Integer low-density lattices based on construction A

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

doi:10.1109/itw.2012.6404707

Cited by 47 publications

(41 citation statements)

References 22 publications

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“…Messages from and to lattice coordinates are converted up and down to the finite field F p as in [17]. It is well-known that iterative decoding needs an underlying graph without small cycles.…”

Section: Choice Of the Component Lattice λmentioning

confidence: 99%

Generalized low-density (GLD) lattices

Boutros

Pietro

Basha

2014

2014 IEEE Information Theory Workshop (ITW 2014)

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We propose the construction of a new family of lattice sphere packings. Given a small-dimensional lattice, we start by building a first lattice in a large dimension by the direct sum of the small lattice. Then, the coordinates of the first large lattice are permuted to yield a second large-dimensional lattice. Finally, our generalized low-density (GLD) lattice is the intersection of the first and the second lattice. We restrict our construction in this paper to integer lattices. GLD lattices are the result of mixing classical lattice theory with modern coding theory. They are potential candidates not only for channel coding as coded modulations, but also for physical-layer network coding and for secure digital communications.

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Section: Choice Of the Component Lattice λmentioning

confidence: 99%

Generalized low-density (GLD) lattices

Boutros

Pietro

Basha

2014

2014 IEEE Information Theory Workshop (ITW 2014)

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“…Improving C won't improve the Euclidean distance of E 8 because p = 2. This simple example shows us that avoiding Construction D with multiple nested binary codes is possible by using a more powerful code C and a larger finite field size p. Hence, LDA lattices [13] refer to the Construction A case where C is an LDPC code and p > 2.…”

Section: Lattices and Construction Amentioning

confidence: 99%

“…A new approach was proposed by the present authors in [13] [14] relying on Construction A together with p-ary LDPC codes [15] [16]. Experiments with reasonable values of p, such as p = 11 and p = 41, showed excellent results with those lattices, referred to as LDA lattices.…”

Section: Introductionmentioning

confidence: 98%

New results on low-density integer lattices

Pietro

Boutros

Zémor

et al. 2013

2013 Information Theory and Applications Workshop (ITA)

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Abstract-A new family of integer lattices built from Construction A and non-binary low-density parity-check (LDPC) codes has been proposed by the authors in 2012. Lattices in this family are referred to as LDA lattices. Previous experimental results revealed excellent performance which clearly single out LDA lattices among the strongest candidates for potential applications in digital communications and networks, such as network coding and information theoretic security at the physical layer level. In this paper, we show that replacing random codes by LDPC codes in Construction A does not induce any structural loss. More precisely, our main theorem states that LDA lattices can achieve Poltyrev capacity limit on an additive white Gaussian noise channel. We present here the detailed proof and its consequences on the lattice dimension, the finite field size, and the parameters of the LDPC ensemble. The latter has a row weight that increases logarithmically in the code length. In a more recent work, it is proved that the Poltyrev limit is attained by a different LDA ensemble having a small constant row weight.

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“…Such lattices are referred to as LDA lattices and they perform close to capacity on the Gaussian channel, in addition to being conceptually simpler than previously proposed lattices based on multiple nested binary codes. In particular, in [19], several integer LDA lattices, all based on a particular (2, 5)-regular LDPC code over the prime field of size 11, were proposed. For dimension 5000, the lattice attains a symbol error rate (under low-complexity iterative decoding) of less than 10 −6 at 1 dB from capacity.…”

Section: Introductionmentioning

confidence: 99%

On adaptive linear programming decoding of nonbinary linear codes over prime fields

Rosnes

Helmling

2016

2016 9th International Symposium on Turbo Codes and Iterative Information Processing (ISTC)

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In this work, we study linear programming (LP) decoding of nonbinary linear codes over prime fields. In particular, we develop a novel separation algorithm for valid inequalities describing the codeword polytope of the so-called constant-weight embedding of a single parity-check (SPC) code over any prime field. The algorithm has linear (in the length of the SPC code) complexity, is structurally different from the one for binary codes, and is based on the principle of dynamic programming. Furthermore, it is the basis of the proposed efficient (relaxed) adaptive LP (ALP) decoder for general (non-SPC) linear codes over any prime field, generalizing the well-known ALP decoding algorithm for binary codes. Numerical results show that the ALP decoding algorithm is very efficient compared to a static approach

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Integer low-density lattices based on construction A

Cited by 47 publications

References 22 publications

Generalized low-density (GLD) lattices

Generalized low-density (GLD) lattices

New results on low-density integer lattices

On adaptive linear programming decoding of nonbinary linear codes over prime fields

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