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## Hardware Complexities of Algebraic Soft-decision Reed-Solomon Decoders and Comparisons

### Citations

577 |
Algebraic coding theory
- Berlekamp
- 1968
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Citation Context ...on process proposed by Guruswami and Sudan [9], [10]. As a result, these algorithms can achieve significant coding gain over hard-decision decoding (HDD), such as the Berlekamp-Massey algorithm (BMA) =-=[11]-=- and modified Euclidean algorithm (MEA) [12], with a complexity that is polynomial with respect to codeword length. ASD algorithms have three steps: multiplicity assignment, interpolation and factoriz... |

338 | Improved decoding of Reed-Solomon and Algebraic-Geometry codes
- Guruswami, Sudan
- 1999
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Citation Context ...[1], [2], [3], [4], [5], [6], [7], [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan [9], =-=[10]-=-. As a result, these algorithms can achieve significant coding gain over hard-decision decoding (HDD), such as the Berlekamp-Massey algorithm (BMA) [11] and modified Euclidean algorithm (MEA) [12], wi... |

273 | Decoding of Reed Solomon codes beyond the error-correction bound
- Sudan
- 1997
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Citation Context ...oped [1], [2], [3], [4], [5], [6], [7], [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan =-=[9]-=-, [10]. As a result, these algorithms can achieve significant coding gain over hard-decision decoding (HDD), such as the Berlekamp-Massey algorithm (BMA) [11] and modified Euclidean algorithm (MEA) [1... |

156 | Algebraic soft-decision decoding of Reed-Solomon codes
- Koetter, Vardy
- 2003
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Citation Context ...ound in many digital communication and storage systems due to their good burst error-correcting capability. Recently, algebraic soft-decision (ASD) decoding algorithms of RS codes have been developed =-=[1]-=-, [2], [3], [4], [5], [6], [7], [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan [9], [10... |

68 | Efficient Decoding of Reed–Solomon Codes beyond Half the Minimum Distance
- Roth, Ruckenstein
- 2000
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Citation Context ...g the combined interpolation technique to the LCC or KV decoding would lead to large area overhead. The factorization problem can be solved by the iterative algorithm proposed by Roth and Ruckenstein =-=[31]-=-. Each iteration of this algorithm involves root computations over finite fields for a polynomial, whose degree can be as large as t. When the degree of the polynomial is higher than one, root computa... |

35 | High-speed architectures for Reed– Solomon decoders
- Sarwate, Shanbhag
- 2001
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Citation Context ...ction fails. To reduce the worst-case latency, the factorization architecture in [34] based on direct root computations can be used when mmax = 4. The BMA can be implemented using the architecture in =-=[37]-=-. This architecture is also scalable. Table II shows the hardware requirement and latency of the blocks in the KV decoder with mmax = 4 for the RS (458, 410) code. In this table, a Mux refers to a 1-b... |

20 | A complexity reducing transformation in algebraic list decoding of Reed-Solomon codes
- Koetter, Vardy
(Show Context)
Citation Context ...ncoder � r' Interpolation Factorization BMA Block diagram of ASD decoder e Erasure decoder The interpolation complexity can be reduced by the reencoding and coordinate transformation techniques [19], =-=[20]-=-. Assume that the received word is r, and denote the k most reliable code positions in r by the set R. The re-encoding is to find a codeword φ that has the same symbols as r in the code positions in R... |

16 | A VLSI architecture for interpolation in soft-decision list decoding of Reed-Solomon codes
- Gross, Kschischang, et al.
(Show Context)
Citation Context ...4. Reencoder � r' Interpolation Factorization BMA Block diagram of ASD decoder e Erasure decoder The interpolation complexity can be reduced by the reencoding and coordinate transformation techniques =-=[19]-=-, [20]. Assume that the received word is r, and denote the k most reliable code positions in r by the set R. The re-encoding is to find a codeword φ that has the same symbols as r in the code position... |

12 |
Interpolation multiplicity assignment algorithms for algebraic soft-decision decoding of Reed-Solomon codes,” AMS-DIMACS volume on Algebraic Coding Theory and
- El-Khamy, McEliece
- 2005
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Citation Context ...ny digital communication and storage systems due to their good burst error-correcting capability. Recently, algebraic soft-decision (ASD) decoding algorithms of RS codes have been developed [1], [2], =-=[3]-=-, [4], [5], [6], [7], [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan [9], [10]. As a re... |

11 | A low-complexity method for Chasetype decoding of Reed-Solomon codes
- Bellorado, Kavčić
- 2006
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Citation Context ... communication and storage systems due to their good burst error-correcting capability. Recently, algebraic soft-decision (ASD) decoding algorithms of RS codes have been developed [1], [2], [3], [4], =-=[5]-=-, [6], [7], [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan [9], [10]. As a result, thes... |

10 | Exponential error bounds for algebraic soft-decision decoding of Reed-Solomon codes
- Ratnakar, Koetter
- 2005
(Show Context)
Citation Context ...gital communication and storage systems due to their good burst error-correcting capability. Recently, algebraic soft-decision (ASD) decoding algorithms of RS codes have been developed [1], [2], [3], =-=[4]-=-, [5], [6], [7], [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan [9], [10]. As a result,... |

10 | Algebraic soft-decision decoding of Reed-Solomon codes using bit-level soft information
- Jiang, Narayanan
- 2008
(Show Context)
Citation Context ...unication and storage systems due to their good burst error-correcting capability. Recently, algebraic soft-decision (ASD) decoding algorithms of RS codes have been developed [1], [2], [3], [4], [5], =-=[6]-=-, [7], [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan [9], [10]. As a result, these alg... |

9 | Efficient VLSI architecture for soft-decision decoding of Reed-Solomon codes
- Zhu, Zhang
(Show Context)
Citation Context ...ES The interpolation problem can be solved by the Kötter’s [15], [16] and Lee-O’Sullivan [17] algorithms. Although the Lee-O’Sullivan algorithm can potentially lead to higher efficiency when mmax ≤ 2 =-=[18]-=-, it does not allow interpolation points or their multiplicities to be changed after the interpolation started. This feature prohibits sharing intermediate interpolation results in the iterative BGMD ... |

8 |
Simplified procedure for correcting both errors and erasures of Reed-Solomon code using Euclidean algorithm
- Truong, Eastman, et al.
- 1988
(Show Context)
Citation Context ...9], [10]. As a result, these algorithms can achieve significant coding gain over hard-decision decoding (HDD), such as the Berlekamp-Massey algorithm (BMA) [11] and modified Euclidean algorithm (MEA) =-=[12]-=-, with a complexity that is polynomial with respect to codeword length. ASD algorithms have three steps: multiplicity assignment, interpolation and factorization. ASD algorithms are different in the m... |

8 |
On Algebraic Decoding of Algebraic-Geometric and Cyclic Codes
- Koetter
- 1996
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Citation Context ... observed that the MLCC decoding can achieve much better performance than the LCC decoding with the same η. III. COMPLEXITY-REDUCING TECHNIQUES The interpolation problem can be solved by the Kötter’s =-=[15]-=-, [16] and Lee-O’Sullivan [17] algorithms. Although the Lee-O’Sullivan algorithm can potentially lead to higher efficiency when mmax ≤ 2 [18], it does not allow interpolation points or their multiplic... |

8 |
An interpolation algorithm using Gröbner bases for soft-decision decoding of Reed-Solomon codes
- Lee, O’Sullivan
- 2006
(Show Context)
Citation Context ...ng can achieve much better performance than the LCC decoding with the same η. III. COMPLEXITY-REDUCING TECHNIQUES The interpolation problem can be solved by the Kötter’s [15], [16] and Lee-O’Sullivan =-=[17]-=- algorithms. Although the Lee-O’Sullivan algorithm can potentially lead to higher efficiency when mmax ≤ 2 [18], it does not allow interpolation points or their multiplicities to be changed after the ... |

8 |
High-speed interpolation architecture for soft-decision decoding of ReedSolomon codes
- Wang, Ma
- 2006
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Citation Context ...th mmax = 4 is significantly higher than that of the BGMD algorithm with mmax = 2 even if three iterations need to be carried out. It is also higher than that of the LCC algorithm with moderate η. In =-=[23]-=-, power representation is used for finite field elements, such that carry-save adders can replace finite field multipliers in the feedback loops for discrepancy coefficients computation to increase th... |

7 |
Backward interpolation architecture for algebraic soft-decision Reed-Solomon decoding
- Zhu, Zhang, et al.
- 2009
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Citation Context ...ation, especially when η is not small. The backward interpolation was also extended for the BGMD algorithm with mmax = 2 to enable the sharing of interpolation results in adjacent decoding iterations =-=[28]-=-. Further speedup of the BGMD interpolation can be achieved by computing the discrepancy coefficients from multiple iterations over the same point in a look-ahead manner, and then combining the polyno... |

6 |
Reduced complexity interpolation architecture for soft-decision Reed-Solomon decoding
- Zhang
- 2006
(Show Context)
Citation Context ...se techniques do not change the relative interpolation complexity of the three ASD algorithms. Although some discrepancy coefficients are zero and the corresponding polynomial updating can be skipped =-=[25]-=-, this property is very hard to be incorporated to simplify the hardware due to the complicated control. By making use of the properties of the points and their multiplicities, further simplifications... |

5 | Soft-decision decoding of Reed-Solomon codes using successive error-and-erasure decoding - Lee, Kumar - 2008 |

4 |
List Decoder of Linear Block Codes
- Nielson
- 2001
(Show Context)
Citation Context ...ved that the MLCC decoding can achieve much better performance than the LCC decoding with the same η. III. COMPLEXITY-REDUCING TECHNIQUES The interpolation problem can be solved by the Kötter’s [15], =-=[16]-=- and Lee-O’Sullivan [17] algorithms. Although the Lee-O’Sullivan algorithm can potentially lead to higher efficiency when mmax ≤ 2 [18], it does not allow interpolation points or their multiplicities ... |

4 | Z.Wang, ―Reencoder design for soft-decision decoding of an (255,239) Reed– Solomon code
- Ma, Vardy
- 2006
(Show Context)
Citation Context ...omials can be initialized the same as before, but (1,-1) weighted degree should be used in the interpolation. A coordinate transformation and re-encoding architecture based on the BMA can be found in =-=[21]-=-. By reformulating the involved equations, a more efficient re-encoder architecture was proposed in [22]. The factorization step can be carried out on the re-encoded and transformed interpolation outp... |

4 |
Fast factorization in soft-decision ReedSolomon decoding
- Zhang, Parhi
- 2005
(Show Context)
Citation Context ...ee can be as large as t. When the degree of the polynomial is higher than one, root computations over finite fields are traditionally carried out by exhaustive search. The prediction-based schemes in =-=[32]-=-, [33] can be employed to circumvent the exhaustive search in most cases. If the prediction is correct, only one field multiplication and constant binary matrix multiplications are required to find th... |

3 |
A Chase-GMD algorithm for soft-decision decoding of Reed-Solomon codes on perpendicular channels
- Xia, Wang, et al.
- 2008
(Show Context)
Citation Context ...tion and storage systems due to their good burst error-correcting capability. Recently, algebraic soft-decision (ASD) decoding algorithms of RS codes have been developed [1], [2], [3], [4], [5], [6], =-=[7]-=-, [8]. In these algorithms, the reliability information from the channel is incorporated into the algebraic interpolation process proposed by Guruswami and Sudan [9], [10]. As a result, these algorith... |

3 |
Factorization-free low-complexity Chase softdecision decoding of Reed-Solomon codes
- Zhu, Zhang
- 2009
(Show Context)
Citation Context ...tive performance of the LCC algorithm is worse than that for the AWGN channel. To improve the performance of the LCC algorithm in applications with ISI, a modified LCC (MLCC) algorithm is proposed in =-=[14]-=-. Similar to the Chase-GMD algorithm in [7], erasures are included in the test vectors. However, in the MLCC algorithm, erasures are assigned to the code positions that have more than one bit with rel... |

3 |
Efficient interpolation architecture for softdecision Reed-Solomon decoding by applying slow-down
- Zhang, Zhu
- 2008
(Show Context)
Citation Context ...pliers in the feedback loops for discrepancy coefficients computation to increase the maximum achievable clock frequency. The clock frequency can be also increased by applying the slow-down technique =-=[24]-=-. However, these techniques do not change the relative interpolation complexity of the three ASD algorithms. Although some discrepancy coefficients are zero and the corresponding polynomial updating c... |

3 |
Algebraic soft-decision decoder architectures for long Reed-Solomon codes
- Zhang, Zhu
(Show Context)
Citation Context ...ctor. Then (α, β ′ ) can be added using one iteration of the Kötter’s interpolation. The backward and Kötter’s forward interpolation can be also carried out by a unified architecture in one iteration =-=[27]-=-. Hence, once the interpolation result of the first test vector is available, the interpolation for each of the other vectors can be done in one single iteration. These techniques greatly reduced the ... |

3 |
Further exploring the strength of prediction in the factorization of soft-decision Reed-Solomon decoding
- Zhang
- 2007
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Citation Context ... be as large as t. When the degree of the polynomial is higher than one, root computations over finite fields are traditionally carried out by exhaustive search. The prediction-based schemes in [32], =-=[33]-=- can be employed to circumvent the exhaustive search in most cases. If the prediction is correct, only one field multiplication and constant binary matrix multiplications are required to find the root... |

3 |
Low latency factorization architecture for algebraic soft-decision decoding of Reed-Solomon codes
- Ma, Vardy, et al.
- 2007
(Show Context)
Citation Context ...ant binary matrix multiplications are required to find the root. When the degree is at most four, the polynomial can be also converted to an affine format, in which the roots can be computed directly =-=[34]-=-. Particularly, when the degree of the polynomial is two, the conversion to the affine format and root computation are much less complicated. In addition, it was discovered in [35] that the factorizat... |

3 |
Partial parallel factorization in soft-decision Reed-Solomon decoding
- Zhang
- 2006
(Show Context)
Citation Context ...an that in the KV decoder with mmax = 4. However, the factorization needs to be carried out three times for the three decoding iterations. The only parallel factorization architecture was proposed in =-=[39]-=-. This architecture can achieve less than two times speed with much more than two times area. Hence, instead of resorting to parallel processing inside the factorization engine, three factorization en... |

2 |
and Z.Wang “Novel interpolation architecture for low-complexity Chase soft-decision decoding of Reed-Solomon codes
- Zhu, Zhang
- 2008
(Show Context)
Citation Context ...different points (α, β) and (α, β ′ ). Given the interpolation result of the current vector, the point (α, β) can be eliminated in one iteration by using the backward interpolation scheme proposed in =-=[26]-=-, regardless of the location of the point in the vector. Then (α, β ′ ) can be added using one iteration of the Kötter’s interpolation. The backward and Kötter’s forward interpolation can be also carr... |

2 | Combined interpolation architecture for soft-decision decoding of Reed–Solomon codes
- Zhu, Zhang, et al.
- 2008
(Show Context)
Citation Context ...rpolation can be achieved by computing the discrepancy coefficients from multiple iterations over the same point in a look-ahead manner, and then combining the polynomial updating in those iterations =-=[29]-=-. In addition, this combined interpolation can be incorporated with the backward interpolation to achieve even higher speed [30]. In the LCC decoding, the interpolation points in a vector all have dif... |

2 |
Area efficient parallel decoder architecture for long BCH codes
- Chen, Parhi
- 2004
(Show Context)
Citation Context ... amount of time. Accordingly, ⌈458/29⌉ = 16-parallel Chien search is required for the root computation in the polynomial selection. The parallel Chien search can be implemented by the architecture in =-=[38]-=-. In addition, plcc parallel Chien search engines need to employed to simultaneously process the outputs of plcc interpolators. The error locations in R, which are the roots of q1(x), have already bee... |

1 |
Generalized minimim distance decoding
- Forney
- 1966
(Show Context)
Citation Context ... bit-level generalized minimum distance (BGMD) [6], and test-vector-based low-complexity Chase (LCC) [5] ASD algorithms. The algorithm in [7] combines the Chase and generalized minimum distance (GMD) =-=[13]-=- decoding. The algorithm in [8] repeats error-and-erasure decoding for every possible pattern of erasures from a large number of unreliable code positions. Despite that these two algorithms may achiev... |

1 |
High-speed re-encoder design for algebraic softdecision Reed-Solomon decoding
- Zhu, Zhang
- 2010
(Show Context)
Citation Context ...tion. A coordinate transformation and re-encoding architecture based on the BMA can be found in [21]. By reformulating the involved equations, a more efficient re-encoder architecture was proposed in =-=[22]-=-. The factorization step can be carried out on the re-encoded and transformed interpolation output directly. In this case, the factorization output can be used as syndromes in HDD, such as the BMA, to... |

1 |
Efficient architecture for generalized minimumdistance decoder of Reed-Solomon codes
- Zhu, Zhang
- 2010
(Show Context)
Citation Context ...d. In the LCC decoding, it would require very high complexity if the rest decoding steps are carried out on the interpolation result for each test vector. A polynomial selection scheme is proposed in =-=[36]-=- to pass only one interpolation output to later decoding steps through testing whether the degree of q1(x) equals its root number. From simulations, this scheme does not result in any performance degr... |