Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following "curve-fitting" problem over a field F : Given n points f(x i :y i )g i=1 , x i

A list decoding algorithm is presented for [n; k] Reed-Solomon (RS) codes over GF (q), which is capable of correcting more than b(n\Gammak)=2c errors. Based on a previous work of Sudan, an extended key equation (EKE) is derived for RS codes, which reduces to the classical key equation when the number of errors is limited to b(n\Gammak)=2c. Generalizing Massey's algorithm that finds the shortest recurrence that generates a given sequence, an algorithm is obtained for solving the EKE in time complexity O(` \Delta (n\Gammak) 2 ), where ` is a design parameter, typically a small constant, which is an upper bound on the size of the list of decoded codewords (the case ` = 1 corresponds to classical decoding of up to b(n\Gammak)=2c errors where the decoding ends with at most one codeword). This improves on the time complexity O(n 3 ) needed for solving the equations of Sudan's algorithm by a naive Gaussian elimination. The polynomials found by solving the EKE are then used for reconstruct...

Abstract This paper provides a survey of the existing literature on the decoding of algebraic-geometric codes. Definitions, theorems and cross references will be given. We show what has been done, discuss what still has to be done and pose some open problems. The following subjects are examined in a more or less historical order.

We present an efficient implementation of Sudan's algorithm for list decoding Hermitian codes beyond half the minimum distance. The main ingredients are a fast way of calculating so-called increasing zero bases, an efficient interpolation algorithm for finding the Q-polynomial, and a reduction of the problem of factoring the Q-polynomial to the problem of factoring a univariate polynomial over a large finite field.

This correspondence presents an algorithmic improvement to Sudan's list-decoding algorithm for Reed--Solomon codes and its generalization to algebraic--geometric codes from Shokrollahi and Wasserman. Instead of completely factoring the interpolation polynomial over the function field of the curve, we compute sufficiently many coefficients of a Hensel development to reconstruct the functions that correspond to codewords. We prove that these Hensel developments can be found efficiently using Newton's method. We also describe the algorithm in the special case of Reed--Solomon codes. Index Terms---Algebraic--geometric codes, Hensel lifting, list decoding, Newton's method, polynomials over algebraic function fields, Reed--Solomon codes. I.

Soft-decision decoding of Reed-Solomon codes delivers significant coding gains over classical minimum distance decoding. In this paper, we present architectures for polynomial interpolation and factorization, the two main steps of the soft-decoding algorithm. We introduce an algorithmic transformation for reducing the iterations required in generating the interpolation polynomial and present efficient architectures by sharing computations. We also describe algorithmic transformations for further reducing the interpolation and factorization latency. An area efficient, folded-pipelined version of the interpolation architecture is also described. Finally, we present an example of a Reed-Solomon soft decoder utilizing the presented architectures, having a 250 Mbps throughput.

4. with R. K"otter, "Error-locating pairs for cyclic codes, " preprint Eindhoven-Link"oping, submitted for publication, March 1993.

A general decoding method for linear codes is investigated for cyclic codes.

The Koetter-Vardy algorithm is an algebraic soft-decision decoder for Reed-Solomon codes which is based on the Guruswami-Sudan list decoder. There are three main steps: 1) multiplicity calculation, 2) interpolation and 3) root finding. The Koetter-Vardy algorithm is challenging to implement due to the high cost of interpolation. We propose a VLSI-oriented improvement to the interpolation algorithm that uses a transformation of the received word to reduce the number of iterations. We show how to reduce the the memory requirements and give an efficient VLSI implementation for the Hasse derivative.

The Koetter-Vardy algorithm is an algebraic softdecision decoder for Reed-Solomon codes. The algorithm is based on an extension to the Guruswami-Sudan list-decoding algorithm with variable multiplicities that are assigned proportional to the reliabilities of the received symbols. There are three steps: (1) multiplicity calculation, (2) interpolation of a bivariate polynomial, and (3) finding the y-roots of this polynomial. A low-complexity algorithm for calculating the multiplicities is proposed. Simulation results indicate that the coding gain is dependent on the code rate and ranges from 0.25 dB to 4.25 dB with a practical upper limit of 1 1.5 dB, assuming binary phase shift keying and additive white Gaussian noise. Higher coding gains of between 2 dB and 6.8 dB can be achieved over a Rayleigh fading channel. The KV algorithm exhibits a performance-complexity tradeoff which is tunable by the choice of m max , n and k. The code parameters should be chosen carefully to take advantage of the "sweet spots" in the performance-complexity profile.