Lattices which are good for (almost) everything

Erez, Uri; Litsyn, Simon; Zamir, Ram

doi:10.1109/itw.2003.1216746

Cited by 130 publications

(254 citation statements)

References 30 publications

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“…Since C is in the region of n-cube [0, p) n , the lattice Λ is obtained by tessellating R n with translations of C, i.e., Λ = C + pZ n , thus Λ is a sub-lattice of Z n . Furthermore, "construction A" of random ensembles of lattices achieve the interference-free AWGN channel capacity [7,2], therefore good lattices can be generated by "construction A". Generally, a sub-lattice Λ 2 of Λ 1 induces a nested lattice partition…”

Section: The Nested Lattices Constructionmentioning

confidence: 99%

Combined shaping and precoding for interference cancellation at low SNR

Philosof

Erez²,

Zamir³

2003

IEEE International Symposium on Information Theory, 2003. Proceedings.

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We investigate the application of the nested coding framework for cancelling known interference at low SNR, that is P X /P Z ≈ 1 and below. We consider multi-dimensional precoding, with anticipation of N s > 1 "future" interference samples. Unlike non-precoded transmission, where capacity can be achieved at low SNR without shaping, optimum precoding at low SNR does require shaping. Eyuboglu and Forney's trellis precoding scheme combines Tomlinson-Harashima precoding and trellis shaping, to achieve both coding and shaping gains in transmission over inter-symbol interference channels with Euclidean distance decoder. However, the standard configuration of this scheme does not support precoding at low SNR, where the capacity is less than 1 bit dim . We propose a low rate precoding scheme which combines MMSE estimation, dithering and a variant of nested codes, based on concatenation of a "syndrome dilution" code and a "syndrome-to-coset" modulation code. We provide simulation results for several configurations of nested trellis codes, which reduce the gap to capacity.

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Section: The Nested Lattices Constructionmentioning

confidence: 99%

Combined shaping and precoding for interference cancellation at low SNR

Philosof

Erez²,

Zamir³

2003

IEEE International Symposium on Information Theory, 2003. Proceedings.

Self Cite

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“…The existence of good lattice codes and good nested lattice codes (for various notions of goodness) has been studied in [46,47] which use the random coding method of [49,52]. In [47], it was shown that there exists lattices which are simultaneously good in both the source and channel coding senses described above.…”

Section: Nested Lattice Codesmentioning

confidence: 99%

“…In [47], it was shown that there exists lattices which are simultaneously good in both the source and channel coding senses described above.…”

Section: Nested Lattice Codesmentioning

confidence: 99%

See 2 more Smart Citations

Lattices for Distributed Source Coding: Jointly Gaussian Sources and Reconstruction of a Linear Function

Krithivasan

Pradhan

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

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Consider a pair of correlated Gaussian sources (X1, X2). Two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder. The decoder is interested in reconstructing a linear combination of X1 and X2 to within a mean-square distortion of D. We obtain an inner bound to the optimal rate-distortion region for this problem. A portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components X1 and X2 first. This results in a better rate region for certain parameter values. Our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory.We then consider the case of linear reconstruction of K sources and provide an inner bound to the optimal rate-distortion region. Some parts of the inner bound are achieved using the following coding structure: lattice vector quantization followed by "correlated" lattice-structured binning.

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The case for structured random codes in network capacity theorems

Nazer

Gastpar

2008

Trans Emerging Tel Tech

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SUMMARYRandom coding arguments are the backbone of most channel capacity achievability proofs. In this paper, we show that in their standard form, such arguments are insufficient for proving some network capacity theorems: structured coding arguments, such as random linear or lattice codes, attain higher rates. Historically, structured codes have been studied as a stepping stone to practical constructions. However, Körner and Marton demonstrated their usefulness for capacity theorems through the derivation of the optimal rate region of a distributed functional source coding problem. Here, we use multicasting over finite field and Gaussian multiple-access networks as canonical examples to demonstrate that even if we want to send bits over a network, structured codes succeed where simple random codes fail. Beyond network coding, we also consider distributed computation over noisy channels and a special relay-type problem.

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Lattices which are good for (almost) everything

Cited by 130 publications

References 30 publications

Combined shaping and precoding for interference cancellation at low SNR

Combined shaping and precoding for interference cancellation at low SNR

Lattices for Distributed Source Coding: Jointly Gaussian Sources and Reconstruction of a Linear Function

The case for structured random codes in network capacity theorems

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