New results on Construction A lattices based on very sparse parity-check matrices

Pietro, Nicola di; Zémor, Gilles; Boutros, Joseph J.

doi:10.1109/isit.2013.6620512

Cited by 25 publications

(17 citation statements)

References 14 publications

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“…The ensemble we will use is the (d l , d r , L ′ ) ensemble introduced 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.…”

Section: Simulation Resultsmentioning

confidence: 99%

Multistage compute-and-forward with multilevel lattice codes based on product constructions

Huang

Narayanan

2014

2014 IEEE International Symposium on Information Theory

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A novel construction of lattices is proposed. This construction can be thought of as Construction A with codes that can be represented as the Cartesian product of L linear codes over F p1 , . . . , F pL , respectively; hence, is referred to as the product construction. The existence of a sequence of such lattices that are good for quantization and Poltyrev-good under multistage decoding is shown. This family of lattices is then used to generate a sequence of nested lattice codes which allows one to achieve the same computation rate of Nazer and Gastpar for computeand-forward under multistage decoding, which is referred to as lattice-based multistage compute-and-forward.Motivated by the proposed lattice codes, two families of signal constellations are then proposed for the separation-based compute-and-forward framework proposed by Tunali et al. together with a multilevel coding/multistage decoding scheme tailored specifically for these constellations. This scheme is termed separation-based multistage compute-and-forward and is shown having a complexity of the channel coding dominated by the greatest common divisor of the constellation size (may not be a prime number) instead of the constellation size itself. I. INTRODUCTIONCompute-and-forward is a novel information forwarding paradigm in wireless communications in which relays in a network decode functions of signals transmitted from multiple transmitters and forward them to a central destination. If these functions are chosen as linear integer combinations, lattice codes are one of the most natural ways to implement a compute-and-forward scheme since a lattice is closed under addition. If the channel state information is not available at the transmitters, compute-and-forward can be implemented effectively by allowing the relay to choose 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 successfully extended the result on achievable rates in [1] to lattices over Eisenstein integers.The lattice codes adopted in [1] are based on those generated by Construction A [3] [4] whose decoding complexity typically depend on decoding of the underlying linear codes. One main drawback of the Construction A lattices is that in order to be Poltyrev-good, the underlying linear codes have to be implemented over very large prime fields which in turn result in high decoding complexity. To alleviate this drawback, in the first part of the paper, inspired by Theorem 2 in [5], we propose a novel lattice construction called product construction that can be thought of as Construction A [3] with codes which can be represented as the Cartesian product of L linear codes over F p 1 , . . . , F p L , respectively. This construction is shown to b...

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Section: Simulation Resultsmentioning

confidence: 99%

Multistage compute-and-forward with multilevel lattice codes based on product constructions

Huang

Narayanan

2014

2014 IEEE International Symposium on Information Theory

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“…While we have particularized to LDLCs herein, we hasten to point out that the techniques presented in this paper can be applied to any coding lattice with a check matrix with a lowertriangular structure, including LDA lattices [10] and onelevel LDPC lattices [24]. The proposed techniques therefore offer a general step towards the practical realization of the performance advances promised by lattice codes.…”

Section: Discussionmentioning

confidence: 99%

“…3.23], we know x 1 has the same NSM as Λ s,m . Therefore, x 1 has the same shaping gain as Λ s,m based on (10). The final lattice codeword can be also represented as…”

Section: Coding Latticementioning

confidence: 99%

Low-Dimensional Shaping for High-Dimensional Lattice Codes

Ferdinand

Kurkoski

Nokleby

et al. 2016

IEEE Trans. Wireless Commun.

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We propose two low-complexity lattice code constructions that have competitive coding and shaping gains. The first construction, named systematic Voronoi shaping, maps short blocks of integers to the dithered Voronoi integers, which are dithered integers that are uniformly distributed over the Voronoi region of a low-dimensional shaping lattice. Then, these dithered Voronoi integers are encoded using a high-dimensional lattice retaining the same shaping and coding gains of low and highdimensional lattices. A drawback to this construction is that there is no isomorphism between the underlying message and the lattice code, preventing its use in applications such as computeand-forward. Therefore we propose a second construction, called mixed nested lattice codes, in which a high-dimensional coding lattice is nested inside a concatenation of low-dimensional shaping lattices. This construction not only retains the same shaping/coding gains as first construction but also provides the desired algebraic structure. We numerically study these methods, for point-to-point channels as well as compute-and-forward using low-density lattice codes (LDLCs) as coding lattices and E8 and Barnes-Wall as shaping lattices. Numerical results indicate a shaping gain of up to 0.86 dB, compared to the state-of-the-art of 0.4 dB; furthermore, the proposed method has lower complexity than state-of-the-art approaches.

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“…Gaborit and Zémor [24] proposed a lattice family of high fundamental gain obtained by Construction A with random non-binary codes. In a more work, di Pietro et al [25] established integer low-density lattices based on Construction A and non-binary LDPC codes, referred to as LDA lattices, and proved that LDA lattice ensemble could attain Poltyrev limit.…”

Section: Constructions Of Lattices and Practical Lattice Codesmentioning

confidence: 99%

Study Status and Prospect of Lattice Codes in Wireless Communication

Duan

Zhao

et al. 2015

2015 Fifth International Conference on Instrumentation and Measurement, Computer, Communication and Control (IMCCC)

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Lattice codes can achieve the capacity of additive white Gaussian noise channel with and without power construction. It is well known that lattice codes can provide a classical information theoretic way to obtain achievable rate and performance gains for point-to-point Gaussian channels. The coding scheme based on lattice codes can be used as building blocks for practical applications, such as point-topoint communication systems, Gaussian networks and MIMO communication systems etc. More importantly, lattice codes can be used to investigate new link adaptive coding and modulation schemes and design sparse code multiple access codebooks. So, lattice codes could be a potential candidate for 5G communication. This paper provides a comprehensive review of the state-of-the-art of lattice codes. It first describes the theory of lattice codes. Then, a comprehensive survey to easily understand channel capacity, error exponents and practical lattice codes is outlined. The application of nested lattice codes is also presented for lossy source coding, dirty paper channel coding and relay communication. Finally, some unresolved issues and ongoing developments related to further improvements of lattice codes are highlighted.

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New results on Construction A lattices based on very sparse parity-check matrices

Abstract: International audienc

Cited by 25 publications

References 14 publications

Multistage compute-and-forward with multilevel lattice codes based on product constructions

Multistage compute-and-forward with multilevel lattice codes based on product constructions

Low-Dimensional Shaping for High-Dimensional Lattice Codes

Study Status and Prospect of Lattice Codes in Wireless Communication

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