Madhu Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-- 193, March 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(in a more general form) by Boneh [3] This algorithm explicitly lists all solutions to our decoding problem in polynomialtime whenever 2h n k, where k = log K log p min , and p min = min 1#i#n p i . We also remark that the result of [25] is a Lee norm analogue of Hamming norm results of [2, 7, 10, 11, 16, 19, 22, 24, 25, 26, 27] on noisy polynomial reconstruction problem and algebraic geometry codes list decoding. Finally, several possible cryptographic applications of noisy polynomial reconstruction have been outlined in [14, 15] It would be interesting to study possible cryptographic applications of noisy Chinese ....

M. Sudan, `Decoding of Reed Solomon codes beyond the errorcorrection bound', J. Complexity , 13 (1997), 180--193.

....set of trajectories we construct the Most Powerful Unfalsified Model. The bivariate polynomial is then derived from a specific representation of the MPUM. 1 Introduction and problem statement In this paper we present a behavioral interpretation of the list decoding approach that was proposed in [1]. We concentrate on the behavioral elements and keep the coding details to a minimum that is just su#cient to appreciate the lines of thought. A more elaborate treatment will be presented in a forthcoming paper. The paper is a follow up of [2, 3, 4] and works out the suggestion made there to put ....

....possibility here. The codeword c is transmitted through a channel where errors may occur so that the received word r is not necessarily equal to the transmitted codeword c. The decoding problem consists of reconstructing the original polynomial m(#) from the received word r. In a recent paper, [1], a list decoding scheme based on bi variate interpolation was proposed. In list decoding a list of possible polynomials m(#) is derived from the received word. The idea put forward in [1] is as follows. Denote the received word by r = # 1 , # n ) Let Q(#, #) F[#, #] be a bivariate ....

[Article contains additional citation context not shown here]

M. Sudan. Decoding of Reed-Solomon codes beyond the error correction bound. Journal of Complexity, 13:180--193, 1997. 12

....codes, with a wide range of applications. A lot of research has been directed towards e#cient decoding of Reed Solomon codes. An important problem in the decoding of Reed Solomon codes is one of decoding beyond the error correction radius. A breakthrough in this area was achieved by Sudan [1] and later extended by Guruswami and Sudan [2] 3] The algorithm of Guruswami Sudan allows the correction of any fraction of # # R erroneous positions, where R is the rate of the ReedSolomon code. This exceeds the earlier known correction ability of a fraction of (1 R) 2 erroneous positions, ....

M.Sudan, "Decoding of reed-solomon codes beyond the error correction bound", J.Complexity, vol. 12, pp. 180--193, 1997.

....the highthroughput implementation with a block pipelining depth of 5, increases throughput by 68 . In addition, the critical path of both the low latency and the high throughput implementation is smaller than that of previously proposed architectures. 1. INTRODUCTION List decoding algorithms [1], 2] and their soft decoding [3] counterparts extend the coding gains obtained from Bounded Minimum Distance (BMD) decoding [4] of Reed Solomon codes. These gains, obtained by decoding beyond half the minimum distance of the code, come at the price of increased complexity compared to BMD ....

....the steps involved in the interpolation algorithm. A block diagram of the process of soft decoding is shown in Fig. 1. The reader is referred to [3] for more details regarding the Koetter Vardy front end. The Guruswami Sudan list decoding process using bivariate polynomials Q(X;Y ) is explained in [1], 2] We now lay out the basic terminology required to understand the interpolation process. For an [n; k] RS code, the input to the list decoder is a set of s ordered triples of the form (x i ; y i ; m i ) where 0 i s. The ordered pair (x i ; y i ) termed an interpolation point is a set of ....

M. Sudan, "Decoding of Reed-Solomon codes beyond the error-correction bound," Journal of Complexity, vol. 13, no. 1, pp. 180-193, March 1997.

....all the codewords within a specified distance from a received vector. The decoder output is a list of possible codewords since the specified distance, which represents the error correction capability, is usually chosen to be larger than half the minimum distance. Based on an algorithm by Sudan [4], Guruswami and Sudan [5] discovered a polynomial time list decoding algorithm which uses interpolation and factorization of bivariate polynomials in order to decode errors upto In vJ symbols for a [n, k] code, where n is the length of the transmitted codeword and k is the number of data symbols ....

....matrix where each row represents a code symbol position i and hence a code evaluation point xi. The elements in the row represent the probabilities of having received each of the possible 2 q symbols, in the code symbol position, corresponding to that row. As a simple example, consider a [7, 4] RS code over GF(8) with primitive element 3 . Let the code evaluation point for the i tn position be 3 i where 0 i 7. Suppose that the transmitted codeword is 3 2, 3 2, 3 5, 3 5, 3 , 3 2, 3 4 , and the computed (7 x 8) size reliability matrix corresponding to the received information is: ....

[Article contains additional citation context not shown here]

M. Sudan, "Decoding of Reed-Solomon codes beyond the error-correction bound," Journal of Complexity, vol. 13, no. 1, pp. 180-193, March 1997.

....interpolation problem is also presented. I. INTRODUCTION Reed Solomon (RS) codes are the most widely used errorcorrecting codes in digital communications and data storage. Recently, major breakthroughs have been achieved in improving the error correction capability of Reed Solomon codes. Sudan [13] and Guruswami Sudan [4] have discovered a listdecoding algorithm, which can correct up to symbol errors. Their methods were later extended by Koetter and Vardy [6] to an algebraic soft decision decoding algorithm for Reed Solomon codes. Both list decoding and algebraic soft decision ....

....points ) # # 79 ) T may be thought of as the transmitted symbols. Then Bezout s theorem implies that any nonzero polynomial Z T [ h K is divisible by d79 . This leads to the interpolationbased decoding algorithms of [4, 6, 8, 13]. The central idea of all these decoding algorithms is to construct a polynomial that passes through a prescribed set of points X # t # # and # # , with prescribed multiplicities b K b K ....

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," J. Complexity, vol. 12, pp. 180--193, 1997.

....ym ) a noise parameter # 0, a degree bound d, and a threshold # 0. We have to find every degree d polynomial h satisfying Such a polynomial will be called a (#, #) fitting polynomial. In the noise free case (i.e. # 0) this is exactly the list decoding problem for which Sudan gave [17] an algorithm that runs in time poly(m,d) and works even for subconstant values of #. However the case we are interested in, i.e. when # 0, the problem as stated becomes ill posed and we need to reformulate it carefully. An immediate objection to Definition 1 is that there could be uncountably ....

.... vision called the Hough transform invented in 1962 to detect particle tracks in bubble chamber images takes precisely such a brute force approach [5, 10, 11] However, it is not clear a priori whether an exponential search is inherent, especially in view of Sudan s list decoding algorithm [17], which solves the problem in polynomial (in d, 1 #) time in the noise free case. In fact, extending Sudan s algorithm was the original motivation of our paper. Remark : Actually, the reconstruction problem in which the data comes from a mixture model was solved even earlier for the noiseless ....

[Article contains additional citation context not shown here]

M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1): 180--193, March 1997.

....error correcting codes in digital communications and data storage. Traditional hard decision decoders correct up to symbol errors for an (n; k) Reed Solomon code. Recently, several breakthroughs have been achieved in improving the error correction capability of a Reed Solomon decoder. Sudan [8] and Guruswami Sudan [3] discovered a list decoding algorithm, which can correct up to n p nk symbol errors. This listdecoding method was later extended by Koetter and Vardy [4] to an algebraic soft decision decoding algorithm, which significantly outperforms hard decision list decoding. Both ....

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," Journal of Complexity, vol. 12, pp. 180--193, 1997.

....decoders for Reed Solomon codes of length n and dimension k can correct up to t = #d min 2# errors where d min = n k 1) is the minimum distance of the code. Recently, a new class of list decoding algorithms has been introduced that can sometimes correct an even greater number of errors [2,3]. The list decoding problem is to find the set of codewords at a Hamming distance of t # from the received word. If t # d min 2 there might not be a unique codeword so the decoder returns a list of candidate codewords. The Guruswami Sudan (GS) list decoding algorithm has t # as large as n # nk ....

....can be used to decode Reed Solomon codes. In the presence of noise (e 0) the interpolation polynomial will pass through some points that are not part of the codeword. The GS algorithm ensures that under certain conditions, the codeword polynomial lives inside the interpolation polynomial [2, 3]. The GS algorithm is an interpolation based list decoder with two main steps: 1. Interpolation Step: Given the set of points L and a positive integer m, compute P (x, y) GF(Q) x, y] 0 of minimal (1, k 1) weighted degree that passes through all the points in L with multiplicity m. 2. ....

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," Journal of Complexity, vol. 13, no. 1, pp. 180--193, 1997.

....than one possible decoding. The problem of nding all possible decodings in such cases is called the list decoding problem. For Reed Solomon codes the number of possible decodings is constant as long as the number of errors is less than N NK (see [GRS95] Using techniques introduced by Sudan [Su97] and subsequently improved in [RR00] and [GS99] one can produce this list with a randomized polynomial time algorithm. In its most ecient form [GS99] the algorithm can handle up to dN NK 1e errors. We could use this list decoding algorithm for our purposes as follows. We pick a Reed Solomon ....

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

....the decoder may attempt to generate a list of the consistent codewords, and then choose among these according 12 to some criterion. This is known as list decoding (see Elias [6] and the references given there) A new technique for list decoding applied to Reed Solomon codes was invented by Sudan [20]. The essential idea is that a received word is used to create a set of points and a 2 variable polynomial interpolating these points is sought. In more detail, let F be a eld and let f(x k ; y k ) 2 F ; k = 1; ng be a set of distinct points. Given integers d; t, Sudan s algorithm ....

M. Sudan, Decoding of Reed-Solomon codes beyond the error correction bound, J. of Complexity, 13 (1997), 180-193. 21

....a small list of codewords which includes every codeword that agrees with the corrupted codeword on a fraction of positions. Several constructions achieving this goal are known by now. These include certain Reed Solomon (RS) and Algebraicgeometric (AG) codes, which achieve a rate of ) [21, 17, 10]. In addition, AG codes can achieve this property over an alphabet of constant size (speci cally, O(1= size) in comparison, the alphabet size of RS codes is at least as large as their blocklength. With respect to those parameters, AG codes are superior to RS codes. Unfortunately, list ....

....C from the following components: i) A Reed Solomon code C1 with rate ( over an alphabet of size q1 = O(n) which is (1=2; O(1= O(1= list recoverable in O(n (log n= time. The existence of a code with these properties follows from the Reed Solomon list decoding algorithms in [21, 10] and their fast implementations from [15, 14] ii) A pseudolinear code C2 with rate ( blocklength d, and alphabet size q2 = O(1= which is ( 1 =2)d; O(1= list decodable. Such a code exists and has an ecient O(d ) time probabilistic construction by the results of [9] see also ....

[Article contains additional citation context not shown here]

Madhu Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

....Classical decoders for Reed Solomon codes of length n and dimension k can correct t = #d min 2# errors where d min = n k 1) is the minimum distance of the code. Recently, a new class of list decoding algorithms for ReedSolomon codes were introduced that can correct t # t errors [1] [2] The list decoding problem is to find the set of codewords at a Hamming distance of t # from the received word. Since there might not be a unique codeword at a distance t # d min 2, the decoder returns a list of candidate codewords. The Guruswami Sudan (GS) algorithm can correct up to n ....

....can be used to decode ReedSolomon codes. In the presence of noise (e 0) the interpolation polynomial will pass through some points that are not part of the codeword. The GS algorithm ensures that under certain conditions, the codeword polynomial lives inside the interpolation polynomial [1] [2] The GS algorithm is an interpolation based list decoder with two main steps: 1. Interpolation Step: Given the set of points L n and a positive integer m, compute P (x, y) GF(Q) x, y] 0 of minimal (1, k 1) weighted degree that passes through all the points in L n with multiplicity m. ....

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," Journal of Complexity, vol. 13, no. 1, pp. 180--193, 1997.

....systematic encodings) The evaluation map method is useful because it provides insight leading to interpolation based decoding algorithms. B. Decoding as an Interpolation Problem In this section, we describe the Guruswami Sudan (GS) algorithm for the hard decision decoding of Reed Solomon codes [5, 6] which is the basis for the Koetter Vardy algorithm. For proofs, please see [5 7] To formally state the algorithm, we need to define the weighted degree of a bivariate polynomial. Let P(x,y) j=0 p i, j x be a bivariate polynomial over GF(Q) and let w x and w y be nonnegative real ....

....leading to interpolation based decoding algorithms. B. Decoding as an Interpolation Problem In this section, we describe the Guruswami Sudan (GS) algorithm for the hard decision decoding of Reed Solomon codes [5, 6] which is the basis for the Koetter Vardy algorithm. For proofs, please see [5 7]. To formally state the algorithm, we need to define the weighted degree of a bivariate polynomial. Let P(x,y) j=0 p i, j x be a bivariate polynomial over GF(Q) and let w x and w y be nonnegative real numbers. The (w x , w y ) weighted degree of P(x,y) deg (w x ,w y ) P) is defined ....

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," J. Complexity, vol. 13, no. 1, pp. 180--193, 1997.

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

No context found.

M. SUDAN. Decoding of Reed-Solomon codes beyond the error-correction diameter. Proceedings of the 35th Annual Allerton Conference on Communication, Control and Computing, 1997.

No context found.

M. SUDAN. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

....size of the decoder s list, an improvement in the RR algorithm s stopping rule, a simplified treatment of the combinatorics of weighted monomial orders, and a proof of the monotonicity of the GS decoding radius as a function of the interpolation multiplicity. I. Introduction In 1997 Madhu Sudan [23], building on previous work of Welch Berlekamp [24] Ar et al. 1] and others, discovered a polynomial time algorithm for decoding certain low rate Reed Solomon (RS) codes beyond the classical d 2 error correcting bound. Two years later, Guruswami and Sudan [9] published a significantly ....

....of correcting virtually all patterns of 13 errors, despite having a conventional error correcting capability of only 12. Example 1 suggests that it might be possible to design a decoding algorithm for RS codes capable of correcting more than t 0 errors. The Guruswami Sudan list decoding algorithm [23,9] does just this. It is apolynomial time algorithm for correcting (in a certain sense) up to t GS errors, where t GS is the largest integer strictly less than n 1)n, i.e. t GS = n 1)n Conservatively, the time complexity is O(n ) where n is the code length and m is the ....

[Article contains additional citation context not shown here]

M. Sudan, "Decoding of Reed--Solomon Codes beyond the Error-Correction Bound," J. Complexity,vol. 13, pp. 180--193, 1997.

....previous work of Ar et al. 1] presented a polynomial time list decoding algorithm for Reed Solomon codes correcting more than LU DOQAYR errors, provided ( A . The exact description of the number of errors ; corrected by this algorithm is rather complicated and can be found in [28] or Figure 1. One lower bound on the number of errors corrected is 5 R H , thus achieving ; 2 R C . A more efficient list decoding algorithm, running in time , correcting the same number of errors has been given by Roth and Ruckenstein [23] For C , this algorithm ....

M. SUDAN. Decoding of Reed-Solomon codes beyond the error-correction diameter. Proceedings of the 35th Annual Allerton Conference on Communication, Control and Computing, 1997.

.... 2 32616 , and a degree parameter , find all univariate polynomials of degree at most # for all but at most values of ) 0 . We give an algorithm that solves this problem for , 10 , which improves over the previous best result [27], for every choice of and . Of particular interest is the case of 2 43 where the result yields the first asymptotic improvement in four decades [21] The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes ....

....benefits were achieved using the list decoding approach to recover from errors. The only improvements known over the algorithm of [21] were decoding algorithms due to Sidelnikov [25] and Dumer [6] which correct F K L O errors, i.e. achieve ;E L CPOQABR L 40 . Recently, Sudan [27], building upon previous work of Ar et al. 1] presented a polynomial time list decoding algorithm for Reed Solomon codes correcting more than LU DOQAYR errors, provided ( A . The exact description of the number of errors ; corrected by this algorithm is rather complicated and ....

[Article contains additional citation context not shown here]

M. SUDAN. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

....y) Thus nding such a bivariate polynomials Q and factoring it, gives a small list of polynomials that includes all the candidates for output of the list decoding algorithm. By picking the degree of Q very carefully, one can improve its performance signi cantly to at least n 2kn errors (see [39] for a more complete analysis of the performance of this algorithm) The interesting aspect of the above algorithm is that it takes some very elementary algebraic concepts, such as unique factorization, Bezout s theorem, and interpolation, and makes algorithmic use of these concepts in developing ....

Madhu Sudan. Decoding of Reed-Solomon codes beyond the error-correction diameter. Proceedings of the 35th Annual Allerton Conference on Communication, Control and Computing, 1997.

.... Rubinfeld, and Sudan [16] Yet no ecient list decoding algorithms were found for codes of decent rate (constant, or even slowly vanishing rate such as R(n) n 1 for some 0) The rst list decoding algorithm correcting (n) errors for 2 for codes of constant rate was due to Sudan [38], who gave such an algorithm for Reed Solomon codes. The algorithm was subsequently extended to algebraic geometric codes by Shokrollahi and Wasserman [35] Yet these results did not decode up to the best known combinatorial bounds on list decoding radius; in fact, they did not correct more than ....

....denotes the Hamming distance. We now give a brief summary of the algorithmic ideas that led to the algorithm in [23] This chain of ideas includes the Welch Berlekamp algorithm [42, 5] an algorithm for a restricted decoding problem due to Ar et al. 1] and the list decoding algorithm of Sudan [38]. Traditional algorithms, starting with those of Peterson [32] attempt to explain y as a function of x. This part becomes explicit in the work of Welch Berlekamp [42, 5] see, in particular, the expositions in [13] or [37, Appendix A] where y is interpolated as a rational function of x, and ....

[Article contains additional citation context not shown here]

Madhu Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

No context found.

Madhu Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-- 193, March 1997.

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. J. Compl., 13:180-193, 1997.

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180--193, 1997. 12

No context found.

M. Sudan, `Decoding of Reed Solomon Codes Beyond The Error-Correction Bound,' Journal of Complexity, vol.13, (no.1), pp. 180-193, March 1997.

No context found.

M. Sudan, "Decoding of Reed-Solomon Codes beyond the Error-Correction Bound," Journal of Complexity, vol. 13, no. 1, pp. 180-193,1997.

No context found.

Madhu Sudan, Decoding of Reed Solomon Codes beyond the Error-Correction Bound. Journal of Complexity 13(1), pp. 180--193, 1997. 18

No context found.

M. Sudan, "Decoding of Reed--Solomon codes beyond the errorcorrection bound," J. Complexity, vol. 13, pp. 180--193, 1997.

No context found.

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," J.Complexity, vol. 12, pp. 180--193, 1997.

No context found.

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," Journal of Complexity, vol. 13, no. 1, pp. 180--193, 1997.

No context found.

M. Sudan, Decoding of Reed Solomon codes beyond the error-correction bound, Journal of Complexity 13 (1997) 180--193.

No context found.

Madhu Sudan, Decoding of Reed Solomon Codes beyond the ErrorCorrection Bound. Journal of Complexity 13(1), pp. 180--193, 1997.

No context found.

Madhu Sudan, Decoding of Reed Solomon Codes beyond the Error-Correction Bound. Journal of Complexity 13(1), pp. 180--193, 1997.

No context found.

M. Sudan, Decoding of Reed Solomon codes beyond the error-correction bound, Journal of Complexity, vol. 13(1), March 1997, pp. 180193.

No context found.

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," Journal of Complexity, vol. 13, no. 1, pp. 180--193, 1997. 25

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, Vol. 13 (1), pages 180--193, 1997.

No context found.

Madhu Sudan, Decoding of Reed Solomon Codes beyond the Error-Correction Bound. Journal of Complexity 13(1), pp. 180--193, 1997. 17

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180--193, 1997. Preliminary version in Proc. of FOCS'96.

No context found.

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," Journal of Complexity, vol. 13, no. 1, pp. 180--193, 1997.

No context found.

M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13, 1997.

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-- 193, 1997. Preliminary version in Proc. of FOCS'96.

No context found.

M. Sudan, "Decoding of Reed-Solomon codes beyond the error correction bound," J.Complexity, vol. 12, pp. 180--193, 1997.

No context found.

M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13, 1997.

No context found.

M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.

No context found.

M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity,13, 1997.

No context found.

M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180--193, 1997. Preliminary version in

No context found.

M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13, 1997.

No context found.

M. Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity, 13, 1997.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC