## Efficient Decoding of Reed-Solomon Codes Beyond Half the Minimum Distance (2000)

### Cached

### Download Links

- [www.cs.technion.ac.il]
- [www.cs.technion.ac.il]
- DBLP

### Other Repositories/Bibliography

Venue: | IEEE Transactions on Information Theory |

Citations: | 49 - 0 self |

### BibTeX

@ARTICLE{Roth00efficientdecoding,

author = {Ron M. Roth and Gitit Ruckenstein},

title = {Efficient Decoding of Reed-Solomon Codes Beyond Half the Minimum Distance},

journal = {IEEE Transactions on Information Theory},

year = {2000},

volume = {46},

pages = {246--257}

}

### Years of Citing Articles

### OpenURL

### Abstract

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...

### Citations

2432 | The Design and Analysis of Computer Algorithms - Aho, Holpcroft, et al. - 1974 |

1940 | The Theory of Error Correcting Codes - Macwilliams, Sloane - 1977 |

525 | Finite Fields - Lidl, Niederreiter - 1997 |

418 |
Theory and Practice of Error Control Codes
- Blahut
- 1984
(Show Context)
Citation Context ... same as that of evaluating a polynomial in F n [x] at the code locators ff j , j 2 [n], and is therefore O(n log 2 n). Step D2 can be carried out through Chien search [4] and Forney's algorithm (see =-=[3]-=-). Both algorithms involve evaluation of polynomials at given points, implying that reconstructing the codewords in Step D2 can be executed in time complexity O(n log 2 n). 2 Writing (x) = P n\Gammak\... |

270 |
Shift-register synthesis and bch decoding
- Massey
- 1969
(Show Context)
Citation Context ...tsv = (v 1 ; v 2 ; : : : ; v n ) be the received word. Several efficient RS decoding algorithms are known for correcting up tos= b(n\Gammak)=2c errors. The Berlekamp-Massey algorithm [2, Section 7.4],=-=[12]-=- and the algorithm of Sugiyama et al. [3, Ch. 7],[18] comprise three steps: Step D0: Computing syndrome elements S 0 ; S 1 ; : : : ; S n\Gammak\Gamma1 from the received wordsv. The syndrome elements a... |

244 | Improved decoding of Reed-Solomon and algebraic-geometric codes
- Guruswami, Sudan
- 1999
(Show Context)
Citation Context ...es to which the algorithm is applied. For example, when ` = 2 we will haves? (n\Gammak)=2 only when ks(n+1)=3. An improvement of Sudan's work [17] has been recently reported by Guruswami and Sudan in =-=[8]-=-, where the constraints on the rates have been relaxed. This work is organized as follows. For the sake of completeness, we review Sudan's algorithm in Section 2. In Section 3, the EKE is derived, and... |

215 | Decoding of Reed-Solomon codes beyond the error correction bound
- Sudan
- 1997
(Show Context)
Citation Context ... executed only after the whole received word has been read but before any output is generated; hence, minimizing the complexity of Step D1 means reducing the latency of the decoder. In a recent paper =-=[17]-=-, Sudan presented a decoding algorithm for [n; k] RS codes of the form (1) that corrects more than b(n\Gammak)=2c errors. In this case, the decoding might not be unique, so the decoder's task is to fi... |

88 |
Effective Polynomial Computation
- Zippel
- 1993
(Show Context)
Citation Context ...factors can be found in time complexity O((` log 2 `) k (n + ` log q)) using root-finders of degree-` univariate polynomials. There are known general algorithms for factoring multivariate polynomials =-=[19]-=-; yet, those algorithms have relatively large complexity when applied to the particular application in this paper. We also mention the recent work of Gao and Shokrollahi [7] where they study the (slig... |

81 | Probabilistic Algorithms in Finite Fields
- Rabin
- 1980
(Show Context)
Citation Context ... y) that Step R3 is applied to at the ith recursion level is at most `. The roots in F = GF (q) of a polynomial of degree u can be found in expected time complexity O((u 2 log 2 u) log q) [10, Ch. 4],=-=[14]-=-; also recall that there are known efficient deterministic algorithms for root extraction when the characteristic of F is small [2, Ch. 10], and root extraction is particularly simple when ` = 2 [11, ... |

53 |
Error-correcting codes for List decoding
- Elias
(Show Context)
Citation Context ...g might not be unique, so the decoder's task is to find the list of codewords that differ from the received word in no more thanslocations. This task is referred to in the literature as list decoding =-=[5]-=-. If Gaussian elimination is used as an equation solver in Sudan's algorithm, then its time complexity is O(n 3 ). The algorithm can be described as a method for interpolating the polynomial f(x) 2 F ... |

42 |
A method for solving key equation for decoding Goppa codes
- Sugiyama, Kasahara, et al.
- 1975
(Show Context)
Citation Context .... Several efficient RS decoding algorithms are known for correcting up tos= b(n\Gammak)=2c errors. The Berlekamp-Massey algorithm [2, Section 7.4],[12] and the algorithm of Sugiyama et al. [3, Ch. 7],=-=[18]-=- comprise three steps: Step D0: Computing syndrome elements S 0 ; S 1 ; : : : ; S n\Gammak\Gamma1 from the received wordsv. The syndrome elements are commonly written in a form of a polynomial S(x) = ... |

33 | Decoding Reed-Solomon codes beyond half the minimum distance,” in Coding Theory, Cryptogrphy and related
- Nielsen, Høholdt
- 2000
(Show Context)
Citation Context ...nnecting the results of this work with the improvements presented in [8], might be possible (Nielsen and Hholdt have been working recently independently on a different approach to accelerate [8]; see =-=[13]-=-). 18 A Appendix We present here the proof of Propositions 4.2, which makes use of Lemma A.1 and Lemma A.2 below. We omit the proof of Lemma A.1 as it is similar to proofs already contained in [12], [... |

31 |
Cyclic decoding procedures for Bose-Chaudhuri-Hocquenghem codes
- Chien
- 1964
(Show Context)
Citation Context ...mplexity of computing (2) is the same as that of evaluating a polynomial in F n [x] at the code locators ff j , j 2 [n], and is therefore O(n log 2 n). Step D2 can be carried out through Chien search =-=[4]-=- and Forney's algorithm (see [3]). Both algorithms involve evaluation of polynomials at given points, implying that reconstructing the codewords in Step D2 can be executed in time complexity O(n log 2... |

31 |
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 ...both Sakata's algorithm and ours generalize Massey's algorithm---the two generalizations are not the same. Our proof of the algorithm in Figure 1 is based on that of an algorithm by Feng and Tzeng in =-=[6]-=-. Let OE denote the (total) order defined in [15] over the set of pairs f(i; t) j i 2 IN; t 2 [`]g; that is, (i; t) OE (i 0 ; t 0 ) if and only if 8 ? ! ? : i + t(k\Gamma1) ! i 0 + t 0 (k\Gamma1) or (... |

21 | Computing roots of polynomials over function fields of curves,” unpublished manuscript
- Gao, Shokrollahi
- 1998
(Show Context)
Citation Context ... multivariate polynomials [19]; yet, those algorithms have relatively large complexity when applied to the particular application in this paper. We also mention the recent work of Gao and Shokrollahi =-=[7]-=- where they study the (slightly different) problem of finding linear factors of bivariate polynomials Q(x; y) where the polynomial arithmetic is carried out modulo a power of x (in which case more sol... |

19 | Addition requirements for matrix and transposed matrix products
- Bshouty, Kaminski, et al.
- 1988
(Show Context)
Citation Context ...syndrome elements in Step D0 are computed by S i = n X j=1 v j j j ff i j (2) where j \Gamma1 j = Y r2[n]nfjg (ff j \Gamma ff r ) (3) (see Proposition 4.1 below). Using a result by Kaminski et al. in =-=[9]-=-, it can be shown that the time complexity of computing (2) is the same as that of evaluating a polynomial in F n [x] at the code locators ff j , j 2 [n], and is therefore O(n log 2 n). Step D2 can be... |

16 | Algebraic-geometric codes and multidimensional cyclic codes: A unified theory and algorithms for decoding using Gröbner bases
- Saints, Heegard
- 1995
(Show Context)
Citation Context ...olynomial S(x; y) is given. Equivalently, this algorithm is a method for solving the EKE (21). Our algorithm and its analysis are based on the approach of Massey [12] and Sakata [16], as presented in =-=[15]-=-. Even though our algorithm bears similarity to Sakata's algorithm---and both Sakata's algorithm and ours generalize Massey's algorithm---the two generalizations are not the same. Our proof of the alg... |

11 |
Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array
- Sakata
- 1988
(Show Context)
Citation Context ...ng that the syndrome polynomial S(x; y) is given. Equivalently, this algorithm is a method for solving the EKE (21). Our algorithm and its analysis are based on the approach of Massey [12] and Sakata =-=[16]-=-, as presented in [15]. Even though our algorithm bears similarity to Sakata's algorithm---and both Sakata's algorithm and ours generalize Massey's algorithm---the two generalizations are not the same... |

7 | Theory and Practice ofError Control Codes - Blahut - 1983 |

2 | Algebraic Coding Theory, Second Edition, Aegean - Berlekamp - 1984 |

2 | Probabilistic algorithms in nite elds - Rabin - 1980 |