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## Multistage compute-and-forward with multilevel lattice codes based on product constructions (2014)

Venue: | in Proc. IEEE ISIT |

Citations: | 5 - 2 self |

### Citations

12423 |
Elements of Information Theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ...is decoder also gives away the knowledge of previously decoded codeword and hence is worse than the previous one. 28 C. Achievable Computation Rate In [37], using the chain rule of mutual information =-=[42]-=-, Wachsmann et al. show that multilevel coding incurs no loss in terms of the achievable information rate for point to point communication. The same proof works for our construction as well, which we ... |

545 |
Channel Coding with Multilevel/Phase Signals
- Ungerboeck
- 1982
(Show Context)
Citation Context ...ion. A. Comparison of Different Mappings For point-to-point communication, one of the most popular and most frequently used labeling strategy for multilevel coding is the one introduced by Ungerboeck =-=[39]-=-. The design guideline of this labeling strategy is to maximize the minimum intra-subset Euclidean distance. However, as mentioned in Section VI-C, this labeling strategy does not guarantee the homomo... |

444 |
Modern Coding Theory
- Richardson, Urbanke
- 2008
(Show Context)
Citation Context ...oding is feasible. This also allows the use of well-developed codes on graphs (e.g., non-binary LDPC) for channel coding and enables one to employ iterative decoding such as message passing algorithm =-=[28]-=- to further reduce the decoding complexity. One key drawback of this scheme is that the channel coding has to work over Fp for a constellation with p elements. Hence, the decoding complexity increases... |

207 | Multilevel codes: theoretical concepts and practical design rules,”
- Wachsmann, Fischer, et al.
- 1999
(Show Context)
Citation Context ...s a Z-module isomorphism between Z/qZ × Z/qZ to Z[ω]/φZ[ω]. In the next section, we propose a novel encoding/decoding pair that incorporates the idea of multilevel coding and multistage decoding [36] =-=[37]-=- which largely relies on Z-module isomorphisms. The proposed encoding/decoding allows us to work over a potentially much smaller field Fq. B. The Second Proposed Family of Constellations Let φ1, φ2, .... |

190 |
A new multilevel coding method using errorcorrecting codes,”
- Imai, Hirakawa
- 1977
(Show Context)
Citation Context ...nduces a Z-module isomorphism between Z/qZ × Z/qZ to Z[ω]/φZ[ω]. In the next section, we propose a novel encoding/decoding pair that incorporates the idea of multilevel coding and multistage decoding =-=[36]-=- [37] which largely relies on Z-module isomorphisms. The proposed encoding/decoding allows us to work over a potentially much smaller field Fq. B. The Second Proposed Family of Constellations Let φ1, ... |

170 | Joint physical layer coding and network coding for bidirectional relaying
- Wilson, Narayanan, et al.
- 2010
(Show Context)
Citation Context ...capacity with lattice decoding [14]. Functional computation in physical layer with such lattices has been realized to asymptotically approach the capacity for the bidirectional relay networks in [21] =-=[22]-=- [23] [24]. The reader is referred to a tutorial paper [25] for more details about using lattices for the bidirectional relay channels. One of the main contribution of [1] is to provide a means to har... |

158 | Compute-and-forward: Harnessing interference through structured codes,” Information Theory
- Nazer, Gastpar
- 2011
(Show Context)
Citation Context ...integer coefficients depending on the channel coefficients and signal-to-noise ratio (SNR). Such a scheme which uses lattices over integers has been analyzed by Nazer and Gastpar for AWGN networks in =-=[1]-=- where achievable computation rates were derived. Based on this approach, in [2], Tunali et al. considered the use of lattices over Eisenstein integers for the compute-and-forward paradigm and success... |

137 | Threshold saturation via spatial coupling: why convolutional LDPC ensembles perform so well over the BEC,” e-print,
- Kudekar, Richardson, et al.
- 2010
(Show Context)
Citation Context ...indeed can approach those theoretic limits with reasonable complexity, we also simulate 31 the proposed scheme with an ensemble of linear codes with iterative decoding. Recently, it has been shown in =-=[43]-=- [44] [45] that the ensemble of spatially-coupled LDPC codes (or LDPC convolutional codes) [46] [47] [48] universally achieve the capacity for the class of binary memoryless symmetric channels under b... |

126 | Lattices which are good for (almost) everything - Erez, Litsyn, et al. - 2005 |

114 |
Achieving 1 log(1 + SNR) on the AWGN channel with lattice encoding and decoding
- Erez, Zamir
- 2004
(Show Context)
Citation Context ... knowledge on lattices and nested lattice codes followed by some preliminaries on abstract algebra. For more details about lattices, lattice codes, and nested lattice codes, the reader is referred to =-=[14]-=- [15] [4]. We then summarize the lattice-based compute-and-forward paradigm and the main result in [1] and briefly mention the separation-based framework proposed in [2, Section V]. A. Lattices An N-d... |

111 | On lattice quantization noise
- Zamir, Feder
- 1996
(Show Context)
Citation Context ...at the distribution of the equivalent noise zeq,m is in general not Gaussian but would become Gaussian in the limit as N → ∞ if Λc is good for quantization due to the Gaussian approximation principle =-=[26]-=- [27, Remark 5]. This results in a computation rate given by R(hm, am, αm) = log + ( P |αm|2 + P‖αmhm − am‖2 ) , (28) where log+(.) , max{0, log(.)}. Intuitively speaking, one can arbitrarily rotate a... |

99 | Averaging bounds for lattices and linear codes
- Loeliger
- 1997
(Show Context)
Citation Context ...algebraic structure of lattices. Using lattices for communication has a rich history in the literature. Typically, a lattice that is Poltyrev-good is required to guarantee reliable communication [16] =-=[19]-=- [20]. In addition to the Poltyrev-goodness, shaping has to be taken into account in order to achieve the AWGN channel capacity. By carefully shaping the lattices with their sublattices, Erez and Zami... |

96 | Capacity of the Gaussian two-way relay channel to within 1/2 bit
- Nam, Chung, et al.
(Show Context)
Citation Context ...ith lattice decoding [14]. Functional computation in physical layer with such lattices has been realized to asymptotically approach the capacity for the bidirectional relay networks in [21] [22] [23] =-=[24]-=-. The reader is referred to a tutorial paper [25] for more details about using lattices for the bidirectional relay channels. One of the main contribution of [1] is to provide a means to harness inter... |

74 |
On coding without restrictions for the AWGN channel
- Poltyrev
- 1994
(Show Context)
Citation Context ...t approach the Poltyrev limit defined in (13). There is a stronger version of Poltyrev-goodness stating that the sequence of lattices achieves an error exponent lower bounded by the Poltyrev exponent =-=[16]-=-. However, the proof of achieving Poltyrev exponent is more involved and is not required to prove the main results in this paper. Hence, we do not pursue it in this paper. The interested reader is ref... |

63 | Iterative decoding threshold analysis for LDPC convolutional codes,”
- Lentmaier, Sridharan, et al.
- 2010
(Show Context)
Citation Context ...cheme with an ensemble of linear codes with iterative decoding. Recently, it has been shown in [43] [44] [45] that the ensemble of spatially-coupled LDPC codes (or LDPC convolutional codes) [46] [47] =-=[48]-=- universally achieve the capacity for the class of binary memoryless symmetric channels under belief propagation (BP) decoding [28]. Motivated by this success, we choose the linear code at the lth lev... |

60 | Capacity bounds for two-way relay channels - Nam, Chung, et al. - 2008 |

59 |
Sphere-bound-achieving coset codes and multilevel coset codes,”
- Forney, Trott, et al.
- 2000
(Show Context)
Citation Context ...raic structure of lattices. Using lattices for communication has a rich history in the literature. Typically, a lattice that is Poltyrev-good is required to guarantee reliable communication [16] [19] =-=[20]-=-. In addition to the Poltyrev-goodness, shaping has to be taken into account in order to achieve the AWGN channel capacity. By carefully shaping the lattices with their sublattices, Erez and Zamir sho... |

55 |
Low-Density Parity Check Codes Over GF(q
- Davey, Mackay
- 1998
(Show Context)
Citation Context ...hile for the proposed product construction lattices, it only depends on the greatest divisor of |Λ∗|. For coding over Fq, we assume that a q-ary LDPC code is implemented and the decoding algorithm in =-=[53]-=- with complexity O(q log(q)) is adopted. In Fig. 22, we provide a comparison of the decoding complexity for Construction A lattices and the proposed product construction lattices. Note that for the pr... |

54 | Reliable physical layer network coding
- Nazer, Gastpar
- 2011
(Show Context)
Citation Context ... in physical layer with such lattices has been realized to asymptotically approach the capacity for the bidirectional relay networks in [21] [22] [23] [24]. The reader is referred to a tutorial paper =-=[25]-=- for more details about using lattices for the bidirectional relay channels. One of the main contribution of [1] is to provide a means to harness interference when there is no channel state informatio... |

54 |
Spatially coupled ensembles universally achieve capacity under belief propagation,”
- Kudekar, Richardson, et al.
- 2012
(Show Context)
Citation Context ...d can approach those theoretic limits with reasonable complexity, we also simulate 31 the proposed scheme with an ensemble of linear codes with iterative decoding. Recently, it has been shown in [43] =-=[44]-=- [45] that the ensemble of spatially-coupled LDPC codes (or LDPC convolutional codes) [46] [47] [48] universally achieve the capacity for the class of binary memoryless symmetric channels under belief... |

41 | An algebraic approach to physical-layer network coding - Feng, Silva, et al. |

41 | Zigangirov, “Convergence analysis of a class of LDPC convolutional codes for the erasure channel
- Sridharan, Lentmaier, et al.
- 2004
(Show Context)
Citation Context ...proposed scheme with an ensemble of linear codes with iterative decoding. Recently, it has been shown in [43] [44] [45] that the ensemble of spatially-coupled LDPC codes (or LDPC convolutional codes) =-=[46]-=- [47] [48] universally achieve the capacity for the class of binary memoryless symmetric channels under belief propagation (BP) decoding [28]. Motivated by this success, we choose the linear code at t... |

40 | Sphere packing and error-correcting codes
- Leech, Sloane
- 1971
(Show Context)
Citation Context ...wn to be crucial for compute-and-forward. For example, 1 and 1 + ω in C correspond to [5, 0] and [1, 0] in F7, respectively, and 1+ (1+ω) = 2+ω in C corresponds to [6, 1] in F27 which is not equal to =-=[3, 0]-=-+ [1, 0] = [4, 0] in F27. The lack of a homomorphism renders the Ungerboeck’s labeling ineffective for compute-and-forward in terms of the achievable computation rate. This coincides with and reinforc... |

33 |
Coset codes - part I: introduction and geometrical classification
- Forney
- 1988
(Show Context)
Citation Context ...M(gi)|ali ∈ F|φ|2 , (38) where all the operations are over C. A variant of Construction D called Construction by Code Formula has attracted a lot of attention since its introduction by Forney in =-=[32]-=-, see for example [33] [34] [35]. Here, we also provide an extension of Construction by Code Formula to the complex field C. It is known that Construction by Code Formula does not always produce a lat... |

30 | Terminated ldpc convolutional codes with thresholds close to capacity
- Lentmaier, Sridharan, et al.
(Show Context)
Citation Context ...sed scheme with an ensemble of linear codes with iterative decoding. Recently, it has been shown in [43] [44] [45] that the ensemble of spatially-coupled LDPC codes (or LDPC convolutional codes) [46] =-=[47]-=- [48] universally achieve the capacity for the class of binary memoryless symmetric channels under belief propagation (BP) decoding [28]. Motivated by this success, we choose the linear code at the lt... |

29 | On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels - Forney - 2003 |

26 |
Algebra, Graduate Texts in Mathematics,
- Hungerford
- 1980
(Show Context)
Citation Context ...ur results in the following sections. All the Lemmas are provided without proofs for the sake of brevity; however, their proofs can be found in standard textbooks of abstract algebra, see for example =-=[17]-=-. We first recall some basic definitions for commutative rings where many of them are covered in [5] (for those not in [5], the reader is referred to [17]). Let R be a commutative ring. Let a, b 6= 0 ... |

22 | Multilevel coding schemes for compute-and-forward with flexible decoding,” Computing Research Repository (CoRR
- Hern, Narayanan
(Show Context)
Citation Context ...y choosing any two linearly independent vectors as generators. Furthermore, as will be discussed later on, for this subclass of constellations, it is possible to include the idea of flexible decoding =-=[38]-=-. In what follows, we provide several interesting examples. Example 27. Let φ = 3 + 2ω with |φ|2 = 7. Thus, we have Z[ω]/φφ̄Z[ω] ∼= F27. The constellation and a Z-module homomorphisms generated by M(v... |

19 |
Performance analysis of a multilevel coded modulation system
- Kofman, Zehavi, et al.
- 1994
(Show Context)
Citation Context ...ension n. Similar to the first level, the decoder then forms the lth level’s estimate as ĉlR = argmax c∈Cl N∏ n=1 P ( c̃lR[n] = c[n]|y[n], ĉ1R[n], . . . , ĉl−1R [n] ) . (80) Remark 31. As shown in =-=[40]-=-, for traditional multilevel coding scheme, adding an interleaver/deinterleaver pair prevents burst error propagation to the next level and hence improves the error probability. Similarly, for the pro... |

19 |
Algebraic construction of shannon codes for regular channels
- Delsarte, Piret
- 1982
(Show Context)
Citation Context ...e the conditional mutual information I(Y;Cl|C1, . . . ,Cl−1). To this end, we follow the proof in [20] and show that the equivalent channel at each level is regular in the sense of Delsarte and Piret =-=[54]-=-. As restated in [20], a channel with transition probabilities {f(y|b), b ∈ B, y ∈ Y } is regular if the input alphabet can be identified with an Abelian group B that acts on the output alphabet Y by ... |

18 | The approximate sum capacity of the symmetric Gaussian K-user interference channel,” arXiv:1206.0197
- Ordentlich, Erez, et al.
- 2012
(Show Context)
Citation Context ...sed scheme are suitable for many other applications that use the lattice structure such as integer-forcing linear receivers [9], precoded compute-and-forward [10], lattice interference alignment [11] =-=[12]-=-, etc. A. Organization The paper is organized as follows. In Section II, we state the compute-and-forward relay network of Nazer and Gastpar [1] and the problem of maximizing the computation rates. In... |

18 | The degrees of freedom of compute-and-forward
- Niesen, Whiting
- 2012
(Show Context)
Citation Context ...ore general model where they can have different rates was considered in [1]. III. PRELIMINARIES It should be noted that using the theory in Diophantine approximation, Niesen and Whiting have shown in =-=[13]-=- that the lattice-based compute-and-forward described above is in general very inefficient in terms of degrees of freedom and a coding scheme relying on the channel state information at transmitters h... |

10 |
Bandwidth efficient low density parity check coding using multilevel coding and iterative multistage decoding
- Narayanan, Li
- 2000
(Show Context)
Citation Context ...level to mitigate the effect of erroneous decoding at the previous first levels. Another way to potentially further lower the error probability is to perform iterative multistage decoding proposed in =-=[41]-=-. However, since the focus of this paper is on the analysis of achievable computation rate instead of error probability, we do not pursue these potential extensions. B. Suboptimal Decoders Motivated b... |

10 | A proof of threshold saturation for spatially-coupled
- Kumar, Young, et al.
- 2012
(Show Context)
Citation Context ... approach those theoretic limits with reasonable complexity, we also simulate 31 the proposed scheme with an ensemble of linear codes with iterative decoding. Recently, it has been shown in [43] [44] =-=[45]-=- that the ensemble of spatially-coupled LDPC codes (or LDPC convolutional codes) [46] [47] [48] universally achieve the capacity for the class of binary memoryless symmetric channels under belief prop... |

9 |
A simple proof for the existence of “good” pairs of nested lattices,”
- Ordentlich, Erez
- 2012
(Show Context)
Citation Context ...h are Poltyrev-good under multistage decoding and good for mean-squared error (MSE) quantization. We then generate a sequence of nested lattice codes by extending the result by Ordentlich and Erez in =-=[6]-=- to the proposed lattices. This sequence of nested lattice codes is adopted for the compute-and-forward problem and a novel strategy called multistage compute-and-forward is proposed which can recover... |

9 | Polar lattices: Where Arıkan meets Forney,” in
- Yan, Ling, et al.
- 2013
(Show Context)
Citation Context ...8) where all the operations are over C. A variant of Construction D called Construction by Code Formula has attracted a lot of attention since its introduction by Forney in [32], see for example [33] =-=[34]-=- [35]. Here, we also provide an extension of Construction by Code Formula to the complex field C. It is known that Construction by Code Formula does not always produce a lattice and it has been shown ... |

9 | Integer low-density lattices based on construction
- Pietro, Boutros, et al.
- 2012
(Show Context)
Citation Context ...troduced in [43, Section II-A] (here we use L′ to denote the coupling length instead of L to avoid confusion with the number of levels L). This can be regarded as the extension of the LDA lattices in =-=[49]-=- [50] or the SCLDA lattices in [51] to the proposed product construction. A. Comparison of Different Mappings For point-to-point communication, one of the most popular and most frequently used labelin... |

8 |
On the robustness of lattice interference alignment,”
- Ordentlich, Erez
- 2013
(Show Context)
Citation Context ...proposed scheme are suitable for many other applications that use the lattice structure such as integer-forcing linear receivers [9], precoded compute-and-forward [10], lattice interference alignment =-=[11]-=- [12], etc. A. Organization The paper is organized as follows. In Section II, we state the compute-and-forward relay network of Nazer and Gastpar [1] and the problem of maximizing the computation rate... |

8 |
Zur Verallgemeinerung des Bertrandschen Postulates, dass zwischen x und 2x stets Primzahlen liegen
- Breusch
- 1932
(Show Context)
Citation Context ...|2 = φ · φ̄ being rational primes congruent to 1 mod 3 (1 mod 4), one can verify that φ and φ̄ are both Eisenstein primes (Gaussian primes) but they are not associates. Moreover, it has been shown in =-=[18]-=- that for every x ≥ 7, there exists a rational prime of this form between x and 2x. Thus, the choices of φ satisfying the above property are abundant. In the following sections, we will focus on the r... |

7 |
Lattices over Eisenstein integers for compute-and-forward
- Tunali, Narayanan, et al.
- 2012
(Show Context)
Citation Context ...ibe the scheme by directly looking at the complex field and Gaussian integers. In fact, this has motivated the generalization of the compute-and-forward paradigm to the ring of Eisenstein integers in =-=[2]-=- where each element in A is chosen from Z[ω] instead of Z[i]. 9c1 FNp C M w1 x1 FN ′ p F N p AN C M cK xK AN . . . . . . channel wK FN ′ p ĉR1 APP C−1 y1 FNpR N . . . . . . û1 FN ′ p ĉRM APP C−1 yM... |

5 | Concatenated signal codes with applications to compute and forward - Tunali, Narayanan |

5 |
New lattice packings of spheres,” Canad
- Barnes, Sloane
- 1983
(Show Context)
Citation Context ...plexity is not determined by the number of elements in Λ∗ but by the greatest divisor in the prime factorization of |Λ∗|. However, the complexity is higher than that of the Construction D lattices in =-=[29]-=- [20] whose complexity is always determined by coding over F2. This is a direct consequence of that all primes should be distinct in the proposed product construction. Remark 17. The possible value of... |

5 | New results on Construction A lattices based on very sparse parity-check matrices
- Pietro, Zémor, et al.
- 2013
(Show Context)
Citation Context ...ced in [43, Section II-A] (here we use L′ to denote the coupling length instead of L to avoid confusion with the number of levels L). This can be regarded as the extension of the LDA lattices in [49] =-=[50]-=- or the SCLDA lattices in [51] to the proposed product construction. A. Comparison of Different Mappings For point-to-point communication, one of the most popular and most frequently used labeling str... |

4 | Structured Lattice Codes for Some TwoUser Gaussian Networks with
- Hong, Caire
- 2013
(Show Context)
Citation Context ...sed construction of lattices and the proposed scheme are suitable for many other applications that use the lattice structure such as integer-forcing linear receivers [9], precoded compute-and-forward =-=[10]-=-, lattice interference alignment [11] [12], etc. A. Organization The paper is organized as follows. In Section II, we state the compute-and-forward relay network of Nazer and Gastpar [1] and the probl... |

4 | Construction of Barnes-Wall Lattices from Linear Codes over Rings
- Harshan, Viterbo, et al.
(Show Context)
Citation Context ... , (38) where all the operations are over C. A variant of Construction D called Construction by Code Formula has attracted a lot of attention since its introduction by Forney in [32], see for example =-=[33]-=- [34] [35]. Here, we also provide an extension of Construction by Code Formula to the complex field C. It is known that Construction by Code Formula does not always produce a lattice and it has been s... |

4 |
Spatially-coupled low density lattices based on construction A with applications to compute-and-forward
- Tunali, Narayanan, et al.
- 2013
(Show Context)
Citation Context ...e we use L′ to denote the coupling length instead of L to avoid confusion with the number of levels L). This can be regarded as the extension of the LDA lattices in [49] [50] or the SCLDA lattices in =-=[51]-=- to the proposed product construction. A. Comparison of Different Mappings For point-to-point communication, one of the most popular and most frequently used labeling strategy for multilevel coding is... |

2 | An algorithm to generate square-free numbers and to compute the Möbius function
- Auil
- 2013
(Show Context)
Citation Context ... A lattices as special cases. The square-free integers are closely related to the Möbius function and can be identified efficiently without factorizing integers. The interested reader is referred to =-=[31]-=-. 12 A. Comparison with Construction D Lattices and its Variant At first glance, due to its multilevel nature, the proposed product construction looks similar to Construction D [29] [4, Page 232]. We ... |

2 |
Integer-forcing linear receiver design with slowest descent method
- Wei, Chen
(Show Context)
Citation Context ...maximizes the achievable rates. Thus, efficient algorithms are called for. So far there have been some work in the literature on efficient algorithms of finding approximately optimal linear functions =-=[52]-=-. It is interesting to design efficient algorithms for the proposed scheme with flexible decoding in which the functions may not be linear. We leave this problem to future work. Nevertheless, the resu... |

1 |
Integer-forcing linear receivers,” arXiv:1003.5966v3 [cs.IT
- Zhan, Nazer, et al.
- 2013
(Show Context)
Citation Context ... on compute-and-forward, the proposed construction of lattices and the proposed scheme are suitable for many other applications that use the lattice structure such as integer-forcing linear receivers =-=[9]-=-, precoded compute-and-forward [10], lattice interference alignment [11] [12], etc. A. Organization The paper is organized as follows. In Section II, we state the compute-and-forward relay network of ... |

1 |
Sequence A005117 in OEIS,” The On-Line Encyclopedia of Integer Sequences. Available online: http://oeis.org/A005117
- Sloane
(Show Context)
Citation Context ...inct in the proposed product construction. Remark 17. The possible value of |Λ∗| is confined in a subset of N. For example, for Z, the proposed constellation allows |Λ∗| to be any square-free integer =-=[30]-=-. Nonetheless, the choices of such |Λ∗| are very rich and absorb Construction A lattices as special cases. The square-free integers are closely related to the Möbius function and can be identified ef... |

1 |
Connections between construction D and related constructions of lattices,” arXiv:1308.6175 [cs.IT
- Kositwattanarerk, Oggier
- 2013
(Show Context)
Citation Context ...ere all the operations are over C. A variant of Construction D called Construction by Code Formula has attracted a lot of attention since its introduction by Forney in [32], see for example [33] [34] =-=[35]-=-. Here, we also provide an extension of Construction by Code Formula to the complex field C. It is known that Construction by Code Formula does not always produce a lattice and it has been shown very ... |