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## The case for structured random codes in network capacity theorems (2007)

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Venue: | in Proceedings of the IEEE Information Theory Workshop (ITW 2007), (Lake Tahoe, CA |

Citations: | 52 - 10 self |

### Citations

12166 | Elements of Information Theory - Cover, Thomas - 1991 |

2071 | Information Theory and Reliable Communication - Gallager - 1968 |

1922 | Weung, “Network information flow
- Ahlswede, Cai, et al.
- 2000
(Show Context)
Citation Context ...ill see that converting an AWGN network into a system of linear equations can be fruitful. VI. NETWORK CODING WITH MACS Recently, there has been a great deal of interest in network coding [29], [30], =-=[44]-=-. The key idea is that for multicasting from one sender to multiple receivers over a network of point-to-point channels, the encoders should sometimes only send a function of their incoming messages (... |

1329 |
Information Theory: Coding Theorems for Discrete Memoryless Systems
- Csiszár, Körner
- 1981
(Show Context)
Citation Context ...ng (which is the case for additive noise), then a matrix with 4selements chosen i.i.d. and uniformly from the finite field will suffice. This is captured in the following lemma from Problem 2.1.11 in =-=[34]-=-. Lemma 1: Let w ∈ Fk be the message and let the channel output be given by y = x+z where x,y,z ∈ Fn and z is an i.i.d. sequence. Then the capacity of the channel is given by C = log |F| − H(Z) and ca... |

1247 |
Noiseless coding of correlated information sources
- Slepian, Wolf
- 2004
(Show Context)
Citation Context ... the expected performance of the best known purely random coding argument for a particular problem class. For example, for a distributed lossless compression problem (such as the Slepian-Wolf problem =-=[27]-=-), we compare to the performance of random binning in expectation. We now turn to codes that allow for the design of both the codebook pdf and algebraic structure. A structured random code (or structu... |

1166 |
El Gamal, “Capacity theorems for the relay channel
- Cover, A
- 1979
(Show Context)
Citation Context ...on theory. It has been successfully used to establish the capacity region of the multiple-access channel [1], [2], stochastically degraded broadcast channel [3], and physically degraded relay channel =-=[4]-=-, just to name a few. However, an elegant multiterminal problem developed by Körner and Marton showed that purely random code constructions are not always sufficient [5]; structured random codes, such... |

1069 | Flows in Networks
- Ford, Fulkerson
- 1962
(Show Context)
Citation Context ... capacity of the point-topoint network. We will also demonstrate that in some cases our achievable rates coincide with the simple upper bound due to the max-flow min-cut theorem of Ford and Fulkerson =-=[45]-=-. 14sWe now briefly review some results for multicasting over point-to-point channel networks. In [44], it was shown that for a unit bit pipe network the multicast capacity is given by the max-flow mi... |

990 | Linear network coding
- Li, Yeung, et al.
- 2003
(Show Context)
Citation Context ... such codes are beyond the scope of this paper. On a related note, in the network coding literature, linear codes have been shown to attain the same performance as random coding in the multicast case =-=[29]-=-, [30]. However, Dougherty, Freiling, and Zeger constructed a (nonmulticast) network for which the linear network coding capacity is significantly lower than the non-linear network coding capacity [31... |

839 | An algebraic approach to network coding
- Koetter, Médard
- 2003
(Show Context)
Citation Context ...codes are beyond the scope of this paper. On a related note, in the network coding literature, linear codes have been shown to attain the same performance as random coding in the multicast case [29], =-=[30]-=-. However, Dougherty, Freiling, and Zeger constructed a (nonmulticast) network for which the linear network coding capacity is significantly lower than the non-linear network coding capacity [31]. A g... |

584 | A Random Linear Network Coding Approach to Multicast
- Ho, Medard, et al.
(Show Context)
Citation Context ... and L receivers. The multicast capacity is given by the max-flow min-cut bound and can be achieved by an algebraic network code over any finite field larger than L (Fq, q > L). For a full proof, see =-=[49]-=-. Now we need to map these linear functions onto the multiple-access channels in the network. We first explore two examples that are a variant of the butterfly network given in [44]. Then, we state ou... |

562 | Broadcast channels
- Cover
- 1972
(Show Context)
Citation Context ...hod takes us quite far in network information theory. It has been successfully used to establish the capacity region of the multiple-access channel [1], [2], stochastically degraded broadcast channel =-=[3]-=-, and physically degraded relay channel [4], just to name a few. However, an elegant multiterminal problem developed by Körner and Marton showed that purely random code constructions are not always su... |

341 | Nested linear/lattice codes for structured multiterminal binning
- Zamir, Shamai, et al.
- 2002
(Show Context)
Citation Context ...1 ⊂ Λ2 and let V0,1 be the Voronoi region of Λ1. Then, a lattice code C can be defined as C = Λ2 ∩ V0,1 = {x : x = y mod Λ1, y ∈ Λ2}. (10) For more on nested lattice codes and their applications, see =-=[28]-=-. Nested lattices are often employed to enable lattice decoding, where the receiver decodes to the nearest lattice point whether or not it is part of the allowable set of the lattice code. More broadl... |

209 |
Multi-way communication channels
- Ahlswede
- 1973
(Show Context)
Citation Context ...f at least one good fixed set of codebooks. This method takes us quite far in network information theory. It has been successfully used to establish the capacity region of the multiple-access channel =-=[1]-=-, [2], stochastically degraded broadcast channel [3], and physically degraded relay channel [4], just to name a few. However, an elegant multiterminal problem developed by Körner and Marton showed tha... |

205 |
Interference alignment and degrees of freedom of the K user interference channel
- Cadambe, Jafar
- 2008
(Show Context)
Citation Context ...tive interferences, one known non-causally at each encoder, dirty paper coding can be successfully implemented with a lattice code but not with an purely random code [12]. Recently, Cadambe and Jafar =-=[13]-=- proposed the concept of interference alignment. In essence, encoders use their knowledge of the channel to align their transmissions so that for each receiver all interferences lie in one subspace an... |

174 | Joint physical layer coding and network coding for bi-directional relaying
- Narayanan, Wilson, et al.
- 2007
(Show Context)
Citation Context ...o the Gaussian case in [18]. Narayanan, Wilson, and Sprintson use similar techniques for the two-way relay channel to allow the relay to decode and retransmit only the sum of the transmitted messages =-=[19]-=-. This strategy was extended to the two-way relay channel with unequal transmit powers by Nam, Chung, and Lee [20]. Structured codes are also useful for parallel relay networks with non-white noise sp... |

171 |
A simple derivation of the coding theorem and some applications
- Gallager
- 1965
(Show Context)
Citation Context ...bol drawn i.i.d. from ˜ X for some N ∈ Z+ and R > 0. The g th receiver upon seeing ˜ Y N g uses a maximum likelihood rule to infer the original message w. Denote this estimate by ˆwg. It follows from =-=[50]-=- that for such a channel and N large enough, the average probability of error over codebooks and messages for receiver g, ¯Pr( ˆ Wg �= W) can be made less than ǫ L if R < I( ˜ X; ˜ Y ). It follows fro... |

159 | Insufficiency of linear coding in network information flow
- Dougherty, Freiling, et al.
- 2005
(Show Context)
Citation Context ...29], [30]. However, Dougherty, Freiling, and Zeger constructed a (nonmulticast) network for which the linear network coding capacity is significantly lower than the non-linear network coding capacity =-=[31]-=-. A great deal of work has gone into showing that linear and lattice codes are sufficient for many channel coding and source coding problems. In the following section, we will briefly review some of t... |

145 | On the capacity of large Gaussian relay networks
- Gastpar, Vetterli
- 2005
(Show Context)
Citation Context ...ise spectra. It has been shown that amplify-and-forward is asymptotically optimal for a parallel relay network with white noise and that it significantly outperforms standard random coding strategies =-=[21]-=-. Using a lattice scheme, Kochman, Khina, Erez, and Zamir have managed to attain similar gains for arbitrary noise spectra [22]. B. Paper Organization In this paper, we survey several of our recent ca... |

139 | Computation over multiple-access channels
- Nazer, Gastpar
- 2007
(Show Context)
Citation Context ...s. By using linear codes at the encoders, they were able to access the full rate region. Note that gains are only seen in this problem over random codes when the underlying sources are correlated. In =-=[8]-=-, we showed that there are large gains, proportional to the number of users, for computing functions over multiple-access channels with structured codes, regardless of the source dependencies. We appl... |

136 |
Coding for noisy channels
- Elias
- 1955
(Show Context)
Citation Context ... can achieve rates all the way up to capacity. Work in this area began with Elias’ proof that there exist binary linear codes which are good for channel coding over the binary symmetric channel (BSC) =-=[32]-=-. Essentially, the proof shows that a codebook generated from a binary matrix with i.i.d. Bernoulli( 1 2 ) entries has pairwise independent codewords that look as if they were drawn elementwise i.i.d.... |

131 |
Multiple access channels
- Liao
- 1972
(Show Context)
Citation Context ...least one good fixed set of codebooks. This method takes us quite far in network information theory. It has been successfully used to establish the capacity region of the multiple-access channel [1], =-=[2]-=-, stochastically degraded broadcast channel [3], and physically degraded relay channel [4], just to name a few. However, an elegant multiterminal problem developed by Körner and Marton showed that pur... |

130 |
A proof of the data compression theorem of Slepian and Wolf for ergodic sources
- Cover
- 1975
(Show Context)
Citation Context ... 2 ) = � n i=1 p(x1,x2). The basic multi-user random coding construction above has been extended to several powerful generalizations in the information theory literature. These include random binning =-=[24]-=-, block Markov coding [4], [25], superposition coding [3], [26], and compress-and-forward [4]. With these tools in hand, most of the currently known achievability results of network information theory... |

123 |
Lattices which are good for (almost) everything
- Erez, Litsyn, et al.
- 2005
(Show Context)
Citation Context ...i.i.d. Gaussian source with variance σ 2 S is: R(D) = 1 2 log Erez, Litsyn, and Zamir proved that there exist lattices which are simultaneously good for AWGN channel coding and Gaussian source coding =-=[39]-=-. These will be very useful to us in proving our theorems. Below we give a formal statement on the existence of lattice codes for channel coding and source coding: Lemma 2: Let s be a length-n Gaussia... |

118 | Polynomial time algorithms for network information flow
- Sanders, Egner, et al.
- 2003
(Show Context)
Citation Context ...ficient to achieve the multicast capacity. Bounds are also given on the required field size. It was independently and concurrently shown by Ho et al. in [46], Jaggi et al. in [47], and Sanders et al. =-=[48]-=- that the field size only needs to be larger than the number of receivers. We reproduce the version from [46] below as it will be useful to us in proving our main theorems. Definition 13: Let G PIPE =... |

113 |
Achieving 1 log (1 + SNR) on the AWGN channel with lattice encoding and decoding
- Erez, Zamir
- 2004
(Show Context)
Citation Context ...Further work by Erez and Zamir showed that nested lattice codes can be capacity-achieving under lattice decoding, i.e. decoding to the closest lattice point whether or not it is an allowable codeword =-=[37]-=-. As in the discrete case, Gaussian source coding can be performed optimally with lattices [38]. Recall that the rate distortion function for mean-squared error for an i.i.d. Gaussian source with vari... |

109 | On lattice quantization noise
- Zamir, Feder
- 1996
(Show Context)
Citation Context ... lattice decoding, i.e. decoding to the closest lattice point whether or not it is an allowable codeword [37]. As in the discrete case, Gaussian source coding can be performed optimally with lattices =-=[38]-=-. Recall that the rate distortion function for mean-squared error for an i.i.d. Gaussian source with variance σ 2 S is: R(D) = 1 2 log Erez, Litsyn, and Zamir proved that there exist lattices which ar... |

103 |
Linear codes for sources and source networks: Error exponents, universal coding
- Csiszár
- 1982
(Show Context)
Citation Context ...used for compressing any discrete alphabet source so long as the rate is higher than the source entropy. Moreover, they can reach any point in the Slepian-Wolf rate region for distributed compression =-=[35]-=-. For additive white Gaussian noise (AWGN) channels, showing that lattice codes are sufficient to reach capacity was considerably more challenging. An AWGN point-to-point channel has an output Y ∈ R w... |

93 | Averaging bounds for lattices and linear codes
- Loeliger
- 1997
(Show Context)
Citation Context ...robability of error goes to 0 as n → ∞. See [39] for a full proof of the existence of such lattices. Note that their goodness for AWGN capacity under ML decoding essentially follows from Theorem 1 in =-=[40]-=-. There Loeliger shows that lattices from “Construction A” (the employed method of randomly generating lattices) satisfy the Minkowski-Hlawka theorem in expectation. This is all that is needed to invo... |

91 |
How to encode the modulo-two sum of binary sources
- Korner, Marton
- 1979
(Show Context)
Citation Context ...sically degraded relay channel [4], just to name a few. However, an elegant multiterminal problem developed by Körner and Marton showed that purely random code constructions are not always sufficient =-=[5]-=-; structured random codes, such as linear or lattice codes, may be required on the achievability side of the proof. This key insight is the inspiration for this paper. Structured random codes are usua... |

87 | An achievable rate region for the multiple-access channel with feedback
- Cover, Leung
- 1981
(Show Context)
Citation Context ...sic multi-user random coding construction above has been extended to several powerful generalizations in the information theory literature. These include random binning [24], block Markov coding [4], =-=[25]-=-, superposition coding [3], [26], and compress-and-forward [4]. With these tools in hand, most of the currently known achievability results of network information theory can be derived. We collectivel... |

62 | Capacity bounds for two-way relay channel
- Nam, Chung, et al.
- 2008
(Show Context)
Citation Context ... to allow the relay to decode and retransmit only the sum of the transmitted messages [19]. This strategy was extended to the two-way relay channel with unequal transmit powers by Nam, Chung, and Lee =-=[20]-=-. Structured codes are also useful for parallel relay networks with non-white noise spectra. It has been shown that amplify-and-forward is asymptotically optimal for a parallel relay network with whit... |

56 | Lattice strategies for the dirty multiple access channel
- Philosof, Khisti, et al.
(Show Context)
Citation Context ...tiple-access channel with two additive interferences, one known non-causally at each encoder, dirty paper coding can be successfully implemented with a lattice code but not with an purely random code =-=[12]-=-. Recently, Cadambe and Jafar [13] proposed the concept of interference alignment. In essence, encoders use their knowledge of the channel to align their transmissions so that for each receiver all in... |

50 | Signaling over MIMO multi-base systems: combination of multi-access and broadcast schemes - Maddah-Ali, Motahari, et al. - 2006 |

44 | Lattices for distributed source coding: Jointly Gaussian sources and reconstruction of a linear function
- Krithivasan, Pradhan
- 2009
(Show Context)
Citation Context ...r@eecs.berkeley.edu) 1sRecently, Krithivasan and Pradhan showed that lattice coding results in an improved inner bound for the distributed compression of a linear function of jointly Gaussian sources =-=[11]-=-. In particular, for two positively correlated Gaussian sources, recovering the difference requires a lower sum rate if one uses a lattice code. For the above scenarios, the computation of a function ... |

44 | Network coding from a network flow perspective
- Ho, Karger, et al.
- 2003
(Show Context)
Citation Context ...ncoding and decoding over a finite field is sufficient to achieve the multicast capacity. Bounds are also given on the required field size. It was independently and concurrently shown by Ho et al. in =-=[46]-=-, Jaggi et al. in [47], and Sanders et al. [48] that the field size only needs to be larger than the number of receivers. We reproduce the version from [46] below as it will be useful to us in proving... |

43 | Lattice codes can achieve capacity on the AWGN channel
- Urbanke, Rimoldi
- 1998
(Show Context)
Citation Context ...g lattices that when intersected with an n-dimensional ball of radius √ nP centered at 0 form a capacity-achieving code. Urbanke and Rimoldi showed that such lattices exist for minimum angle decoding =-=[36]-=-. Further work by Erez and Zamir showed that nested lattice codes can be capacity-achieving under lattice decoding, i.e. decoding to the closest lattice point whether or not it is an allowable codewor... |

37 | Rethinking information theory for mobile ad hoc networks
- Andrews, Shakkottai, et al.
- 2008
(Show Context)
Citation Context ...s on another. For a manyto-one interference channel, Bresler, Parekh and Tse use a lattice to ensure that all the interferences seen at the relevant decoder are aligned [14]. It was also suggested in =-=[15]-=- that structured codes will be useful for implementing interference alignment and similar schemes in general multi-user networks. Finally, Sanderovich, Peleg, and Shamai find a scaling law for distrib... |

23 | Lattice coding increases multicast rates for gaussian multiple-access networks
- Nazer, Gastpar
- 2007
(Show Context)
Citation Context ...e a structured code to compute functions reliably on multiple-access channels. We developed such network coding strategies for discrete channels in [17] and extended our ideas to the Gaussian case in =-=[18]-=-. Narayanan, Wilson, and Sprintson use similar techniques for the two-way relay channel to allow the relay to decode and retransmit only the sum of the transmitted messages [19]. This strategy was ext... |

21 |
Asymptotic optimality of group and systematic codes for some channels
- Dobrushin
- 1963
(Show Context)
Citation Context ... to be capacity-achieving are often more restrictive than those for unrestricted random codes. For instance, linear codes achieve capacity for point-to-point channels only when the noise is symmetric =-=[6]-=-, [7]. Thus, it is tempting to believe that random codes are a strictly more powerful tool for proving capacity theorems. We will show that, in a network setting, structured random codes can be more p... |

19 | The Gaussian manyhelp-one distributed source coding problem - Tavildar, Viswanath, et al. - 2010 |

13 | Rematch and forward for parallel relay networks
- Kochman, Khina, et al.
- 2008
(Show Context)
Citation Context ... and that it significantly outperforms standard random coding strategies [21]. Using a lattice scheme, Kochman, Khina, Erez, and Zamir have managed to attain similar gains for arbitrary noise spectra =-=[22]-=-. B. Paper Organization In this paper, we survey several of our recent capacity results based on structured codes to show that even if we simply want to send bits across a network, random coding argum... |

13 |
Low complexity algebraic network codes
- Jaggi, Chou, et al.
- 2003
(Show Context)
Citation Context ...ver a finite field is sufficient to achieve the multicast capacity. Bounds are also given on the required field size. It was independently and concurrently shown by Ho et al. in [46], Jaggi et al. in =-=[47]-=-, and Sanders et al. [48] that the field size only needs to be larger than the number of receivers. We reproduce the version from [46] below as it will be useful to us in proving our main theorems. De... |

10 |
Group Codes do not Achieve Shannon’s Channel Capacity for General Discrete
- Ahlswede
- 1971
(Show Context)
Citation Context ...e capacity-achieving are often more restrictive than those for unrestricted random codes. For instance, linear codes achieve capacity for point-to-point channels only when the noise is symmetric [6], =-=[7]-=-. Thus, it is tempting to believe that random codes are a strictly more powerful tool for proving capacity theorems. We will show that, in a network setting, structured random codes can be more powerf... |

7 |
Scaling laws in decentralized processing of interfered Gaussian channels
- Sanderovich, Peleg, et al.
- 2008
(Show Context)
Citation Context ...nting interference alignment and similar schemes in general multi-user networks. Finally, Sanderovich, Peleg, and Shamai find a scaling law for distributed interference cancellation using lattices in =-=[16]-=-. Another instance of structural gain is seen in network coding for wireless networks. Instead of avoiding interference from other users, we can use a structured code to compute functions reliably on ... |

6 | Structured Random Codes and Sensor Network Coding Theorems
- Nazer, Gastpar
- 2008
(Show Context)
Citation Context ..., for computing functions over multiple-access channels with structured codes, regardless of the source dependencies. We applied these results to joint source-channel sensor network scenarios in [9], =-=[10]-=-. To appear in European Transactions on Telecommunications: Special Issue on New Directions in Information Theory. The authors are with the Department of Electrical Engineering and Computer Sciences, ... |

6 |
Computation over Gaussian multiple-access channels
- Nazer, Gastpar
(Show Context)
Citation Context ...h transmission whereas the lower bound allows for cooperation. Therefore, it seems that the lower bound is quite loose for large ℓ. We presented a slightly improved achievable scheme as part of [23], =-=[43]-=- but there was an error in the proof. 3) Performance of Best Known Random Code: For the problems considered above, random coding arguments perform quite poorly. As in the Körner-Marton problem, the be... |

5 |
The approximate capacity of a one-sided interference channel
- Bresler, Parekh, et al.
- 2007
(Show Context)
Citation Context ...pace and the desired signal lies on another. For a manyto-one interference channel, Bresler, Parekh and Tse use a lattice to ensure that all the interferences seen at the relevant decoder are aligned =-=[14]-=-. It was also suggested in [15] that structured codes will be useful for implementing interference alignment and similar schemes in general multi-user networks. Finally, Sanderovich, Peleg, and Shamai... |

4 | Spatial filtering in sensor networks with computation codes - Sarwate, Nazer, et al. - 2007 |

4 | Joint Wyner-Ziv/dirty-paper coding by analog modulo-lattice modulation
- Kochman, Zamir
(Show Context)
Citation Context ...th n = ℓk, ℓ ∈ Z+ channel uses: Dℓ = Mσ 2 � �� � N MN ℓ − 1 S + δ. (38) N + MP N + MP Furthermore, the distortion can be lower bounded by: � ℓ . (39) Dℓ ≥ Mσ 2 � N S N + MP Proof: (Achievability.) In =-=[41]-=-, Kochman and Zamir develop an elegant joint source-channel lattice scheme for sending a Wyner-Ziv Gaussian source over a dirty paper channel. Our distributed refinement scheme consists of two main st... |

1 | Communication over MIMO X channels: Signaling and multiplexing gain,” tech - Maddah-Ali, Motahari, et al. - 2006 |