Low Density Lattice Codes

Abstract

Abstract-Low-density lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the n-dimensional Euclidean space as a linear transformation of a corresponding integer message vector b, i.e., x = G G Gb, where H H H = G G G 01 is restricted to be sparse. The fact that H H H is sparse is utilized to develop a linear-time iterative decoding scheme which attains, as demonstrated by simulations, good error performance within 0.5 dB from capacity at block length of n =100,000 symbols. The paper also discusses convergence results and implementation considerations.

Fault Identification Via Nonparametric Belief Propagation

“…NBP allows the algorithm to reason about real-valued variables. Variants of NBP have been used in CS [20] and low-density lattice codes (codes defined over real-valued alphabets) [27][28][29]. Our method constructs a relaxation of the fault pattern prior using a mixture of Gaussians, which takes into account both the binary nature of the problem as well as the sparsity of the fault pattern.…”

“…In our continuous model, both sets of factors f i andĝ s consist of mixtures of Gaussians; thus their product, and the messages computed during BP, will also be representable as mixtures of Gaussians [27,33]. Unfortunately, at each step of the algorithm, the number of mixture components required to represent the messages will increase at an exponential rate, and must be approximated by a smaller mixture.…”

“…Unfortunately, at each step of the algorithm, the number of mixture components required to represent the messages will increase at an exponential rate, and must be approximated by a smaller mixture. To handle the exponential growth of mixture components, approximations to BP over continuous variables were proposed in NBP [33] and variants in a broad array of problem domains [20,27,28,42]. Those approximations use sampling to limit the number of components in each mixture.…”

“…Thus, it is both computationally efficient and sufficiently accurate to use a simple uniform discretization over possible values, allowing our functions over x s to be represented by fixed-length vectors [20,27]. Although typically the term "non-parametric BP" refers to an algorithm using stochastic samples and Gaussian mixture approximations to the messages, rather than a fixed discrete quantization, here we use it more generically to distinguish BP in our fully continuous model from a standard discrete BP performed directly on (3).…”

“…Although an arbitrary function over d variables would require O(n d ) computation, where n is the number of discretization bins, the messages can be computed more efficiently, because f i is a function over a linear combination of the x s [27,49]. In essence, one uses a change of variables to separate each integrand, defining variables for the cumulative sum,x sj = a isj x sj +x sj+1 , and scaled messagesm sj i (x) = m sj i (x/a isj ).…”