A Displacement Structure Approach to List Decoding of Reed-Solomon and Algebraic-Geometric Codes∗

Using the method of displacement we shall develop a unified framework for derivation of efficient list decoding algorithms for algebraic-geometric codes. We will demonstrate our method by accelerating Sudan’s list decoding algorithm for Reed-Solomon codes [22], its generalization to algebraicgeometric codes by Shokrollahi and Wasserman [21], and the recent improvement of Guruswami and Sudan [8] in the case of Reed-Solomon codes. The main problem we study in this paper is that of efficiently finding nonzero elements in the kernel of a structured matrix. The structure of such an n × nmatrix allows it to be “compressed” to αn parameters where the so-called displacement rank α is usually a small constant. The compressed representation is obtained by applying to the original matrix a displacement operator, a technique that allows us to derive fast efficient algorithms. In particular, we can find a PLU -decomposition of the original matrix in time O(αn2), which is quadratic in n for constant α. We will describe appropriate displacement operators for matrices that occur in the context of list decoding, and apply our general algorithm to them. For example, we will obtain algorithms that use O(n2`) and O(n7/3`) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively, where ` is the length of the list. Assuming that ` is constant, this gives algorithms of running time O(n2) and O(n7/3), which is the same as the running time of conventional decoding algorithms. We will also sketch methods to parallelize our algorithms. ∗This work was supported by NSF grant CCR 9732355.