We generalize Sudan's results for Reed-Solomon codes to the class of algebraic-geometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error-correction bound (d\Gamma1)=2, d being the minimumdistance of the code. Our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields. For the latter problem we design a polynomial-time algorithm and show that the resulting overall list-decoding algorithm runs in polynomial time under some mild conditions. Several examples are included.

A list decoding for an error-correcting code is a decoding algorithm that generates a list of codewords within a Hamming distance t from the received vector, where t can be greater than the error-correction bound. In [18], a list-decoding procedure for Reed-Solomon codes [19] was generalized to algebraic-geometric codes. A recent work [8] gives improved list decodings for Reed-Solomon codes and algebraic-geometric codes that work for all rates and have many applications. However, these list-decoding algorithms are rather complicated. In [17], Roth and Ruckenstein proposed an e#cient implementation of the list decoding of Reed-Solomon codes. In this correspondence, extending Roth and Ruckenstein's fast algorithm for finding roots of univariate polynomials over polynomial rings, i.e., the Reconstruct Algorithm, we will present an e#cient algorithm for finding the roots of univariate polynomials over function fields. Based on the extended algorithm, we give an e#cient list-decoding algorithm for algebraicgeometric codes. Index Terms: Root-finding algorithm, algebraic-geometric codes, list decoding. # Xin-Wen Wu was with the Department of Electrical and Computer Engineering, Mail Code 0407, University of California, San Diego, La Jolla, CA 92093-0407, he is currently with the Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China. Email: wxw@math08.math.ac.cn. Paul H. Siegel is with the Center for Magnetic Recording Research, Mail Code 0401, University of California, San Diego, La Jolla, CA 92093-0401. Email: psiegel@ucsd.edu. This work was supported by the National Science Foundation under grant NCR-9612802 and by the National Storage Industry Consortium. The material in this paper was presented in part at the 2000 IEEE Internatio...

In 1948 Shannon developed fundamental limits on the efficiency of communication over noisy channels. The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero. Fifty years later, codes for the Gaussian channel have been discovered that come close to these fundamental limits. There is now a substantial algebraic theory of error-correcting codes with as many connections to mathematics as to engineering practice, and the last 20 years have seen the construction of algebraic-geometry codes that can be encoded and decoded in polynomial time, and that beat the Gilbert–Varshamov bound. Given the size of coding theory as a subject, this review is of necessity a personal perspective, and the focus is reliable communication, and not source coding or cryptography. The emphasis is on connecting coding theories for Hamming and Euclidean space and on future challenges, specifically in data networking, wireless communication, and quantum information theory.

Abstract The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition. 1

Abstract. In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at P together with functions for each value in this semigroup, provided P is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes. 1.

Using methods originating in numerical analysis, we will develop a unified framework for derivation of efficient list decoding algorithms for algebraicgeometric codes. We will demonstrate our method by accelerating Sudan's list decoding algorithm for Reed-Solomon codes [22], its generalization to algebraic-geometric codes by Shokrollahi and Wasserman [21], and the recent improvement of Guruswami and Sudan [8] in the case of ReedSolomon codes. The basic problem we attack in this paper is that of efficiently finding nonzero elements in the kernel of a structured matrix. The structure of such an n \Theta n- matrix allows it to be "compressed" to ffn parameters for some ff which is usually a constant in applications. The concept of structure is formalized using the displacement operator. The displacement operator allows to perform matrix operations on the compressed version of the matrix. In particular, we can find a PLU-decomposition of the original matrix in time O(ffn 2 ), which is q...

Abstract. In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results we construct algebraicgeometric codes which are an extension to surfaces of the well-known differential codes on curves. We also study some properties of these codes and extend to them some known properties for codes on curves. AMS Classification: 14J99, 14J20, 14G50, 94B27.

A new effective decoding algorithm is presented for arbitrary algebraicgeometric codes on the basis of solving a generalized key equation with the majority coset scheme of Duursma. It is an improvement of Ehrhard’s algorithm, since the method corrects up to the half of the Goppa distance with complexity order O(n 2.81), and with no further assumption on the degree of the divisor G. Key words – AG codes, Ehrhard’s key equation, majority coset decoding.

We present a new algorithm for decoding AG-codes significantly beyond the error-correction bound. Specifically, given a word y whose distance to the AG-code is at most e, where e is a parameter depending on the block length and the dimension of the code, our algorithm produces all codewords that have distance e from y. We also discuss modifications of our general algorithm and show how to obtain similar algorithms for binary codes using concatenated codes. I. Introduction A linear error-correcting [n; k]q -code C is a k-dimensional subspace of Fq n . The elements of C are called codewords. If the minimum Hamming distance between any two distinct codewords is d, then C is called an [n; k; d]q -code. n, k, and d are called the block length, dimension, and minimum distance of C, respectively. Following a construction of Goppa, one can use algebraic curves over finite fields to design linear error-correcting codes called algebraic-geometric codes or AG-codes. These codes can be viewed...

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to “compress,” in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant). In this paper we demonstrate the power of the displacement structure approach by deriving in a unified way efficient algorithms for a number of decoding problems. We accelerate the Sudan’s list decoding algorithm for Reed-Solomon codes, its generalization to algebraic-geometric codes by Shokrollahi and Wasserman, and the improvement of Guruswami and Sudan in the case of Reed-Solomon codes. In particular, we notice that matrices that occur in the context of list decoding have low displacement rank, and use this fact to derive algorithms that use O(n 2 ℓ)andO(n 7/3 ℓ) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively. Here ℓ denotes the length of the list; assuming that ℓ is constant, this gives algorithms of running time O(n 2)andO(n 7/3), which is the same as the running time of conventional decoding algorithms. We also present efficient parallel algorithms for the above tasks. To the best of our knowledge this is the first application of the concept of displacement rank to the unified derivation of several decoding algorithms; the technique can be useful in finding efficient and fast methods for solving other decoding problems.