## Decoding Reed Solomon Codes beyond the Error-Correction Bound (1997)

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Citations: | 215 - 17 self |

### BibTeX

@MISC{Sudan97decodingreed,

author = {Madhu Sudan},

title = {Decoding Reed Solomon Codes beyond the Error-Correction Bound },

year = {1997}

}

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### Abstract

We present a randomized algorithm which takes as input n distinct points f(xi; yi)g n i=1 from F \Theta F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., yi = f (xi) for at least t values of i), provided t = \Omega (

### Citations

1941 |
The theory of error-correcting codes
- MacWilliams, Sloane
- 1977
(Show Context)
Citation Context ...n easily, this problem captures the bounded distance decoding problem for Reed Solomon codes. There is a rich history of work associated with this problem. The classical work of Berlekamp-Massey (cf. =-=[2, 9]-=-), corrects upto b(d , 1)=2c errors. Sidelnikov [11] and Dumer [4] have constructed algorithms which correct up to b(d , 1)=2c + c log n errors for any constant c [4, 11]. We give an algorithm that im... |

670 |
Completeness Theorems for NonCryptographic Fault-Tolerant Distributed Corn putation (extended abstract
- Goldwasser, Wigderson
- 1988
(Show Context)
Citation Context ...like" codes have occurred repeatedly. Examples include (1) In determining the hardness of the permanent on random instances [16, 7], (2) Fault tolerant computing in distributed computing environm=-=ents [3], and-=- (3) Many result involving \probabilistically-checkable proofs". (See survey by Feigenbaum [8] for a detailed look at many connections. ) In particular, in the case of applications to probabilist... |

439 | Algebraic Coding Theory - Berlekamp - 1968 |

254 |
Introduction to Coding Theory
- Lint
- 1992
(Show Context)
Citation Context ... codewords are the strings f(p(0); p(w); p(w 2 ); : : : ; p(w jF j1 ))g p , where w is somesxed primitive element of F and p ranges over all polynomials of degree at most d over F (see, for instance, =-=[18], page 8-=-6). The algorithmic tasks of relevance to this paper are the tasks of \errorcorrection " and \maximum-likelihood decoding". The problem of -errorcorrection for an [n; k; ]-code is dened for ... |

233 |
Some connections between nonuniform and uniform complexity classes
- Karp, Lipton
- 1980
(Show Context)
Citation Context ... easily presented), for which they show that the existence of small size maximum likelihood decoding circuits would imply the collapse of the polynomial hierarchy (using a result of Karp and Lipton's =-=[14]-=-). However the codes presented by Bruck and Naor do not have large distance. It still remains open if the maximum likelihood decoding problem is hard for any constant distance code. The Reed Solomon c... |

224 |
Tilborg. On the inherent intractability of certain coding problems (corresp
- Berlekamp, McEliece, et al.
- 1978
(Show Context)
Citation Context ...e another interesting question. Lastly we speculate on the complexity of the maximum-likelihood problem (or the nearest codeword problem). This problem is known to be NP-hard for general linear codes =-=[-=-5]. The hardness of the problem considered in [5] could be due to one of two reasons: (1) It is a maximum likelihood decoding problem rather than a -error correction problem; (2) The code is specied a... |

183 | Learning decision trees using the Fourier spectrum
- Kushilevitz, Mansour
- 1993
(Show Context)
Citation Context ...on problem has found interesting solutions for some errorcorrecting code. Thesrst, due to Goldreich and Levin [10], provides a solution for certain families of Hadamard Codes. Kushilevitz and Mansour =-=[15]-=-, provide a variant of this algorithm which applies to the same codes. The second instance involves a generalization of the codes and algorithm given by [10], and is due to [11]. Both the codes given ... |

107 |
Error correction of algebraic block codes
- Welch, Berlekamp
- 1986
(Show Context)
Citation Context ...for any constant c [4, 11]. We give an algorithm that improves over these results when k=n is su ciently small (i.e., less than 1=3). Our algorithm is motivated by an algorithm of Welch and Berlekamp =-=[14, 3]-=- which corrects b(d , 1)=2c + 1 errors. In this article we describe the algorithm of Welch and Berlekamp and use it motivate our decoding algorithm. We also describe a crucial intermediate step from [... |

85 | Learning polynomials with queries: The highly noisy case
- Goldreich, Rubinfeld, et al.
- 2000
(Show Context)
Citation Context ...or instance, is there an algorithm that can decode from more errors (than (d , 1)=2) when = k=n > 1=3? A nice target would be a decoding algorithm that works for ( ) 1 , p . In this case we know (cf. =-=[5, 10]-=-) that the number of codewords within a distance of ( )n is bounded by a polynomial in n. One does expect that the problem will become harder as ( ) ! 1, .Itwould be interesting to see if the problem ... |

72 |
A HardCore Predicate for any One-Way Function
- Goldreich, Levin
(Show Context)
Citation Context ... [11]. To the best of our knowledge, there are only two instances where the -reconstruction problem has found interesting solutions for some errorcorrecting code. Thesrst, due to Goldreich and Levin [=-=10]-=-, provides a solution for certain families of Hadamard Codes. Kushilevitz and Mansour [15], provide a variant of this algorithm which applies to the same codes. The second instance involves a generali... |

50 | Reconstructing algebraic functions from noisy data
- Ar, Lipton, et al.
- 1992
(Show Context)
Citation Context ...[12] corrects ( ( ),o(1))n errors in polynomial time, where ( )=1, 1 1+ , k 2 where = $r % 2 1 1 + , : 4 2 We also present the following two (hopefully) useful lower bounds on ( ): 1 Introduction 8 2 =-=[0; 1]-=- ( ) 1 , s 2 + 2 4 + 2 1 , p 2 : For integers n; k and q such that a nite eld of size q exists, the Reed Solomon codes are [n; k; d = n , k] codes over the alphabet F = GF(q) (the Galois eld of order ... |

46 | Highly resilient correctors for polynomials
- Gemmell, Sudan
- 1992
(Show Context)
Citation Context ...when t satises this property then there is a unique function f satisfying (1). A particularly simple algorithm for this case is given by Berlekamp and Welch [4] (see, for instance, Gemmell and Sudan [=-=9]-=-). In this paper we present an algorithm which solves the problem given above for t =sp nd). Notice that forsxed d, as n !1, the fraction of agreement required by our algorithm approaches 0 (and not 1... |

45 |
Polynomial factorization 1987-1991
- Kaltofen
- 1992
(Show Context)
Citation Context ...n Part 1 above and output all p such that y , p(x) divides Q. The polynomial Q can be factored in time polynomial in its degree using the algorithm of Kaltofen [7] or Grigoriev [6] (see also Kaltofen =-=[8]-=-). Proof: For Part 1, we observe as in the proof of Lemma 2 that for any i, the condition Q(xi;yi) = P jl qjlx j y l = 0 is a linear constraint on the unknowns qjl. Thus a solution satisfying (1) can ... |

34 | Algorithmic Complexity in Coding Theory and the Minimum Distance Problem - Vardy - 1997 |

34 | On the hardness of computing the permanent of random matrices
- Feige, Lund
- 1992
(Show Context)
Citation Context ...xity theory. It is in this area that numerous application for \Reed Solomon like" codes have occurred repeatedly. Examples include (1) In determining the hardness of the permanent on random insta=-=nces [16, 7], (2)-=- Fault tolerant computing in distributed computing environments [3], and (3) Many result involving \probabilistically-checkable proofs". (See survey by Feigenbaum [8] for a detailed look at many ... |

30 |
Bounded distance +1 soft-decision Reed-Solomon decoding
- Berlekamp
- 1996
(Show Context)
Citation Context ...for any constant c [4, 11]. We give an algorithm that improves over these results when k=n is su ciently small (i.e., less than 1=3). Our algorithm is motivated by an algorithm of Welch and Berlekamp =-=[14, 3]-=- which corrects b(d , 1)=2c + 1 errors. In this article we describe the algorithm of Welch and Berlekamp and use it motivate our decoding algorithm. We also describe a crucial intermediate step from [... |

29 |
New directions in testing. Distributed Computing and Cryptography
- Lipton
- 1991
(Show Context)
Citation Context ...xity theory. It is in this area that numerous application for \Reed Solomon like" codes have occurred repeatedly. Examples include (1) In determining the hardness of the permanent on random insta=-=nces [16, 7], (2)-=- Fault tolerant computing in distributed computing environments [3], and (3) Many result involving \probabilistically-checkable proofs". (See survey by Feigenbaum [8] for a detailed look at many ... |

26 |
The hardness of decoding linear codes with preprocessing
- Bruck, Naor
- 1990
(Show Context)
Citation Context ...is more standard. It would be nice to know which of the two causes is responsible for the hardness, since for all well-known codes, the -error correction problem seems to well-solved. Bruck and Naor [=-=6]-=- present a code (not well-known, but nevertheless easily presented), for which they show that the existence of small size maximum likelihood decoding circuits would imply the collapse of the polynomia... |

16 |
Factorization of polynomials over a finite field and the solution of systems of algebraic equations
- Grigoriev
(Show Context)
Citation Context ...polynomial Q obtained in Part 1 above and output all p such that y , p(x) divides Q. The polynomial Q can be factored in time polynomial in its degree using the algorithm of Kaltofen [7] or Grigoriev =-=[6]-=- (see also Kaltofen [8]). Proof: For Part 1, we observe as in the proof of Lemma 2 that for any i, the condition Q(xi;yi) = P jl qjlx j y l = 0 is a linear constraint on the unknowns qjl. Thus a solut... |

13 | A polynomial-time reduction from bivariate to univariate integral polynomial factorization
- Kaltofen
- 1982
(Show Context)
Citation Context ...imply factor the polynomial Q obtained in Part 1 above and output all p such that y , p(x) divides Q. The polynomial Q can be factored in time polynomial in its degree using the algorithm of Kaltofen =-=[7]-=- or Grigoriev [6] (see also Kaltofen [8]). Proof: For Part 1, we observe as in the proof of Lemma 2 that for any i, the condition Q(xi;yi) = P jl qjlx j y l = 0 is a linear constraint on the unknowns ... |

4 |
Two algorithms for the decoding of linear codes
- Dumer
- 1989
(Show Context)
Citation Context ... for Reed Solomon codes. There is a rich history of work associated with this problem. The classical work of Berlekamp-Massey (cf. [2, 9]), corrects upto b(d , 1)=2c errors. Sidelnikov [11] and Dumer =-=[4]-=- have constructed algorithms which correct up to b(d , 1)=2c + c log n errors for any constant c [4, 11]. We give an algorithm that improves over these results when k=n is su ciently small (i.e., less... |

4 |
Decoding Reed-Solomon codes beyond (d \Gamma 1)=2 errors and zeros of multivariate polynomials. Problems of Information Transmission
- Sidelnikov
- 1994
(Show Context)
Citation Context ...ng problem for Reed Solomon codes. There is a rich history of work associated with this problem. The classical work of Berlekamp-Massey (cf. [2, 9]), corrects upto b(d \Gamma 1)=2c errors. Sidelnikov =-=[11]-=- and Dumer [4] have constructed algorithms which correct up to b(d \Gamma 1)=2c + c log n errors for any constant c [4, 11]. We give an algorithm that improves over these results when k=n is sufficien... |

3 |
Decoding Reed-Solomon codes beyond (d − 1)/2 and zeros of multivariate polynomials
- Sidelnikov
- 1994
(Show Context)
Citation Context ...ecoding problem for Reed Solomon codes. There is a rich history of work associated with this problem. The classical work of Berlekamp-Massey (cf. [2, 9]), corrects upto b(d , 1)=2c errors. Sidelnikov =-=[11]-=- and Dumer [4] have constructed algorithms which correct up to b(d , 1)=2c + c log n errors for any constant c [4, 11]. We give an algorithm that improves over these results when k=n is su ciently sma... |

2 |
The Use of Coding Theory
- Feigenbaum
- 1995
(Show Context)
Citation Context ...nent on random instances [16, 7], (2) Fault tolerant computing in distributed computing environments [3], and (3) Many result involving \probabilistically-checkable proofs". (See survey by Feigen=-=baum [8]-=- for a detailed look at many connections. ) In particular, in the case of applications to probabilistically checkable proofs, it becomes useful to be able to characterize functions that are not close ... |

1 | Completeness theorems for non-cryptographic fault-tolerant distributed computation - Manuscript - 1996 |