## Improved Decoding of Reed-Solomon and Algebraic-Geometry Codes (1999)

### Cached

### Download Links

- [theory.lcs.mit.edu]
- [gladstone.systems.caltech.edu]
- [theory.csail.mit.edu]
- [theory.csail.mit.edu]
- [theory.lcs.mit.edu]
- [people.csail.mit.edu]
- [people.csail.mit.edu]
- [people.csail.mit.edu]
- [theory.lcs.mit.edu]
- [theory.lcs.mit.edu]
- [theory.lcs.mit.edu]
- [theory.lcs.mit.edu]
- [people.csail.mit.edu]
- [theory.lcs.mit.edu]
- [theory.lcs.mit.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | IEEE TRANSACTIONS ON INFORMATION THEORY |

Citations: | 261 - 41 self |

### BibTeX

@ARTICLE{Guruswami99improveddecoding,

author = {Venkatesan Guruswami and Madhu Sudan},

title = {Improved Decoding of Reed-Solomon and Algebraic-Geometry Codes },

journal = {IEEE TRANSACTIONS ON INFORMATION THEORY},

year = {1999},

volume = {45},

pages = {1757--1767}

}

### Years of Citing Articles

### OpenURL

### Abstract

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

### Citations

2068 |
The theory of error correcting codes
- MacWilliams, Sloane
- 1988
(Show Context)
Citation Context ...well-known: in provement for the case of algebraic-geometric codes extends particular the classical algorithms of Berlekamp and Massey the methods of [19] and improves upon their bound for every (see =-=[14]-=- for a description) achieve such running time bounds. choice ofnandd0. We also present some other consequences of It is also easily seen that 2then there may exist our algorithm including a solution t... |

974 |
A course in computational algebraic number theory. Graduate Texts in Mathematics, 138
- Cohen
- 1993
(Show Context)
Citation Context ... coefficients inF[X]). ThereforeT2F[X], and 2dxdywheredx,dyare the degrees ofQinxandyrespectively. This bound on the degree ofTfollows easily from the definition of the discriminant (see for instance =-=[5]-=-), and it is also easy to prove that the discriminantTcan be computed inO(dxd4y)field operations. Next we find an2Fsuch thatT()6=0. This can be done deterministically by trying out an arbitrary set of... |

461 |
Algebraic Coding Theory
- Berlekamp
- 1968
(Show Context)
Citation Context ...em in polynomial time, as long as 2(i.e. achieves=(1�)=2). Faster algorithms, with running timeO(N2)or better, are also well-known: in particular the classical algorithms of Berlekamp and Massey (see =-=[2, 19]-=- for a description) achieve such running time bounds. It is also easily seen that 2then there may exist several different codewords within distanceeof a received word, and so the decoding algorithm ca... |

441 | Theory and Practice of Error Control Codes - Blahut - 1983 |

381 | A hard-core predicate for all one-way functions
- Goldreich, Levin
- 1989
(Show Context)
Citation Context ...cted a number of errors that were of the form `(n) = ( 1 2 + o(1))ffi(n) for some families of Reed-Solomon codes. This problem was introduced to the computer science literature by Goldreich and Levin =-=[14]-=- who gave a 1 This applies to bounds that apply for all codes, regardless of their alphabet size. For small alphabets, eg. for binary codes, a better bound can be proven. Since our primary focus is Re... |

359 |
Algebraic Function Fields and Codes
- Stichtenoth
- 1993
(Show Context)
Citation Context ...s up toe<n�p2n(n�d)�g+1errors. 4.1 Definitions An algebraic-geometry code is built over a structure termed an algebraic function field. Definitions and basic properties of these codes can be found in =-=[15, 26]-=-; for purposes of self-containment and ease of exposition, we now develop a slightly different notation to express our results. An algebraic function field is described by a six-tupleA=(Fq;X;X;K;g;ord... |

262 | Introduction to Coding Theory - Lint - 1982 |

220 | Decoding of Reed-Solomon codes beyond the error-correction bound
- Sudan
- 1997
(Show Context)
Citation Context ...stant, or even slowly vanishing rate such as R(n) = n \Gamma 1+" for some " ? 0). The first list decoding algorithm correcting ffffi(n) errors for ff ? 1 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 ... |

147 | Improved low-degree testing and its applications
- Arora, Sudan
(Show Context)
Citation Context ...any simple way to decode Reed-Muller codes up to their list decoding radius, or for that matter even beyond half the distance for all rates, and this remains an interesting open question. However, in =-=[2, 40]-=-, using clever reductions to the univariate case, an algorithm to list decode Reed-Muller codes well beyond half the distance is presented for codes of low rate. A consequence of Theorem 2 is that, fo... |

115 | Algbraic soft-decision decoding of reed-solomon codes
- Koetter, Vardy
- 2003
(Show Context)
Citation Context ... as an intermediate step with a suitable choice of weights, one gets an algorithm that decodes algebraic-geometric codes beyond half the minimum distance for every value of rate. In fact, as noted in =-=[27]-=-, a careful choice of weights enables decoding up to the combinatorial bound on list decoding radius. 2.2 Other algorithmic results A rich body of algorithmic results concerning list decoding have app... |

109 |
Error-correction for algebraic block codes
- Welch, Berlekamp
- 1986
(Show Context)
Citation Context ...ta (\Delta ; \Delta ) 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" ... |

98 | List decoding for noisy channels
- Elias
- 1957
(Show Context)
Citation Context ... in t 0 ? t places, the decoder may simply throw it hands up in the air and cite the above paragraph. Or, in an alternate notion of decoding, called list decoding, proposed in the late 1950s by Elias =-=[10]-=- and Wozencraft [43], the decoder could try to output a list of codewords within distance t 0 of the received vector. If t 0 is not much larger than t and the errors are caused by a probabilistic (non... |

92 | Learning polynomials with queries: The highly noisy case
- Goldreich, Rubinfeld, et al.
(Show Context)
Citation Context ...abet size. 2shighly efficient randomized list decoding algorithm for Hadamard codes, when the received vector was given implicitly. This work led to some extensions by Goldreich, Rubinfeld, and Sudan =-=[16]-=-. Yet no efficient list decoding algorithms were found for codes of decent rate (constant, or even slowly vanishing rate such as R(n) = n \Gamma 1+" for some " ? 0). The first list decoding algorithm ... |

81 |
Lower bounds to error probability for coding on discrete memoryless channels
- Shannon, Gallager, et al.
- 1967
(Show Context)
Citation Context ...d coding theory. Elias [10] used this notion to get a better handle on the error-exponent in the strong forms of Shannon's coding theorem. The notion also plays a dominant role in the Elias-Bassalygo =-=[34, 4]-=- upper bound on the rate of a code as a function of its relative distance. Through the decades the notion has continued to be investigated in a combinatorial context; and more recently has seen a spur... |

66 | Efficient checking of polynomials and proofs and the hardness of approximation problems - Sudan - 1992 |

66 | Generalized Minimum Distance Decoding - Forney - 1966 |

56 |
Error correcting codes for list decoding
- Elias
- 1991
(Show Context)
Citation Context ...de towards resolving it in [25, 20, 18]. ?From the point of usage, it is more useful to compare the rate of a code with its list decoding radius. This question has been investigated over the years by =-=[6, 7, 45, 11, 20]-=-. It follows from the converse to Shannon's coding theorem that a q-ary code of relative list decoding radius `(n) has rate at most R(n) ss 1 \GammasHq(`(n)). The above mentioned works show that there... |

56 | List Decoding of Error-Correcting Codes - Guruswami - 2004 |

55 |
Encoding and error-correction procedures for the BoseChauduri codes
- Peterson
- 1960
(Show Context)
Citation Context ...erlekamp 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 interpolate... |

51 | List decoding algorithms for certain concatenated codes
- Guruswami, Sudan
(Show Context)
Citation Context ...ate what the algorithm achieves for a general set of weights. For this part, we will just assume that the weight vector is an arbitrary vector of non-negative reals, and get the following: Theorem 4 (=-=[23, 24]-=-) Given vectors x; y 2 F n q , a weight vector w 2 R n *0 , and a real number " ? 0, a list of all polynomials p 2 F k q [x] satisfying P ijp(xi)=yi wi ? q k(" + Pn i=1 w 2 i ) can be found in time po... |

50 | Reconstructing algebraic functions from noisy data
- Ar, Lipton, et al.
- 1992
(Show Context)
Citation Context ...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... |

50 | Efficient decoding of Reed-Solomon codes beyond half the minimum distance
- Roth, Ruckenstein
- 2000
(Show Context)
Citation Context ...rrected by this algorithm, see [23] or Figure 1.) A more efficient list decoding algorithm, running in timeO(n2log2n), correcting the same number of errors has also been given by Roth and Ruckenstein =-=[17]-=-. For!0, this algorithm corrects an error rate!1, thus allowing for nearly twice as many errors as the classical approach. For codes of rate greater than1=3, however, this algorithm does not improve o... |

46 | Extractor codes - Ta-Shma, Zuckerman |

45 |
Polynomial factorization 1987–1991
- Kaltofen
- 1992
(Show Context)
Citation Context ...lds in time polynomial in the size of the field or probabilistically in time polynomial in the logarithm of the size of the field and can also be solved deterministically over the rationals and reals =-=[10, 12, 13]-=-. Thus our algorithm ends up solving the curve-fitting problem over fairly general fields. It is interesting to contrast our algorithm with results which show bounds on the number of codewords that ma... |

43 | Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization
- Kaltofen
- 1985
(Show Context)
Citation Context ...tion some of the significant advances made in the complexity of factoring multivariate polynomials that were made in the 1980's. These algorithms, discovered independently by Grigoriev [17], Kaltofen =-=[26]-=-, and Lenstra [28], form the technical foundations of the decoding algorithm above. Modulo these algorithms, the decoding algorithm and its proof rely only on elementary algebraic concepts. Exploiting... |

43 | Expander-based constructions of efficiently decodable codes - Guruswami, Indyk - 2001 |

42 | List decoding of algebraic-geometric codes
- Shokrollahi, Wassermann
- 1999
(Show Context)
Citation Context ... 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 ffi(n)=2 errors for any code of rate greater than 1=3. The... |

42 |
List Decoding. Quarterly Progress Report
- Wozencraft
- 1958
(Show Context)
Citation Context ...the decoder may simply throw it hands up in the air and cite the above paragraph. Or, in an alternate notion of decoding, called list decoding, proposed in the late 1950s by Elias [10] and Wozencraft =-=[43]-=-, the decoder could try to output a list of codewords within distance t ′ of the received vector. If t ′ is not much larger than t and the errors are caused by a probabilistic (non-malicious) channel,... |

41 | Chinese remaindering with errors
- Goldreich, Ron, et al.
(Show Context)
Citation Context ...y mentioned the works that addressed the question of more efficient implementations of the list decoding algorithms for Reed-Solomon and algebraic-geometric codes from [23]. Goldreich, Ron, and Sudan =-=[15]-=- considered the question of list decoding a number-theoretic code called the Chinese Remainder code (henceforth, CRT code). Here, the messages are identified with integers m in the range 0 ^ m ! K and... |

41 |
Salil Vadhan. Pseudorandom generators without the XOR lemma
- Sudan, Trevisan
- 2001
(Show Context)
Citation Context ...any simple way to decode Reed-Muller codes up to their list decoding radius, or for that matter even beyond half the distance for all rates, and this remains an interesting open question. However, in =-=[2, 40]-=-, using clever reductions to the univariate case, an algorithm to list decode Reed-Muller codes well beyond half the distance is presented for codes of low rate. A consequence of Theorem 2 is that, fo... |

39 | Combinatorial bounds for list decoding
- Guruswami, H̊astad, et al.
(Show Context)
Citation Context ...02139, USA. madhu@mit.edu. 1s1 Combinatorics of list decoding We start by defining the notion of the list decoding radius of an (infinite family of) codes. This notion is adapted from a definition in =-=[20]-=-, who term it the "polynomial list decoding radius". Definition 1 A family of codes C has a list decoding radius L : Z + ! Z + if there exists a polynomial p(\Delta ) such that for every code C 2 C of... |

36 | Algorithmic complexity in coding theory and the minimum distance problem
- Vardy
- 1997
(Show Context)
Citation Context ... extension of the algorithm of Section 2 is to the case of weighted curve fitting. This case is somewhat motivated by a decoding problem called the soft-decision decoding problem havePni=1�rwi+1 (see =-=[26]-=- for a formal description), as one might use the reliability information on the individual symbols in the received word more flexibly by encoding them appropriately as the weights below instead of dec... |

32 |
A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes
- Feng, Tzeng
- 1991
(Show Context)
Citation Context ... decoding algorithms for alternant codes given in classical texts seem to correctd0=2errors. For the more restricted BCH codes, there are algorithms that decode beyond half the designed distance (cf. =-=[9]-=- and also [4, Chapter 9]). 3.2 Errors and Erasures decoding The algorithm of Section 2 is also capable of dealing with other notions of corruption of information. A much weaker notion of corruption (t... |

30 | Finding Smooth integers in short intervals using CRT decoding
- Boneh
(Show Context)
Citation Context ...n indicate that such a code can be list decoded with small lists up to about n \Gammasp kn errors. Goldreich et al. [15] initiated the study of list decoding CRT codes and this was continued in Boneh =-=[8]-=-. However, these algorithms corrected only about n \Gammas\Omegas\Gamma q kn log pn log p1 \Deltaserrors and therefore their performance was poor when the pi's had widely different magnitudes. Subsequ... |

29 |
Bounded distance + 1 soft decision Reed-Solomon coding
- Berlekamp
- 1996
(Show Context)
Citation Context ...ta (\Delta ; \Delta ) 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" ... |

28 | Soft-decision decoding of Chinese Remainder Codes
- Guruswami, Sahai, et al.
- 2000
(Show Context)
Citation Context ... these algorithms corrected only about n \Gammas\Omegas\Gamma q kn log pn log p1 \Deltaserrors and therefore their performance was poor when the pi's had widely different magnitudes. Subsequently, in =-=[22]-=-, it was realized that algebraic and number-theoretic codes can be unified under the umbrella of ideal-based codes. Loosely speaking, the messages of an ideal-based code are all elements of small "siz... |

27 |
Algebraic geometry codes
- Høholdt, Lint, et al.
- 1998
(Show Context)
Citation Context ...s up toe<n�p2n(n�d)�g+1errors. 4.1 Definitions An algebraic-geometry code is built over a structure termed an algebraic function field. Definitions and basic properties of these codes can be found in =-=[15, 26]-=-; for purposes of self-containment and ease of exposition, we now develop a slightly different notation to express our results. An algebraic function field is described by a six-tupleA=(Fq;X;X;K;g;ord... |

19 | Computing roots of polynomials over function fields of curves,” unpublished manuscript
- Gao, Shokrollahi
- 1998
(Show Context)
Citation Context ... step in O(n 2 ) time, when the output list size is a constant [31, 33]. Similar running times are also known for the root finding problem (which suffices for the second step in the algorithms above) =-=[3, 12, 29, 31, 33, 44]-=-. Together these algorithms lead to the possibility that a good implementation of list decoding may actually even be able to compete with the classical Berlekamp-Massey decoding algorithm in terms of ... |

18 |
Highly resilient correctors for multivariate polynomials
- Gemmell, Sudan
- 1992
(Show Context)
Citation Context ...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 this leads to the efficient decoding. (Specifically a rational function a(x)=b(x) can be computed such that for every i 2... |

18 |
Bounds on list decoding of MDS codes
- Justesen, Høholdt
(Show Context)
Citation Context ...f non-linear codes which match this bound. The question of whether the bound is the best one can prove for linear codes remains open, though significant progress has been made towards resolving it in =-=[25, 20, 18]-=-. ?From the point of usage, it is more useful to compare the rate of a code with its list decoding radius. This question has been investigated over the years by [6, 7, 45, 11, 20]. It follows from the... |

17 |
Factorization of polynomials over a finite field and the solution of systems of algebraic equations
- Grigoriev
- 1986
(Show Context)
Citation Context ...od point to mention some of the significant advances made in the complexity of factoring multivariate polynomials that were made in the 1980's. These algorithms, discovered independently by Grigoriev =-=[17]-=-, Kaltofen [26], and Lenstra [28], form the technical foundations of the decoding algorithm above. Modulo these algorithms, the decoding algorithm and its proof rely only on elementary algebraic conce... |

16 | Decoding Hermitian codes with Sudan’s algorithm
- Nielsen, Høholdt
- 1999
(Show Context)
Citation Context ...roblem has been considered in the literature, with significant success. In particular, it is now known how to implement the interpolation step in O(n 2 ) time, when the output list size is a constant =-=[31, 33]-=-. Similar running times are also known for the root finding problem (which suffices for the second step in the algorithms above) [3, 12, 29, 31, 33, 44]. Together these algorithms lead to the possibil... |

14 | A Hensel lifting to replace factorization in list decoding of algebraic-geometric and Reed-Solomon codes
- Augot, Pecquet
- 2000
(Show Context)
Citation Context ... step in O(n 2 ) time, when the output list size is a constant [31, 33]. Similar running times are also known for the root finding problem (which suffices for the second step in the algorithms above) =-=[3, 12, 29, 31, 33, 44]-=-. Together these algorithms lead to the possibility that a good implementation of list decoding may actually even be able to compete with the classical Berlekamp-Massey decoding algorithm in terms of ... |

13 | A polynomial-time reduction from bivariate to univariate integral polynomial factorization
- Kaltofen
- 1982
(Show Context)
Citation Context ...lds in time polynomial in the size of the field or probabilistically in time polynomial in the logarithm of the size of the field and can also be solved deterministically over the rationals and reals =-=[10, 12, 13]-=-. Thus our algorithm ends up solving the curve-fitting problem over fairly general fields. It is interesting to contrast our algorithm with results which show bounds on the number of codewords that ma... |

13 | Decoding algebraic-geometric codes beyond the error-correction bound
- Shokrollahi, Wasserman
- 1998
(Show Context)
Citation Context ...The im- with running timeO(n2)or better, are also well-known: in provement for the case of algebraic-geometric codes extends particular the classical algorithms of Berlekamp and Massey the methods of =-=[19]-=- and improves upon their bound for every (see [14] for a description) achieve such running time bounds. choice ofnandd0. We also present some other consequences of It is also easily seen that 2then th... |

12 |
New Upper Bounds for Error Correcting Codes, Probl. Peredachi Inf
- Bassalygo
- 1965
(Show Context)
Citation Context ...d coding theory. Elias [10] used this notion to get a better handle on the error-exponent in the strong forms of Shannon's coding theorem. The notion also plays a dominant role in the Elias-Bassalygo =-=[34, 4]-=- upper bound on the rate of a code as a function of its relative distance. Through the decades the notion has continued to be investigated in a combinatorial context; and more recently has seen a spur... |

12 |
Modular curves, Shimura curves, and codes better than the Varshamov-Gilbert bound
- Tsfasman, Vlădut, et al.
- 1982
(Show Context)
Citation Context ...ude the Reed-Solomon codes as a special case. These codes are of significant interest because they yield explicit construction of codes that beat the Gilbert-Varshamov bound over small alphabet sizes =-=[24]-=- (i.e., achieve higher value ofdfor infinitely many choices ofnandkthan that given by the probabilistic method). Decoding algorithms for algebraic-geometric codes are typically based on decoding algor... |

12 |
Bounds for codes in the case of list decoding of finite volume
- Blinovsky
- 1986
(Show Context)
Citation Context ...de towards resolving it in [25, 20, 18]. ?From the point of usage, it is more useful to compare the rate of a code with its list decoding radius. This question has been investigated over the years by =-=[6, 7, 45, 11, 20]-=-. It follows from the converse to Shannon's coding theorem that a q-ary code of relative list decoding radius `(n) has rate at most R(n) ss 1 \GammasHq(`(n)). The above mentioned works show that there... |

11 |
List cascade decoding
- Zyablov, Pinsker
- 1981
(Show Context)
Citation Context ...de towards resolving it in [25, 20, 18]. ¿From the point of usage, it is more useful to compare the rate of a code with its list decoding radius. This question has been investigated over the years by =-=[6, 7, 45, 11, 20]-=-. It follows from the converse to Shannon’s coding theorem that a q-ary code of relative list decoding radius ℓ(n) has rate at most R(n) ≈ 1 − Hq(ℓ(n)). The above mentioned works show that there exist... |

10 |
Asymptotic Combinatorial Coding Theory
- Blinovsky
- 1997
(Show Context)
Citation Context ...de towards resolving it in [25, 20, 18]. ?From the point of usage, it is more useful to compare the rate of a code with its list decoding radius. This question has been investigated over the years by =-=[6, 7, 45, 11, 20]-=-. It follows from the converse to Shannon's coding theorem that a q-ary code of relative list decoding radius `(n) has rate at most R(n) ss 1 \GammasHq(`(n)). The above mentioned works show that there... |