## Stopping set distribution of LDPC code ensembles (2005)

Venue: | IEEE Trans. Inform. Theory |

Citations: | 53 - 1 self |

### BibTeX

@ARTICLE{Orlitsky05stoppingset,

author = {Alon Orlitsky and Krishnamurthy Viswanathan and Junan Zhang and Student Member},

title = {Stopping set distribution of LDPC code ensembles},

journal = {IEEE Trans. Inform. Theory},

year = {2005},

volume = {51},

pages = {929--953}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—Stopping sets determine the performance of low-density parity-check (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tanner-graph ensembles, including the following. An expression for the normalized average stopping set distribution, yielding, in particular, a critical fraction of the block length above which codes have exponentially many stopping sets of that size. A relation between the degree distribution and the likely size of the smallest nonempty stopping set, showing that for a I

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Citation Context ...parity-check (LDPC) codes have been shown to perform extremely well even with computationally efficient iterative decoding [1]–[3]. These algorithms involve message-passing on the codes’ Tanner graph =-=[4]-=-. Of these, Gallager’s soft-decoding algorithm [1] is equivalent to maximumlikelihood (ML) decoding when the Tanner graph of the code has no cycles of length smaller than or equal to twice the number ... |

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477 | Design of capacity-approaching irregular low-density parity-check codes
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Citation Context ...defined in Section IV.) By using the inclusion–exclusion principle, we can then show that the asymptotic average block probability is if Note that the inequality is usually called stability condition =-=[17]-=- and the theorem predicts that when , the ensembles have a constant average error floor, however for individual codes, their block error floor may approach zero. In Fig. 2, we plot the average block e... |

417 | Urbanke, “The capacity of low-density parity-check codes under message-passing decoding
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Citation Context ...ach the capacity of the erasure channel with linear decoding complexity. The problem of analyzing finite-length LDPC codes was considered in [5] which used the concept of stopping sets, introduced in =-=[6]-=-, to evaluate the average error probability under iterative decoding over the erasure channel. The role played by stopping set distributions in iterative decoding is analogous to that played by distan... |

282 | Efficient erasure correcting codes
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Citation Context ...les of length smaller than or equal to twice the number of iterations. For the binary erasure channel (BEC), Gallager’s soft-decoding algorithm reduces to the iterative decoding algorithm proposed in =-=[2]-=- where it was shown that, as the block length tends to infinity, appropriately designed irregular LDPC codes can approach the capacity of the erasure channel with linear decoding complexity. The probl... |

128 |
Low-Density Parity Check Codes
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Citation Context ...ck (LDPC) codes, minimum distance, stopping set. I. INTRODUCTION LOW-density parity-check (LDPC) codes have been shown to perform extremely well even with computationally efficient iterative decoding =-=[1]-=-–[3]. These algorithms involve message-passing on the codes’ Tanner graph [4]. Of these, Gallager’s soft-decoding algorithm [1] is equivalent to maximumlikelihood (ML) decoding when the Tanner graph o... |

81 |
Finite length analysis of low-density parity-check codes on the binary erasure channel
- Di, Proietti, et al.
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Citation Context ...nity, appropriately designed irregular LDPC codes can approach the capacity of the erasure channel with linear decoding complexity. The problem of analyzing finite-length LDPC codes was considered in =-=[5]-=- which used the concept of stopping sets, introduced in [6], to evaluate the average error probability under iterative decoding over the erasure channel. The role played by stopping set distributions ... |

52 |
On Ensembles of Low-Density Parity-Check Codes: Asymptotic Distance Distributions
- Litsyn, Shevelev
- 2002
(Show Context)
Citation Context ...al LDPC code ensembles. In his monograph [1], Gallager calculated the distance spectrum of regular Gallager ensembles. Litsyn and Shevelev extended these calculations to seven other regular ensembles =-=[10]-=- (including Tanner-graph ensembles) and to irregular Gallager ensembles [11]. Burshtein et al. first studied the stopping set distributions for Tanner-graph ensembles in [12]. In this paper, we derive... |

43 | Asymptotic enumeration methods for analyzing LDPC codes
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Citation Context ...n other regular ensembles [10] (including Tanner-graph ensembles) and to irregular Gallager ensembles [11]. Burshtein et al. first studied the stopping set distributions for Tanner-graph ensembles in =-=[12]-=-. In this paper, we derive simpler expressions for the distributions. Di et al. [8], [9], studied the minimum distance of Tannergraph LDPC code ensembles. In [13], Richardson et al. studied the stoppi... |

14 |
Weight distributions: How deviant can you be
- Di, Urbanke, et al.
- 2001
(Show Context)
Citation Context ...lock length, when the degree distributions satisfy , there always exists a sequence of LDPC codes whose stopping number grows linearly with . This is analogous to results shown for min• imum distance =-=[8]-=-, [9]. The behavior of the average block error as a function of the channel erasure probability .We define an error-floor threshold below which stopping sets of linear size make exponentially small co... |

9 |
Efficient encoding of low density parity check codes
- Richardson, Urbanke
- 2001
(Show Context)
Citation Context ...LDPC) codes, minimum distance, stopping set. I. INTRODUCTION LOW-density parity-check (LDPC) codes have been shown to perform extremely well even with computationally efficient iterative decoding [1]–=-=[3]-=-. These algorithms involve message-passing on the codes’ Tanner graph [4]. Of these, Gallager’s soft-decoding algorithm [1] is equivalent to maximumlikelihood (ML) decoding when the Tanner graph of th... |

2 |
Finite length analysis of LDPC codes with large left degrees
- Zhang, Orlitsky
- 2002
(Show Context)
Citation Context ... variable nodes. This fact was used in [5] to derive a recursive formula for calculating the average error probability of regular LDPC codes. The recursion was subsequently simplified and extended in =-=[7]-=-. The above results focus on exact expressions for the average error probabilities of finite-length codes. In this paper, we bound the error probability of long LDPC codes by analyzing the asymptotic ... |

2 |
distributions in ensembles of irregular low-density parity-check codes
- “Distance
- 2003
(Show Context)
Citation Context ...ce spectrum of regular Gallager ensembles. Litsyn and Shevelev extended these calculations to seven other regular ensembles [10] (including Tanner-graph ensembles) and to irregular Gallager ensembles =-=[11]-=-. Burshtein et al. first studied the stopping set distributions for Tanner-graph ensembles in [12]. In this paper, we derive simpler expressions for the distributions. Di et al. [8], [9], studied the ... |

1 |
distribution of low-density parity-check codes
- “Weight
(Show Context)
Citation Context ...length, when the degree distributions satisfy , there always exists a sequence of LDPC codes whose stopping number grows linearly with . This is analogous to results shown for min• imum distance [8], =-=[9]-=-. The behavior of the average block error as a function of the channel erasure probability .We define an error-floor threshold below which stopping sets of linear size make exponentially small contrib... |

1 |
Finite length analysis of varous low-density parity-check ensembles for the binary erasure channel
- Richarson, Shokrollahi, et al.
(Show Context)
Citation Context ...butions for Tanner-graph ensembles in [12]. In this paper, we derive simpler expressions for the distributions. Di et al. [8], [9], studied the minimum distance of Tannergraph LDPC code ensembles. In =-=[13]-=-, Richardson et al. studied the stopping number of several code ensembles. Their results can be summarized as follows. If , then most codes in the ensemble have logarithmic minimum distance and stoppi... |

1 |
Weight distribtution of LDPC code ensembles: Combinatorics meets statistical physics
- Di, Montanari, et al.
(Show Context)
Citation Context ...ain result of the section, a lower bound on the stopping number of code ensembles after expurgation of codes with small stopping sets. Note that further work is still required on the upper bound (see =-=[15]-=-). Theorem 13: For the ensemble , if , then Proof: It follows from Theorem 7 that if then . By the definition of , for . For any and , stopping sets of size where make an exponentially small contribut... |

1 |
Large Deviations for Perfomrnace Analysis
- Shwartz, Weiss
- 1995
(Show Context)
Citation Context ...set [2], i.e.,sORLITSKY et al.: STOPPING SET DISTRIBUTION OF LDPC CODE ENSEMBLES 943 We bound from above by considering stopping sets of different sizes. For any and Clearly and by the Chernoff bound =-=[16]-=- and and (42) Therefore, the third term in (42) vanishes exponentially. Taking expectation on both sides of (42) over the ensemble and (43) Let denote the set of degree- variable nodes. We can bound f... |