effective only for low rate codes. In a very recent paper [4], Guruswami and Sudan proposed an improved polynomialtime algorithm for Reed-Solomon and algebraic geometric codes. The algorithm has a better error-correction rate than well-known algorithms for every choice of the code rates.