List Decoding Of Reed-Solomon Codes Research Articles

연판정 Reed-Solomon 리스트 디코딩을 위한 저복잡도 Interpolation 구조

Reed-Solomon (RS) codes are powerful error-correcting codes used in diverse applications. Recently, algebraic soft-decision decoding algorithm for RS codes that can correct the errors beyond the error correcting bound has been proposed. The algorithm requires very intensive computations for interpolation, therefore an efficient VLSI architecture, which is realizable in hardware with a moderate hardware complexity, is mandatory for various applications. In this paper, we propose an efficient architecture with low hardware complexity for interpolation in soft-decision list decoding of Reed-Solomon codes. The proposed architecture processes the candidate polynomial in such a way that the terms of X degrees are processed in serial and the terms of Y degrees are processed in parallel. The processing order of candidate polynomials adaptively changes to increase the efficiency of memory access for coefficients; this minimizes the internal registers and the number of memory accesses and simplifies the memory structure by combining and storing data in memory. Also, the proposed architecture shows high hardware efficiency, since each module is balanced in terms of latency and the modules are maximally overlapped in schedule. The proposed interpolation architecture for the (255, 239) RS list decoder is designed and synthesized using the DongbuHitek standard cell library, the number of gate counts is 25.1K and the maximum operating frequency is 200 MHz.

Error correcting codes for robust color wavelet watermarking

Abstract This article details the conception, design, development and analysis of invisible, blind and robust color image watermarking algorithms based on the wavelet transform. Using error correcting codes, the watermarking algorithms are designed to be robust against intentional or unintentional attacks such as JPEG compression, additive white Gaussian noise, low pass filter and color attacks (hue, saturation and brightness modifications). Considering the watermarking channel characterized by these attacks, repetition, Hamming, Bose Chaudhuri Hocquenghem and Reed-Solomon codes are used in order to improve the robustness using different modes and appropriate decoding algorithms. The article compares the efficiency of different type of codes against different type of attacks. To the best of our knowledge this is the first time that the effect of error-correcting codes against different attacks are detailed in a watermarking context in such a precise way: describing and comparing the effect of different classes of codes against different type of attacks. This article clearly shows that list decoding of Reed-Solomon codes using the algorithm of Sudan exhibits good performance against hue and saturation attacks. The use of error correcting codes in a concatenation mode allows the non-binary block codes to show good performance against JPEG compression, noise and brightness attacks.

Reliability-Based Forward Recursive Algorithms for Algebraic Soft-Decision Decoding of Reed--Solomon Codes

We propose efficient forward recursive algorithms for algebraic soft-decision list decoding of Reed-Solomon codes, which utilize channel reliability information, and outperform the Koetter-Vardy (KV) algorithm with lower decoding latency. We evaluate the performance of the proposed decoding algorithms on additive white Gaussian noise and partial response channels. Simulation results show that we can achieve better performance on a modified extended-extended partial response class 4 channel than on the best possible performance of the KV algorithm, as given by the asymptotic bound for high-rate codes.

Linear Diophantine Equations Over Polynomials and Soft Decoding of Reed–Solomon Codes

This paper generalizes the classical Knuth-Schoumlnhage algorithm computing the greatest common divisor (gcd) of two polynomials for solving arbitrary linear Diophantine systems over polynomials in time, quasi-linear in the maximal degree. As an application, the following weighted curve fitting problem is considered: given a set of points in the plane, find an algebraic curve (satisfying certain degree conditions) that goes through each point the prescribed number of times. The main motivation for this problem comes from coding theory, namely, it is ultimately related to the list decoding of Reed-Solomon codes. This paper presents a new fast algorithm for the weighted curve fitting problem, based on the explicit construction of a Groebner basis. This gives another fast algorithm for the soft decoding of Reed-Solomon codes different from the procedure proposed by Feng, which works in time (w/r) O(1)nlog2n, where r is the rate of the code, and w is the maximal weight assigned to a vertical line

Guessing More Secrets via List Decoding

We consider the following game introduced by Chung, Graham, and Leighton in [Chung et al. 01]. One player, **A** , picks _k_ > 1 secrets from a universe of _N_ possible secrets, and another player, **B** , tries to gain as much information about this set as possible by asking binary questions ƒ : [_N_] → {0, 1}. Upon receiving a question ƒ, **A** adversarially chooses one of her _k_ secrets, and answers ƒ according to it. In this paper we present an explicit set of 2^{<em>O(k)</em>}(log _N_) questions, along with a 2^{<em>O(k</em> ²)}(log²_N_) recovery algorithm that achieves **B**'s goal in this game. This, in particular, completely solves the problem for any constant number of secrets _k_. Our strategy is based on the list decoding of Reed-Solomon codes, and it extends and generalizes ideas introduced by Alon, Guruswami, Kaufman, and Sudan in [Alon et al. 02].

Extractor Codes

We study error-correcting codes for highly noisy channels. For example, every received signal in the channel may originate from some half of the symbols in the alphabet. Our main conceptual contribution is an equivalence between error-correcting codes for such channels and extractors. Our main technical contribution is a new explicit error-correcting code based on Trevisan's extractor that can handle such channels, and even noisier ones. Our new code has polynomial-time encoding and polynomial-time soft-decision decoding. We note that Reed-Solomon codes cannot handle such channels, and our study exposes some limitations on list decoding of Reed-Solomon codes. Another advantage of our equivalence is that when the Johnson bound is restated in terms of extractors, it becomes the well-known Leftover Hash Lemma. This yields a new proof of the Johnson bound which applies to large alphabets and soft decoding. Our explicit codes are useful in several applications. First, they yield algorithms to extract many hardcore bits using few auxiliary random bits. Second, they are the key tool in a recent scheme to compactly store a set of elements in a way that membership in the set can be determined by looking at only one bit of the representation. Finally, they are the basis for the recent construction of high-noise, almost-optimal rate list-decodable codes over large alphabets.

Behavioral Models for List Decoding

Recently it has been shown that list decoding of Reed-Solomon codes may be translated into a bivariate interpolation problem. The data consist of pairs in a finite field and the aim is to find a bivariate polynomial that interpolates the given pairs and is minimal with respect to some criterion. We present a systems theoretic approach to this interpolation problem. With the data points we associate a set of time series, also called trajectories. For this set of trajectories we construct the Most Powerful Unfalsified Model (MPUM). This is the smallest possible model that explains these trajectories. The bivariate polynomial is then derived from a specific polynomial representation of the MPUM.

Efficient root-finding algorithm with application to list decoding of algebraic-geometric codes

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 previous work by M. Shokrollahi and H. Wasserman (see ibid., vol.45, p.432-7, March 1999) a list-decoding procedure for Reed-Solomon codes was generalized to algebraic-geometric codes. Recent work by V. Guruswami and M. Sudan (see ibid., vol.45, p.1757-67, Sept. 1999) 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. R. Roth and G. Ruckenstein (see ibid., vol.46, p.246-57, Jan. 2000) proposed an efficient 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 present an efficient algorithm for finding the roots of univariate polynomials over function fields. Based on the extended algorithm, we give an efficient list-decoding algorithm for algebraic-geometric codes.