Construction πA\pi_AπA and πD\pi_DπD Lattices: Construction, Goodness, and Decoding Algorithms (original) (raw)
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Construction πA and πD Lattices: Construction, Goodness, and Decoding Algorithms
IEEE Transactions on Information Theory, 2017
A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of L linear codes over F p1 ,. .. , F pL , respectively, and hence is referred to as Construction π A. The existence of a sequence of such lattices that is good for channel coding (i.e., Poltyrev-limit achieving) under multistage decoding is shown. A new family of multilevel nested lattice codes based on Construction π A lattices is proposed and its achievable rate for the additive white Gaussian channel is analyzed. A generalization named Construction π D is also investigated which subsumes Construction A with codes over prime fields, Construction D, and Construction π A as special cases.
Set Partitioning and Multilevel Coding for Codes Over Gaussian Integer Rings
This work demonstrates that the concept of set partitioning can be applied to Gaussian integer rings. It is shown that it is always possible to partition the Gaussian integer rings into subsets in a manner that the minimum Euclidean distance of each subset is strictly larger than in the original set. This enables multilevel code constructions for codes over Gaussian integers.
A Multilevel Framework for Lattice Network Coding
arXiv (Cornell University), 2015
We present a general framework for studying the multilevel structure of lattice network coding (LNC), which serves as the theoretical fundamental for solving the ring-based LNC problem in practice, with greatly reduced decoding complexity. Building on the framework developed, we propose a novel lattice-based network coding solution, termed layered integer forcing (LIF), which applies to any lattices having multilevel structure. The theoretic foundations of the developed multilevel framework lead to a new general lattice construction approach, the elementary divisor construction (EDC), which shows its strength in improving the overall rate over multiple access channels (MAC) with low computational cost. We prove that the EDC lattices subsume the traditional complex construction approaches. Then a soft detector is developed for lattice network relaying, based on the multilevel structure of EDC. This makes it possible to employ iterative decoding in lattice network coding, and simulation results show the large potential of using iterative multistage decoding to approach the capacity. is called a prime in R when p | ab for some a, b ∈ R * , then either p | a or p | b. An ideal I of R is a non-empty subset of R that is closed under subtraction (which implies that I is a group under addition), and is defined by: 1) ∀a, b ∈ I, a − b ∈ I. 2) ∀a ∈ I, ∀r ∈ R, then ar ∈ R and ra ∈ R. If A = {a 1 , • • • , a m } is a finite non-empty subset of R, we use a 1 , • • • , a m to represent the ideal generated by A, i.e.
The index coding problem involves a sender with K messages to be transmitted across a broadcast channel, and a set of receivers each of which demands a subset of the K messages while having prior knowledge of a different subset as side information. We consider the specific instance of noisy index coding where the broadcast channel is Gaussian and every receiver demands all the messages from the source. We construct lattice index codes for this channel by encoding the K messages individually using K modulo lattice constellations and transmitting their sum modulo a shaping lattice. We introduce a design metric called side information gain that measures the advantage of a code in utilizing the side information at the receivers, and hence its goodness as an index code. Based on the Chinese remainder theorem, we then construct lattice index codes with large side information gains using lattices over the following principal ideal domains: rational integers, Gaussian integers, Eisenstein integers, and the Hurwitz quaternions. Among all lattice index codes constructed using any densest lattice of a given dimension, our codes achieve the maximum side information gain.
Adaptive compute-and-forward with lattice codes over algebraic integers
2015 IEEE International Symposium on Information Theory (ISIT), 2015
We consider the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all principal ideal domains (PID). A novel scheme called adaptive compute-andforward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. This is enabled by generalizing the famous Construction A from PID to other rings of imaginary quadratic integers which may not form PID and by showing such the construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive computeand-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.
A bounded-distance decoding algorithm for lattices obtained from a generalized code formula
IEEE Transactions on Information Theory, 1994
Abstrucf-A multistage decoding algorithm is given for lattices obtained from a multilevel code formula. The algorithm is shown to have the same effective error-correcting radius as maximum-likelihood decoding, so that the performance loss is essentially determined by the increase in the effective error coefficient, for which an expression is given. The code formula generalizes some previous multilevel constructions to constructions of known single-level binary lattices with many levels, and then to decoders for them with the proposed algorithm. The trade-off between complexity reduction and performance loss is evaluated for several known lattices and two new ones, indicating that the approach is effective provided the binary codes involved in the code formula are not too short.
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...
Lattices from Codes for Harnessing Interference: An Overview and Generalizations
2014
In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D, and Construction π_A (previously called product construction) recently proposed by the authors. We then discuss two generalizations where the first one is a general construction of lattices named Construction π_D subsuming the above three constructions as special cases and the second one is to go beyond principal ideal domains and build lattices over algebraic integers.