Wiretap encoding of lattices from number fields using codes over F&lt;inf&gt;p&lt;/inf&gt;

Kositwattanarerk, Wittawat; Ong, Soon Sheng; Oggier, Frédérique

doi:10.1109/isit.2013.6620699

Cited by 9 publications

(14 citation statements)

References 6 publications

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“…7, respectively. Based on these designed code rates, the corresponding information rates are calculated by using (19) as R 1 = 1.741 bits/s/Hz, R 2 = 1.548 bits/s/Hz and R 3 = 1.161 bits/s/Hz, respectively. The corresponding unrestricted Shannon limit and uniform input capacity for each information rate are plotted in each figure.…”

Section: Simulation Resultsmentioning

confidence: 99%

On the design of multi-dimensional irregular repeat-accumulate lattice codes

Qiu

Yang

Xie

et al. 2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Most multi-dimensional (more than two dimensions) lattice partitions only form additive quotient groups and lack multiplication operations. This prevents us from constructing lattice codes based on multi-dimensional lattice partitions directly from non-binary linear codes over finite fields. In this paper, we design lattice codes from Construction A lattices where the underlying linear codes are non-binary irregular repeataccumulate (IRA) codes. Most importantly, our codes are based on multi-dimensional lattice partitions with finite constellations. We propose a novel encoding structure that adds randomly generated lattice sequences to the encoder's messages, instead of multiplying lattice sequences to the encoder's messages. We prove that our approach can ensure that the decoder's messages exhibit permutation-invariance and symmetry properties. With these two properties, the densities of the messages in the iterative decoder can be modeled by Gaussian distributions described by a single parameter. With Gaussian approximation, extrinsic information transfer (EXIT) charts for our multi-dimensional IRA lattice codes are developed and used for analyzing the convergence behavior and optimizing the decoding thresholds. Simulation results show that our codes can approach the unrestricted Shannon limit within 0.46 dB and outperform the previously designed lattice codes with two-dimensional lattice partitions and existing lattice coding schemes for large codeword length.

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Section: Simulation Resultsmentioning

confidence: 99%

On the design of multi-dimensional irregular repeat-accumulate lattice codes

Qiu

Yang

Xie

et al. 2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…In [10], Ebeling considers the construction when K is a cyclotomic field Q(ζ p ), and we will outline this construction in Example 1. Its maximal real subfield Q(ζ p + ζ −1 p ) is studied in [12] and [19]. In [12], the reduction is done by the ideal (2m), yielding codes over a ring of polynomials with coefficients modulo 2m whereas in [19] the reduction is done by the ideal (2 − ζ p + ζ −1 p ) and the resulting codes are over F p .…”

Section: A General Lattice Constructionmentioning

confidence: 99%

“…Its maximal real subfield Q(ζ p + ζ −1 p ) is studied in [12] and [19]. In [12], the reduction is done by the ideal (2m), yielding codes over a ring of polynomials with coefficients modulo 2m whereas in [19] the reduction is done by the ideal (2 − ζ p + ζ −1 p ) and the resulting codes are over F p . Alternatively, quadratic extensions K = Q( √ −l) are considered for example in [13,20] where the reduction is done by the ideal (p e ) and the resulting codes are over the ring O K /p e O K .…”

Section: A General Lattice Constructionmentioning

confidence: 99%

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Construction A of Lattices Over Number Fields and Block Fading (Wiretap) Coding

Kositwattanarerk

Ong

Oggier

2015

IEEE Trans. Inform. Theory

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We propose a lattice construction from totally real and complex multiplication fields, which naturally generalizes Construction A of lattices from p-ary codes obtained from the cyclotomic field Q(ζ p ), p a prime, which in turn contains the so-called Construction A of lattices from binary codes as a particular case. We focus on the maximal totally real subfield Q(ζ p r + ζ −r p ) of the cyclotomic field Q(ζ p r ), r ≥ 1. Our construction has applications to coset encoding of algebraic lattice codes for block fading channels, and in particular for block fading wiretap channels.

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“…In effect, let x = γ ρ x π + r x and y = γ ρ y π + r y we have that ϕ π (x + y) = γ ρ x π + r x + γ ρ y π + r y (mod π) so that ϕ π (x + y) = r x + r y , which is clearly equivalent to ϕ π (x) + ϕ π (y) = r x + r y . As for the claim of the generator matrix, compute a lattice point x using the matrix and show that µ π (ϕ−1 π (x)) is in C [k, n, d H ] p and conclude using a volume argument as in [15] Definition 12: Volume of a lattice. The volume of the lattice Λ (volume of the fundamental parallelotope) is Vol (Λ) = |det (G)| .…”

Section: The Five Families Of (Nested) Lattice Codes Over Euclidementioning

confidence: 99%

Arithmetic geometry of compute and forward

Vázquez-Castro

2014

2014 IEEE Information Theory Workshop (ITW 2014)

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We propose the joint study of function computation (arithmetics) and lattice coding gain (geometry) to derive the (complex modulo) arithmetics of compute and forward over Euclidean geometry. First, we demonstrate that only five families of complex alphabets exist that admit euclidean complex modulo arithmetics. Second, we prove that the (per-dimension) euclidean division algorithm is equivalent to a closest vector algorithm hence a natural framework for compute and forward. Third, we derive the nominal coding gains of the five resulting families of (nested) lattice codes obtained as preimages of linear block codes. Finally we apply the proposed arithmetic geometry framework to the MAC channel and show graphical illustration of the 2-user case over the Eisenstein numbers.

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Wiretap encoding of lattices from number fields using codes over F<inf>p</inf>

Cited by 9 publications

References 6 publications

On the design of multi-dimensional irregular repeat-accumulate lattice codes

On the design of multi-dimensional irregular repeat-accumulate lattice codes

Construction A of Lattices Over Number Fields and Block Fading (Wiretap) Coding

Arithmetic geometry of compute and forward

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