## Asymptotic enumeration methods for analyzing LDPC codes (2004)

Venue: | IEEE Trans. Inform. Theory |

Citations: | 41 - 2 self |

### BibTeX

@ARTICLE{Burshtein04asymptoticenumeration,

author = {David Burshtein and Gadi Miller},

title = {Asymptotic enumeration methods for analyzing LDPC codes},

journal = {IEEE Trans. Inform. Theory},

year = {2004},

volume = {50},

pages = {1115--1131}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show how asymptotic estimates of powers of polynomials with non-negative coefficients can be used in the analysis of low-density parity-check (LDPC) codes. In particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular LDPC code ensembles. We then consider the binary erasure channel (BEC). Using these estimates we derive lower bounds on the error exponent, under iterative decoding, of LDPC codes used over the BEC. Both regular and irregular code structures are considered. These bounds are compared to the corresponding bounds when optimal (maximum likelihood) decoding is applied.

### Citations

8609 |
Elements of information theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ...ts. Variants of these results have appeared in [5, 11, 12, 13, 8, 9]. III.1 The single variable case Throughout the paper we use extensively the following property of binomial coefficients (see, e.g. =-=[4]-=-[Eq. (12.53)]) log � � n = n[h(α) + o(1)] αn 4sHere log is the base-2 logarithm function, h(x) = −x log(x) − (1 − x) log(1 − x) is the binary entropy function, and o(1) is a function of n that approac... |

438 | Design of capacity-approaching irregular low-density parity check codes
- Richardson, Shokrollahi, et al.
- 2001
(Show Context)
Citation Context ...pectrum for the irregular bipartite graph based ensemble. This ensemble is widely used since on the one hand it has remarkable performance when decoded using the belief propagation decoding algorithm =-=[17, 22]-=-, and on the other hand it is easy to produce a random element from this ensemble. In [15] the average spectrum was found for a related irregular ensemble, in which the parity check matrices have know... |

433 |
A recursive approach to low complexity codes
- Tanner
- 1981
(Show Context)
Citation Context ... BEC is provided in Section V. Section VI concludes the paper. 2sII Background II.1 LDPC codes and graph representations It is convenient to specify LDPC codes using their Tanner graph representation =-=[27]-=-. The Tanner graph is a bipartite graph, where the nodes on the left side are associated with the codeword bits (variable nodes) and the nodes on the right are associated with the parity-check equatio... |

423 |
Low density parity check codes
- Gallager
- 1963
(Show Context)
Citation Context ...rial problems can be solved using enumerating functions. These problems involve powers of polynomials with non-negative coefficients. For example, the spectrum of lowdensity parity-check (LDPC) codes =-=[10]-=- can often be expressed using an enumerating function. Another example is the analysis of the error probability of LDPC codes over the binary erasure channel (BEC) when iterative decoding is applied [... |

280 | Expander codes
- Sipser, Spielman
- 1996
(Show Context)
Citation Context ...roves (the error exponent approaches the random coding error exponent) as c grows [18]. V.5 Discussion V.5.1 Expander graph arguments In [2] we showed how expander graph arguments similar to those in =-=[26]-=- can be used to analyze various message passing decoding algorithms, including Gallager’s hard decoding algorithm and 26sGallager’s soft decoding algorithm with appropriate clipping of the messages. C... |

252 | Efficient erasure correcting codes
- Luby, Mitzenmacher, et al.
- 2001
(Show Context)
Citation Context ...rties of Gallager’s soft decoding algorithm are best understood for the special case of the BEC. In fact, to date this is the only channel for which it was shown that channel capacity can be achieved =-=[16]-=-. In [6] a recursion for obtaining the decoding error probability of regular LDPC codes under iterative decoding was proposed. Although this recursion yields the exact error probability, there is no o... |

170 | Improved Low-Density Parity-Check Codes Using Irregular Graphs
- Luby, Mitzenmacher, et al.
- 2001
(Show Context)
Citation Context ...pectrum for the irregular bipartite graph based ensemble. This ensemble is widely used since on the one hand it has remarkable performance when decoded using the belief propagation decoding algorithm =-=[17, 22]-=-, and on the other hand it is easy to produce a random element from this ensemble. In [15] the average spectrum was found for a related irregular ensemble, in which the parity check matrices have know... |

169 | The Capacity of Low-Density Parity Check Codes under Message-Passing Decoding
- Richardson, Urbanke
- 2001
(Show Context)
Citation Context ... optimal (Maximum Likelihood, ML) decoding. Other bounds were obtained in [18]. The analysis of iterative decoding is in general more complicated. A consequence of the concentration result derived in =-=[17, 20]-=- is that if an outer code with arbitrarily high rate is used in a concatenated coding scheme, then the error probability is exponentially decreasing (see [20][footnote 4]). However, the error exponent... |

129 |
Urbanke, "Efficient encoding of low-density parity-check codes
- Richardson, L
- 2001
(Show Context)
Citation Context ...s, such that each check node neighbor of S is connected to S by at least two edges. In Figure 2 we show a stopping set S, which is connected to a set of check neighbors denoted by T . As explained in =-=[21, 6]-=-, the iterative decoding of LDPC codes succeeds if and only if the set of variable nodes which correspond to erasures does not contain a subset which is a stopping set. We use this fact to derive boun... |

111 |
Finitelength analysis of low-density parity-check codes on the binary erasure channel
- Di, Proietti, et al.
- 2002
(Show Context)
Citation Context ...] can often be expressed using an enumerating function. Another example is the analysis of the error probability of LDPC codes over the binary erasure channel (BEC) when iterative decoding is applied =-=[6]-=-. ∗ IEEE Transactions on Information Theory, volume 50, no. 6, pp. 1115–1131, June 2004. This research was supported by the Israel Science Foundation, grant no. 22/01–1. 1sIn this paper we discuss asy... |

62 |
Bounds on the decoding error probability of binary linear codes via their spectra
- Poltyrev
(Show Context)
Citation Context ...random code ensemble. Once the spectrum is obtained one can obtain upper bounds on the probability of decoding error for various channels, both for the given ensemble and for the expurgated one, e.g. =-=[10, 19, 25, 18, 23]-=-. Note that (30) is sufficient for the application of these bounds, and Theorem 4 shows that the bound on the spectrum (30) is tight. V Lower bounds on the error exponent for the BEC V.1 Iterative dec... |

62 | New sequences of linear time erasure codes approaching the channel capacity
- Shokrollahi
- 1999
(Show Context)
Citation Context ...t can be shown [16] that Gallager’s belief propagation algorithm can correct a δ fraction of losses (erasures) in the channel if δλ (1 − ρ(1 − x)) < x (2) for x ∈ (0, δ]. Furthermore, it can be shown =-=[16, 24]-=- that there exist sequences of LDPC codes that achieve channel capacity when using the belief propagation algorithm, by explicitly constructing these codes. III Asymptotic estimates of powers of polyn... |

47 |
On ensembles of low-density parity-check codes: asymptotic distance distributions
- Litsyn, Shevelev
- 2002
(Show Context)
Citation Context ...nf x>0 d + (1 − x) d 2xαd − h (α) (c − 1) Note: If d is odd then the degree of (1+x) d +(1−x) d is d−1. Thus, in this case, if α > (d−1)/d we have limN→∞ log ¯ SαN/N = −∞. This result was obtained in =-=[14]-=- using a different approach. 9sIV.1.2 Gallager’s ensemble Let c < d be two integers, and denote by A d N the matrix with N columns and N/d rows obtained by horizontally concatenating d size-N/d identi... |

35 | Bounds on the maximum-likelihood decoding error probability of low-density parity-check codes
- Miller, Burshtein
- 2001
(Show Context)
Citation Context ...f LDPC codes over the binary erasure channel (BEC). In [10] Gallager obtained bounds on the error exponent of LDPC codes under optimal (Maximum Likelihood, ML) decoding. Other bounds were obtained in =-=[18]-=-. The analysis of iterative decoding is in general more complicated. A consequence of the concentration result derived in [17, 20] is that if an outer code with arbitrarily high rate is used in a conc... |

28 | Random coding techniques for nonrandom codes
- Shulman
- 1999
(Show Context)
Citation Context ...random code ensemble. Once the spectrum is obtained one can obtain upper bounds on the probability of decoding error for various channels, both for the given ensemble and for the expurgated one, e.g. =-=[10, 19, 25, 18, 23]-=-. Note that (30) is sufficient for the application of these bounds, and Theorem 4 shows that the bound on the spectrum (30) is tight. V Lower bounds on the error exponent for the BEC V.1 Iterative dec... |

16 |
Some results on the asymptotic behaviour of coefficients of large powers of functions
- Gardy
- 1995
(Show Context)
Citation Context ...ight. The derivation of tighter bounds is left for further research. Another possibility for further research is the generalization of the results to channels other then the BEC. Finally we note that =-=[5, 11, 12, 8, 9]-=- can be used to derive higher order approximations to the average distance spectrum and to the distribution of the spectrum of stopping sets. Acknowledgment The authors would like to thank Herbert Wil... |

14 | The average case analysis of algorithms: multivariate asymptotics and limit distributions, Rapport de recherche no
- Flajolet, Sedgewick
- 1997
(Show Context)
Citation Context .... 1sIn this paper we discuss asymptotic properties of enumerating functions and show their usefulness in analyzing low-density parity-check (LDPC) codes. Variants of these properties have appeared in =-=[5, 11, 12, 13, 8, 9]-=-. We first derive the asymptotic distance spectrum of various LDPC code ensembles. The resulting expressions are either new, or otherwise require much more laborious methods to obtain. In particular w... |

14 |
Distance distributions in ensembles of irregular low-density parity-check codes
- Litsyn, Shevelev
(Show Context)
Citation Context ...n the one hand it has remarkable performance when decoded using the belief propagation decoding algorithm [17, 22], and on the other hand it is easy to produce a random element from this ensemble. In =-=[15]-=- the average spectrum was found for a related irregular ensemble, in which the parity check matrices have known row and column weight profiles. The average spectrum of irregular LDPC codes was also co... |

12 | Bounds on the performance of belief propagation decoding
- Burshtein, Miller
- 2002
(Show Context)
Citation Context ...h a waterfall region. It can also be seen that the gap between iterative and ML decoding is larger for a (5, 10) code compared to a (3, 6) code. The last observation is consistent with our results in =-=[3]-=-, where we have considered, for an arbitrary binary-input symmetric-output channel, the performance under iterative decoding of a (c, d) regular code with fixed planned rate, R = 1 − c/d, as c grows. ... |

10 | Expander graph arguments for message passing algorithms - Burshtein, Miller - 2001 |

7 | The average case analysis of algorithms: saddle point asymptotics, Rapport de recherche no
- Flajolet, Sedgewick
- 1994
(Show Context)
Citation Context .... 1sIn this paper we discuss asymptotic properties of enumerating functions and show their usefulness in analyzing low-density parity-check (LDPC) codes. Variants of these properties have appeared in =-=[5, 11, 12, 13, 8, 9]-=-. We first derive the asymptotic distance spectrum of various LDPC code ensembles. The resulting expressions are either new, or otherwise require much more laborious methods to obtain. In particular w... |

7 |
On Improved bounds on decoding error probability of block codes over interleaved fading channels, with applications to Turbo-like codes
- Sason, Shamai
- 2001
(Show Context)
Citation Context ...random code ensemble. Once the spectrum is obtained one can obtain upper bounds on the probability of decoding error for various channels, both for the given ensemble and for the expurgated one, e.g. =-=[10, 19, 25, 18, 23]-=-. Note that (30) is sufficient for the application of these bounds, and Theorem 4 shows that the bound on the spectrum (30) is tight. V Lower bounds on the error exponent for the BEC V.1 Iterative dec... |

4 |
Saddlepoint approximations in statistics”, Ann
- Daniels
- 1954
(Show Context)
Citation Context .... 1sIn this paper we discuss asymptotic properties of enumerating functions and show their usefulness in analyzing low-density parity-check (LDPC) codes. Variants of these properties have appeared in =-=[5, 11, 12, 13, 8, 9]-=-. We first derive the asymptotic distance spectrum of various LDPC code ensembles. The resulting expressions are either new, or otherwise require much more laborious methods to obtain. In particular w... |

4 |
Weight distribution of iterative coding systems: How deviant can you be
- DI, RICHARDSON, et al.
- 2001
(Show Context)
Citation Context ... spectrum was found for a related irregular ensemble, in which the parity check matrices have known row and column weight profiles. The average spectrum of irregular LDPC codes was also considered in =-=[7]-=-. We then consider the problem of bounding the error exponent of LDPC codes over the binary erasure channel (BEC). In [10] Gallager obtained bounds on the error exponent of LDPC codes under optimal (M... |

4 |
A generalisation of Stirling’s formula, J. für die reine und angewandte Mathematik 196
- Hayman
- 1956
(Show Context)
Citation Context |

3 |
Saddle point methods for the multinomial distribution
- Good
- 1957
(Show Context)
Citation Context |

1 |
Convex Optimization, to be published
- Boyd, Vandenberghe
- 2004
(Show Context)
Citation Context ...er of variables (large m). 6 (6) (7)s2. Consider the right hand side of (7) as a function of α and β. Define g(α, β) = inf x>0,y>0 p(x, y) log xα = inf {log p(x, y) − α log x − β log y} yβ x>0,y>0 By =-=[1]-=-[Section 3.2.3], g(α, β) is concave with respect to (w.r.t.) α and β in its domain (g(α, β) > −∞). We now show that this domain is convex and closed. Recall that in the theorem we use strong Lagrangia... |