## Modern coding theory: the statistical mechanics and computer science point of view (2007)

Venue: | SUMMER SCHOOL ON COMPLEX SYSTEMS, LES HOUCHES |

Citations: | 9 - 0 self |

### BibTeX

@INPROCEEDINGS{Montanari07moderncoding,

author = {Andrea Montanari and Rüdiger Urbanke},

title = {Modern coding theory: the statistical mechanics and computer science point of view },

booktitle = {SUMMER SCHOOL ON COMPLEX SYSTEMS, LES HOUCHES},

year = {2007},

pages = {704--2857},

publisher = {Online]. Available}

}

### OpenURL

### Abstract

These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on ‘Complex Systems’ in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as ‘large graphical models’. While most of the lectures are devoted to the classical channel coding problem over simple memoryless channels, we present a discussion of more complex channel models. We conclude with an overview of the main open challenges in the field.

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Citation Context ...egree two and therefore connect exactly two factors). Such a graphical representation is also known as normal graph or as Forney-style factor graph (FSFG), in honor of Dave Forney who introduced them =-=[15]-=-. Let us now look at a concrete example. x1 x2 x3 xn σ0 σ1 σ2 σ3 σn−1 σn p(σ0) p(x2,y2,σ2 | σ1) p(x1,y1,σ1 | σ0) p(x3,y3,σ3 | σ2) Figure 17: The FSFG corresponding to (6.2). p(xn,yn,σn | σn−1) Example... |

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Citation Context ...ormly. These are well known #-P complete problems. 14 The currently best algorithms for solving random instances for the K-SAT problem are based on a variant of BP, which is called survey propagation =-=[11, 33, 11]-=-. 5.1 Proving Correctness through Correlation Decay A simple trick to bound the error incurred by BP consists in using the correlation decay properties [1, 57] of the measure µ( · ). Let i ∈ {1, . . .... |

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Citation Context ... the general situation: as long as the channel memory can be described by a Markov chain the factor graph approach applies and we can use message-passing schemes to construct efficient coding schemes =-=[14, 4, 21, 22, 25, 43]-=-. 6.3 Asymmetric Channels - The Z Channel Let us now consider the second generalization, namely the case of non-symmetric channels. Consider the channel depicted on the right of Fig. 1. For obvious re... |

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Citation Context ...se, are these close to each other or very different? And what is their relationship with the transmitted codeword? The connection to statistical mechanics emerges in the study of the decoding problem =-=[56, 51]-=-. To make it completely transparent we rewrite the distribution µ( · ) in Boltzmann form µC,y(x) = EC,y(x) = 1 Z(C, y) e−EC,y(x) , (2.9) { ∑N − i=1 log Q(yi|xi), if x ∈ C , (2.10) +∞, otherwise. The w... |

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Citation Context ... the general situation: as long as the channel memory can be described by a Markov chain the factor graph approach applies and we can use message-passing schemes to construct efficient coding schemes =-=[14, 4, 21, 22, 25, 43]-=-. 6.3 Asymmetric Channels - The Z Channel Let us now consider the second generalization, namely the case of non-symmetric channels. Consider the channel depicted on the right of Fig. 1. For obvious re... |

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Citation Context ... cavity equations. With respect to Bethe-Peierls, the cavity approach includes a hierarchy of (‘replica symmetry breaking’) refinements of such equations that aim at capturing long range correlations =-=[37]-=-. This will briefly described in Section 5. We should finally mention that several improvements over Bethe approximation have been developed within statistical physics. Among them, Kikuchi’s cluster v... |

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Citation Context ...ing a random code from the ensemble, provided N is large enough. By the optimality of MAP decoding, pd ≤ pc. In coding theory some techniques have been developed to prove upper and lower bounds on pc =-=[19, 52]-=-. In particular it is easy to find ensembles for which there exist a gap between the two thresholds (namely pd < pc strictly). Consider for instance (k, l) regular ensembles with a fixed ratio l/k = 1... |

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Citation Context ...The typical convergence of the performance to the asymptotic limit is shown in Fig. 21. The points correspond to simulation results whereas the solid curves correspond to a general scaling conjecture =-=[2]-=-. Roughly speaking, this scaling conjecture states that around the threshold the error probability behaves as follows: Let the channel be parameterized by ǫ with increasing ǫ indicating a worsening of... |

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Citation Context ...es into a non-uniform input distribution at the channel input in order to bring the mutual information closer to capacity. For a detailed discussion on coding for the Z-channel we refer the reader to =-=[34, 59, 5]-=-. 7 Open Problems Let us close by reviewing some of the most important open challenges in the channel coding problem. 7.1 Order of Limits Density evolution computes the limit lim t→∞ lim N→∞ E[P(N,t) ... |

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Citation Context ...es into a non-uniform input distribution at the channel input in order to bring the mutual information closer to capacity. For a detailed discussion on coding for the Z-channel we refer the reader to =-=[34, 59, 5]-=-. 7 Open Problems Let us close by reviewing some of the most important open challenges in the channel coding problem. 7.1 Order of Limits Density evolution computes the limit lim t→∞ lim N→∞ E[P(N,t) ... |

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Citation Context ...mmetric channel BSC(p) is defined by letting A = {0,1} and Q(0|0) = Q(1|1) = 1−p; the normalization then enforces Q(1|0) = Q(0|1) = p. In words, the channel “flips” the input bit with probability p ∈ =-=[0, 1]-=-. Since flips are introduced for each bit independently we 1 It is worth mentioning that an alternative approach would be to consider the noise as ‘adversarial’ (or worst case) under some constraint o... |

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Citation Context ... corresponding derivations. Finally, it would be interesting to have a sharp estimate for pc. It turns out that a sharp characterization of pc can be obtained through statistical mechanics techniques =-=[42, 58, 38]-=-. The statistical mechanics result has been proved to be a rigorous upper bound for general code ensembles, and it is conjectured to be tight [39, 35]. The starting point is to consider the conditiona... |

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Citation Context ...n for this convergence is the importance of probabilistic models and/or probabilistic techniques in each of these domains. This has long been obvious in information theory [53], statistical mechanics =-=[10]-=-, and statistical inference [45]. In the last few years it has also become apparent in coding theory and theoretical computer science. In the first case, the invention of Turbo codes [7] and the re-in... |

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Citation Context ...uctural properties of µ( · ) that are commonly regarded as incompatible with replica symmetry breaking. Replica symmetry breaking is instead necessary to describe the structure of ‘metastable states’ =-=[17]-=-. This can be loosely described as very deep local minima in the energy landscape introduced in Section 4.1. Here ‘very deep’ means that Θ(N) bit flips are necessary to lower the energy (number of uns... |

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Citation Context ...se, are these close to each other or very different? And what is their relationship with the transmitted codeword? The connection to statistical mechanics emerges in the study of the decoding problem =-=[56, 51]-=-. To make it completely transparent we rewrite the distribution µ( · ) in Boltzmann form µC,y(x) = EC,y(x) = 1 Z(C, y) e−EC,y(x) , (2.9) { ∑N − i=1 log Q(yi|xi), if x ∈ C , (2.10) +∞, otherwise. The w... |

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Citation Context ... corresponding derivations. Finally, it would be interesting to have a sharp estimate for pc. It turns out that a sharp characterization of pc can be obtained through statistical mechanics techniques =-=[42, 58, 38]-=-. The statistical mechanics result has been proved to be a rigorous upper bound for general code ensembles, and it is conjectured to be tight [39, 35]. The starting point is to consider the conditiona... |

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10 |
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Citation Context ... bold continuous lines to the limit t → ∞. In the inset we plot the expected conditional entropy E H(Xi|ν (t) i ). efficient alternatives in the coding context (mainly based on Fourier transform, see =-=[49, 48]-=-), we shall describe population dynamics because it is easily programmed. Let us describe the algorithm within the setting of a general message passing decoder, cf. Eq. (4.17). Given an integer N ≫ 1,... |

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Citation Context ...the correct exponential rate. This claim has been supported rigorously by the results of [47, 3, 31]. Let us finally comment on the relation between distance enumerator and the Franz-Parisi potential =-=[18]-=-, introduced in the study of glassy systems. In this context the potential is used to probe the structure of the Boltzmann measure. One considers a system with energy function E(x), a reference config... |

8 |
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Citation Context ...n is at least a fraction 1 2e ln(2) ≈ 0.924 of capacity over the entire range of p (with equality when p approaches one). Even more fortunate, from this perspective the Z channel is the extremal case =-=[32, 54]-=-: the information rate of any binary-input memoryless channel when the input distribution is the uniform one is at least a fraction 1 2e ln(2) of its capacity. From the above discussion we conclude th... |

7 |
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Citation Context ...es into a non-uniform input distribution at the channel input in order to bring the mutual information closer to capacity. For a detailed discussion on coding for the Z-channel we refer the reader to =-=[34, 59, 5]-=-. 7 Open Problems Let us close by reviewing some of the most important open challenges in the channel coding problem. 7.1 Order of Limits Density evolution computes the limit lim t→∞ lim N→∞ E[P(N,t) ... |

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Citation Context ...is smaller than x), and α is a constant which depends on the channel as well as on the channel. For the BEC this scaling law has been shown to be correct [2]. In fact, even a refined version is known =-=[13]-=-: define z = √ 2 − n(ǫ − ǫd + βn 3) where β is a constant depending on the ensemble. Then for z fixed and n tending to infinity we have PB(n, ǫ) = Q (z/α) ( 1 + O ( n −1/3)) . For general channels on ... |

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Citation Context ... − µ(xi|x t ′ t )| ≤ e−κt (5.10) can be proved if the clause density α stays below a threshold αu(k) that is estimated to behave as 2 log k αu(k) = k [1 + ok(1)]. While we refer to the original paper =-=[40]-=- for the details of the proof we limit ourselves to noticing that the left hand side of Eq. (5.10) can be estimated efficiently using a density evolution procedure. This allows to estimate the thresho... |

6 |
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Citation Context ...es of some fixed size. Within statistical physics this is sometimes called the population dynamics algorithm (and made its first appearance in the study of the localization transition on Cayley trees =-=[46]-=-). Although there exist more 11 The suspicious reader will notice that this is not exactly a particular case of the previous statement, because f(x) = I(x < 0) + 1 I(x = 0) is not a continuous functio... |

5 |
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Citation Context ...e other hand, for regular ensembles as the ones considered here, φq(ω) = φ(ω): the annealed calculation yields the correct exponential rate. This claim has been supported rigorously by the results of =-=[47, 3, 31]-=-. Let us finally comment on the relation between distance enumerator and the Franz-Parisi potential [18], introduced in the study of glassy systems. In this context the potential is used to probe the ... |

5 |
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Citation Context ...ined through statistical mechanics techniques [42, 58, 38]. The statistical mechanics result has been proved to be a rigorous upper bound for general code ensembles, and it is conjectured to be tight =-=[39, 35]-=-. The starting point is to consider the conditional entropy of the channel input x given the output y, HN(X|Y ). As shown in Eq. (2.16) this is given by the expectation of the log partition function a... |

4 |
the asymptotic weight and stopping setdistribution ofregular LDPCensembles
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Citation Context ...e other hand, for regular ensembles as the ones considered here, φq(ω) = φ(ω): the annealed calculation yields the correct exponential rate. This claim has been supported rigorously by the results of =-=[47, 3, 31]-=-. Let us finally comment on the relation between distance enumerator and the Franz-Parisi potential [18], introduced in the study of glassy systems. In this context the potential is used to probe the ... |

3 |
Correlation inequalities: a useful tool in the theory of LDPC codes
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Citation Context ...e other hand, for regular ensembles as the ones considered here, φq(ω) = φ(ω): the annealed calculation yields the correct exponential rate. This claim has been supported rigorously by the results of =-=[47, 3, 31]-=-. Let us finally comment on the relation between distance enumerator and the Franz-Parisi potential [18], introduced in the study of glassy systems. In this context the potential is used to probe the ... |

3 |
bounds for LDPC and LDGM codes under MAP decoding
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Citation Context ...ined through statistical mechanics techniques [42, 58, 38]. The statistical mechanics result has been proved to be a rigorous upper bound for general code ensembles, and it is conjectured to be tight =-=[39, 35]-=-. The starting point is to consider the conditional entropy of the channel input x given the output y, HN(X|Y ). As shown in Eq. (2.16) this is given by the expectation of the log partition function a... |