Encoding and Decoding Construction D' Lattices for Power-Constrained Communications (original) (raw)

Power-constrained communications using LDLC lattices

2009 IEEE International Symposium on Information Theory, 2009

An explicit code construction for using low-density lattice codes (LDLC) on the constrained power AWGN channel is given. LDLC lattices can be decoded in high dimension, so that the code relies on the Euclidean distance between codepoints. A sublattice of the coding lattice is used for code shaping. Lattice codes are designed using the continuous approximation, which allows separating the contribution of the shaping region and coding lattice to the total transmit power. Shaping and lattice decoding are both performed using a belief-propagation decoding algorithm. At a rate of 3 bits per dimension, a dimension 100 code which is 3.6 dB from the sphere bound is found.

Low-Density Parity-Check Lattices: Construction and Decoding Analysis

IEEE Transactions on Information Theory, 2000

Low-density parity-check codes (LDPC) can have an impressive performance under iterative decoding algorithms. In this paper we introduce a method to construct high coding gain lattices with low decoding complexity based on LDPC codes. To construct such lattices we apply Construction D 0 , due to Bos, Conway, and Sloane, to a set of parity checks defining a family of nested LDPC codes. For the decoding algorithm, we generalize the application of max-sum algorithm to the Tanner graph of lattices. Bounds on the decoding complexity are derived and our analysis shows that using LDPC codes results in low decoding complexity for the proposed lattices. The progressive edge growth (PEG) algorithm is then extended to construct a class of nested regular LDPC codes which are in turn used to generate low density parity check lattices. Using this approach, a class of two-level lattices is constructed. The performance of this class improves when the dimension increases and is within 3 dB of the Shannon limit for error probabilities of about 10 06. This is while the decoding complexity is still quite manageable even for dimensions of a few thousands. Index Terms-Additive white Gaussian noise (AWGN) channel, iterative decoding, lattice codes, lattices, low-density parity-check (LDPC) codes, low-density parity-check (LDPC) lattices, min-sum algorithm, progressive-edge growth (PEG) construction, Tanner graphs.

Girth-10 LDPC Codes Based on 3-D Cyclic Lattices

IEEE Transactions on Vehicular Technology, 2000

In this paper, we propose a new method based on combinatorial designs for constructing high-girth low-density parity-check (LDPC) codes. We use a 3-D lattice to generate balanced incomplete block designs based on planes and lines in the lattice. This gives families of regular LDPC codes with girths of at least 6, 8, and 10, whose parity-check matrices are all block circulant. The main advantage of this construction is that the algebraic structure leads to efficient encoders and decoders. Based on the block-circulant structure of a parity-check matrix, we present an efficient encoder that can be parallelized to improve the speed of encoding. The simulation results show that these families of LDPC codes perform very well on additive-white-Gaussian-noise channels (roughly 0.45 dB from the channel capacity) and Rayleigh fading channels (roughly 0.51 dB from the channel capacity). , he was with UTStarcom Inc. R&D., Hefei, where he worked on the design and implementation of wideband code-division multiple access and multiple-inputmultiple-output orthogonal frequency-division multiplexing systems. His current research interests include information theory, error-correcting codes, signal processing, and hardware implementation of communication systems.

Construction of Full-Diversity LDPC Lattices for Block-Fading Channels

ArXiv, 2016

LDPC lattices were the first family of lattices which have an efficient decoding algorithm in high dimensions over an AWGN channel. Considering Construction D' of lattices with one binary LDPC code as underlying code gives the well known Construction A LDPC lattices or 1-level LDPC lattices. Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. Frequency-hopping schemes and orthogonal frequency division multiplexing (OFDM) can conveniently be modelled as block-fading channels. Applying lattices in this type of channel entails dividing a lattice point into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, have been generalized to a lattice constructio...

Lattices and Block Codes

2017

Block codes have been wildly used in error-correcting area of information communication for many years. Recently, some researchers found that the using of lattices may reduce the bottleneck of block codes, the lattices codes may be considered for the future 5G. However, the researches on this topic are still in its infancy. In this article, we considered a different encode/decode method by using lattices theory. We first introduced and studied a lattice-valued function on a set, by which we can generate binary block codes. Moreover we discuss how to get the lattices arising from binary block codes. We introduce the notion of semigroup codes and prove that any binary semigroup code V is a lattice in the order " ≤c ". From such lattice we can construct a lattice function f which determines a binary block code V1 and (V1, ≤c) is isomorphic to (V, ≤c). For the special semigroup code V , we can get a lattice function f such that f determines a binary block code V1 and V1 = V. T...

Design criteria for lattice network coding

2011

The compute-and-forward (C-F) relaying strategy proposed by Nazer and Gastpar is a powerful new approach to physical-layer network coding. Nazer-Gastpars construction of C-F codes relies on asymptotically-good lattice partitions that require the dimension of lattices to tend to infinity. Yet it remains unclear how such C-F codes can be constructed and analyzed under practical constraints. Motivated by this, an algebraic approach was taken to compute-and-forward, which provides a framework to study C-F codes constructed from finite-dimensional lattice partitions. Building on the algebraic framework, this paper moves one step further; it aims to derive the design criteria for the C-F codes constructed from finite-dimensional lattice partitions (also referred to as lattice network codes). It is shown that the receiver parameters {aℓ} and α should be chosen such that the quantity Q = |α|2 + SNRΣℓ=1L ||αhℓ - αℓ||2 is minimized, and the lattice partition should be designed such that the minimum inter-coset distance is maximized. These design criteria imply that finding the optimal receiver parameters is equivalent to solving a shortest vector problem, and designing good lattice partitions can be reduced to the design of good linear codes for complex Construction A.

Coding gain and lattice codes with small number of symbols

Constellation diagrams are an important factor in channel coding, representing the physical properties of symbols when they are mapped to electrical impulses. An abstract phenomenon is thus converted into an electrical impulse. Transmission efficiency can be improved by increasing the number of orthogonal carriers, by shaping the whole constellation diagram and by correct positioning of symbols (points) in the diagram. Most common coding schemes do not employ optimum constellation diagram point distributions, so it is obvious that the common method of using equidistant independent direction symbol packing is not desirable. Symbols with added noise are geometrically modelled as spheres, so the coding problem is reduced to the problem of sphere packing of n-dimensional hyperspheres. This problem can be solved with the use of lattices, which can be quite efficiently represented with the algebraic groups. Because lattices are used, this is called "lattice coding". Simple mathe...

Reduced-Memory Decoding of Low-Density Lattice Codes

IEEE Communications Letters, 2010

This letter describes a belief-propagation decoder for low-density lattice codes of finite dimension, in which the messages are represented as single Gaussian functions. Compared to previously-proposed decoders, memory is reduced because each message consists of only two values, the mean and variance. Complexity is also reduced because the check node operations are on single Gaussians, avoiding approximations needed previously, and because the variable node performs approximations on a smaller number of Gaussians. For lattice dimension =1000 and 10,000, this decoder looses no more than 0.1 dB in SNR, compared to the decoders which use much more memory.