Lattice Codes Research Articles

Macro-pico amplitude-space sharing with layered interference alignment

Inter-cell interference leads to severe performance degradation in cellular networks, and the study of multi-user interference channel is the corner stone for solving this problem. Amplitude-space layered interference alignment (IA), as an effective complementation to the vector-space IA, is a promising method to increase the data rate in static interference channels. However, recent studies of layered IA has been focused on analyzing the degrees of freedom (DoF) or the achievable rate under specific channel constraints. In this paper, we propose a layered IA scheme that can work with arbitrary channel coefficients. We develop a layer partitioning method and optimize the active layer assignment through linear programming. An implementation scheme is then introduced with multi-level nested lattice codes where the signal and interference are nested in amplitude space, and the interference from different users is nestedly aligned. The performance of the proposed scheme is finally evaluated in homogeneous and heterogeneous cellular networks with practical settings.

Weak Secrecy in the Multiway Untrusted Relay Channel With Compute-and-Forward

We investigate the problem of secure communications in a Gaussian multiway relay channel applying the compute-and-forward scheme under usage of nested lattice codes. All nodes employ half-duplex operation and can exchange confidential messages only via an untrusted relay. The relay is assumed to be honest but curious, i.e., an eavesdropper that conforms to the system rules and applies the intended relaying scheme. We start with the general case of the single-input multiple-output L-user multiway relay channel and provide an achievable secrecy rate region under a weak secrecy criterion. We show that the securely achievable sum rate is equivalent to the difference between the computation rate and the multiple access channel (MAC) capacity. In particular, we show that all nodes must encode their messages such that the common computation rate tuple falls outside the MAC capacity region of the relay. We provide results for the single-input single-output and the multiple-input single-input L-user multiway relay channel as well as the two-way relay channel. We discuss these results and show the dependence between channel realization and achievable secrecy rate. We further compare our result to available results in the literature for different schemes and show that the proposed scheme operates close to the compute-and-forward rate without secrecy.

Structured Lattice Codes for Some Two-User Gaussian Networks With Cognition, Coordination, and Two Hops

We study a number of two-user interference networks with multiple-antenna transmitters/receivers (MIMO), transmitter side information in the form of linear combinations (over an appropriate finite-field) of the information messages, and two-hop relaying. We start with a cognitive interference channel (CIC) where one of the transmitters (noncognitive) has knowledge of a rank-1 linear combination of the two information messages, while the other transmitter (cognitive) has access to a rank-2 linear combination of the same messages. This is referred to as the network-coded CIC, since such linear combination may be the consequence of some random linear network coding scheme implemented in the backbone wired network. For such channel we develop an achievable region based on a few novel concepts: precoded compute-and-forward (PCoF) with channel integer alignment (CIA), combined with standard dirty-paper coding. We also develop a capacity region outer bound and find the symmetric generalized degrees of freedom (GDoF) region of the network-coded CIC. Through the GDoF characterization, we show that knowing mixed data (linear combinations of the information messages) provides a unbounded spectral efficiency gain over the classical CIC counterpart, if the ratio (in decibel) of signal-to-noise (SNR) to interference-to-noise is larger than certain threshold. Then, we consider a Gaussian relay network having the two-user MIMO IC as the main building block. We use PCoF with CIA to convert the MIMO IC into a deterministic finite-field IC. Then, we use a linear precoding scheme over the finite-field to eliminate interference in the finite-field domain. Using this unified approach, we derive the symmetric sum rate of the two-user MIMO IC with coordination, cognition, and two-hops. We also provide finite-SNR results (not just DoF) which show that the proposed coding schemes are competitive against state-of-the-art interference avoidance scheme based on orthogonal access, for standard randomly generated Rayleigh fading channels.

Low-Density Lattice Codes for Full-Duplex Relay Channels

We propose a class of practical efficient lattice codes for real-valued full-duplex one- and two-way relay channels. First, we investigate the problem from a theoretical perspective, proposing lattice-coding instantiations of superposition block Markov encoding. Our encoding/decoding strategies recover the well-known decode-and-foward rates for the one-way relay channel and a previously-proven rate region for the two-way relay channel. Then, we construct practical, low-complexity implementations of these schemes using low-density lattice codes. Simulations show that our schemes achieve performance as close as 2.5 dB away from theoretical limits. Finally, we show that, due to features inherent to full-duplex relaying and practical codes, the gap to theoretical limits depends on the channel gains and transmit power of the relay relative to the source(s). We characterize this gap analytically, providing insight into the design of practical full-duplex relay systems.

Parallel Concatenated Convolutional Lattice Codes With Constrained States

Convolutional lattice codes, also known as signal codes, have been proposed as a technique to generate structured codes that have good performance. While in principle optimal decoding can be achieved using the Viterbi Algorithm, in practice due to Tomlinson-Harashima precoding, the size of the state space is too large, and one must resort to suboptimal techniques such as sequential decoding. In this paper, we take an alternate approach. By employing a judicious selection of tap coefficients and in combination with precoding, we show that the state space can be constrained to a relatively small set such that Viterbi decoding is practical. The performance of such codes still exhibits a large gap to capacity, and we further propose a parallel concatenation similar to that of turbo codes, resulting in a “turbo signal code.” Due to the relatively small state space, iterative decoding based on the BCJR algorithm is now possible. The gaps between the SNR for a frame error rate of 1% and the optimal performance theoretically achievable for a code of the same rate over an AWGN channel are found by simulation to be within 0.75-0.85 dB with a block length of 8192.

Achievable rate regions for many‐to‐one Gaussian interference channel with a fusion centre

This paper considers a many-to-one Gaussian interference channel with a fusion centre (FC) where there is a K -user interference channel in which only one relay (receiver) faces interference while the remaining K-1 receivers are interference free. All the relays communicate information about their observed sequence to the FC through noiseless links at communication rate R 0. First, by analysing traditional relaying schemes, i.e., Decode and Forward, Compress and Forward and Compute and Forward, three rate-regions for this setting are derived. Then, based on nested lattice codes, a new achievable rate-region is provided. Based on the proposed scheme, one can design a transmission scheme that can recover both integer and non-integer linear combination of messages. Numerical examples show that if channel gains are integer, the proposed scheme performs similarly to the compute-and-forward scheme. In the case of non-integer channel gains, the proposed scheme outperforms other relaying schemes at high signal-to-noise ratios. Finally, it is shown that if the channel gains are larger than one and if the rate of each relay-to-FC link equals to the capacity of an AWGN channel, then the proposed scheme can achieve the capacity region in high SNR regime.

Lattice Codes for Many-to-One Interference Channels With and Without Cognitive Messages

A new achievable rate region is given for the Gaussian cognitive many-to-one interference channel. The proposed novel coding scheme is based on the compute-and-forward approach with lattice codes. Using the idea of decoding sums of codewords, our scheme improves considerably upon the conventional coding schemes which treat interference as noise or decode messages simultaneously. Our strategy also extends directly to the usual many-to-one interference channels without cognitive messages. Comparing to the usual compute-and-forward scheme where a fixed lattice is used for the code construction, the novel scheme employs scaled lattices and also encompasses key ingredients of the existing schemes for the cognitive interference channel. With this new component, our scheme achieves a larger rate region in general. For some symmetric channel settings, new constant gap or capacity results are established, which are independent of the number of users in the system.

On Product Distance of Lattice Code over a Biquadratic Number Field

Lattice code is a type of error correcting codes composed of the image of embedding map of an integral ideal in a certain algebraic number field with a bilinear form. In a mobile communication, transmitted signals are attenuated by phenomenon called fading, and the Rayleigh fading of multi-path radio wave is one of important and essential type of fading. Some lattice codes are applicable in order to mitigate the influence of the fading and to extract the most feasible source information from the received signal. The minimum product distance is one of key indicators of lattice code along with the modulation diversity. In this paper, we show some experimental results on the minimum product distance for some lattices of prime ideal in a composition fields of two real quadratic fields using a free formula manipulation system, PARI/GP.

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

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ), 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> + ζ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-r</sup> ) of the cyclotomic field Q(ζ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> ), 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.

Computation Over Gaussian Networks With Orthogonal Components

Function computation over Gaussian networks with orthogonal components is studied for arbitrarily correlated discrete memoryless sources. Two classes of functions are considered: 1) the arithmetic sum function and 2) the type function. The arithmetic sum function in this paper is defined as a set of multiple weighted arithmetic sums, which includes averaging of the sources and estimating each of the sources as special cases. The type or frequency histogram function counts the number of occurrences of each argument, which yields various fundamental statistics, such as mean, variance, maximum, minimum, median, and so on. The proposed computation coding first abstracts Gaussian networks into the corresponding modulo sum multiple-access channels via nested lattice codes and linear network coding and then computes the desired function using linear Slepian-Wolf source coding. For orthogonal Gaussian networks (with no broadcast and multiple-access components), the computation capacity is characterized for a class of networks. For Gaussian networks with multiple-access components (but no broadcast), an approximate computation capacity is characterized for a class of networks.

On the capacity of state-dependent Gaussian cognitive interference channel

A Gaussian cognitive interference channel with state (G-CICS) is studied. In this paper, we focus on the two-sender, two-receiver case and consider the communication situation in which two senders transmit a common message to two receivers. Transmitter 1 knows only message W1, and transmitter 2, referred to as the cognitive user, knows both messages W1 and W2 and also the channel’s states sequence non-causally. Receiver 1 needs to decode only W1 while receiver 2 needs to decode both messages. In this paper, we investigate the weak and moderate interference case where we assume that the channel gain a satisfies |a|≤1. In addition, inner and outer bounds on the capacity region are derived in the regime of high state power, i.e., the channel state sequence has unbounded variance. First, we show that the achievable rate by Gelfand-Pinsker coding vanishes in the high state power regime under a condition over the channel gain. In contrast, we propose a transmission scheme (based on lattice codes) that can achieve positive rates, independent of the interference. Our transmission scheme can achieve the capacity region in a high signal-to-noise ratio (SNR) regime. Also, regardless of all channel parameters, the gap between the achievable rate region and the outer bound is at most 0.5 bits.

Progress toward Monte Carlo–thermal hydraulic coupling using low-order nonlinear diffusion acceleration methods

A new approach for coupled Monte Carlo (MC) and thermal hydraulics (TH) simulations is proposed using low-order nonlinear diffusion acceleration methods. This approach uses new features such as coarse mesh finite difference diffusion (CMFD), multipole representation for fuel temperature feedback on microscopic cross sections, and support vector machine learning algorithms (SVM) for iterations between CMFD and TH equations. The multipole representation method showed small differences of about 0.3% root mean square (RMS) error in converged assembly source distribution compared to a conventional MC simulation with ACE data at the same temperature. This is within two standard deviations of the real uncertainty. Eigenvalue differences were on the order of 10pcm. Support vector machine regression was performed on-the-fly during MC simulations. Regression results of macroscopic cross sections parametrized by coolant density and fuel temperature were successful and eliminated the need of partial derivative tables generated from lattice codes. All of these new tools were integrated together to perform MC–CMFD–TH–SVM iterations. Results showed that inner iterations between CMFD–TH–SVM are needed to obtain a stable solution.

Joint Transceiver and Power Splitter Design Over Two-Way Relaying Channel With Lattice Codes and Energy Harvesting

This letter considers a compute-and-forward two-way relaying channel with simultaneous wireless information and power transfer based on nested lattice codes. Two single-antenna nodes exchange information via a multi-antenna intermediate relay station. Meanwhile, energies in wireless signals are beamed and flow from the relay to terminals for static circuit consumption P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> and uplink transmission power P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> . The optimization problem is formulated to minimize the transmit power P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> of relay, while guaranteeing the minimal transmission rate at each terminal. We propose an efficient iterative algorithm to jointly design the beam-former, receiver and power splitter, based on semi-definite relaxation and programming. Numerical results of transmission power comparisons validate our analysis.

A New Achievable Rate Region for the Gaussian Two-Way Relay Channel via Hybrid Broadcast Protocol

In this letter, we study the Gaussian two-way relay channel (GTWRC) with a direct link in which two nodes want to exchange information with each other with the help of a relay node in the presence of a direct link between two nodes. Specially, we focus on a protocol with four phases, which is called the hybrid broadcast (HBC) protocol. In the HBC protocol, sequential transmissions from both users are followed by a transmission from the relay. Using nested lattice codes, we obtain a new achievable rate region for this protocol. In fact, utilizing a four-stage lattice partition chain, we introduce an intermediate lattice in our lattice-based coding scheme, and by using it, we split the codebook of each node judiciously into two parts. This enables us to decode a linear combination of lattice codewords partially at the relay. Using our proposed strategy, we show that the achievable rate region is superior to the obtained achievable rate region by Kim, Mitran, and Tarokh.

Lattice Coding for the Two-Way Line Network

Full-duplex allows for the simultaneous flow of information in two directions and, in point-to-point Gaussian two-way channels, doubles capacity. A two-way line network where two sources exchange messages through multiple serial relays is considered. It is shown that when all nodes are full duplex, one may achieve to within a constant gap, independent of the number of relays, of the capacity of two one-way line networks. This shows that, even in the presence of relays that carry information in two directions, full duplex is able to approximately double capacity. A novel lattice coding scheme is developed for the two-way line network with two relays, which may be extended to an arbitrary number of relays and to half-duplex scenarios. The key technical contribution is the achievability strategy, where each relay decodes the sum of several signals (using lattice codes) and then re-encodes it into another lattice codeword. This allows other nodes to again decode sums of codewords. The presented lattice-coding-based scheme ensures that both directions simultaneously fully utilize the relays' powers, even for asymmetric channels. The symmetric rate achieved by the proposed scheme is within 0.5 log 5 bit/Hz/s of the symmetric rate capacity regardless of the number of relays.

On Interference Networks Over Finite Fields

We present a framework to study linear deterministic interference networks over finite fields. Unlike the popular linear deterministic models introduced to study Gaussian networks, we consider networks where the channel coefficients are general scalars over some extension held F(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) (scalar mth extension held models), m x m diagonal matrices over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (m-symbol extension ground held models), and m x m general nonsingular matrices (multiple-input and multiple-output (MIMO) ground held models). We use the companion matrix representation of the extension held to convert mth extension scalar models into MIMO ground held models, where the channel matrices have special algebraic structure. For such models, we consider the 2×2×2 topology (two-hop two-flow) and the three-user interference network topology. We derive achievability results and feasibility conditions for certain schemes based on the precoding-based network alignment (PBNA) approach, where intermediate nodes use random linear network coding (i.e., propagate random linear combinations of their incoming messages) and nontrivial precoding/decoding is performed only at the network edges, at the sources and destinations. Furthermore, we apply this approach to the scalar 2 × 2 × 2 complex Gaussian interference channel with fixed channel coefficients and show two competitive schemes outperforming other known approaches at any SNR, where we combine finite field linear precoding/decoding with lattice coding and the compute and forward approach at the signal level. As a side result, we also show significant advantages of vector linear network coding both in terms of feasibility probability (with random coding coefficients) and in terms of coding latency, with respect to standard scalar linear network coding, in PBNA schemes.

Achievable Rate Regions for Two-Way Relay Channel Using Nested Lattice Coding

This paper studies a Gaussian two-way relay channel where two communication nodes exchange messages with each other via a relay. It is assumed that all nodes operate in half-duplex mode without any direct link between the communication nodes. A compress-and-forward relaying strategy using nested lattice codes is first proposed. Then, the proposed scheme is improved by performing layered coding: A common layer is decoded by both receivers, and a refinement layer is recovered only by the receiver that has the best channel conditions. The achievable rates of the new scheme are characterized and are shown to be higher than those provided by the decode-and-forward strategy in some regions.

Multiuser Lattice Coding for the Multiple-Access Relay Channel

This paper considers the multiantenna multiple-access relay channel (MARC), in which multiple users transmit messages to a common destination with the assistance of a relay. In a variety of MARC settings, the dynamic decode-and-forward (DDF) protocol is very useful due to its outstanding rate performance. However, the lack of good structured codebooks so far hinders practical applications of DDF for MARC. In this work, two classes of structured MARC codes are proposed: 1) one-to-one relay-mapper-aided multiuser lattice coding (O-MLC); and 2) modulo-sum relay-mapper-aided multiuser lattice coding (MS-MLC). The former enjoys better rate performance, whereas the latter provides more flexibility to tradeoff between the complexity of the relay mapper and the rate performance. It is shown that, in order to approach the rate performance achievable by an unstructured codebook with maximum-likelihood decoding, it is crucial to use a new K-stage coset decoder for structured O-MLC instead of the one-stage decoder proposed in previous works. However, if O-MLC is decoded with the one-stage decoder only, it can still achieve the optimal DDF diversity-multiplexing gain tradeoff in the high signal-to-noise ratio regime. As for MS-MLC, its rate performance can approach that of the O-MLC by increasing the complexity of the modulo-sum relay-mapper. Finally, for practical implementations of both O-MLC and MS-MLC, practical short-length lattice codes with linear mappers are designed, which facilitate efficient lattice decoding. Simulation results show that the proposed coding schemes outperform existing schemes in terms of outage probabilities in a variety of channel settings, especially when the users-to-relay links are better than the other channel links.

The Approximate Sum Capacity of the Symmetric Gaussian &lt;inline-formula&gt; &lt;tex-math notation="TeX"&gt;$K$ &lt;/tex-math&gt;&lt;/inline-formula&gt;-User Interference Channel

Interference alignment has emerged as a powerful tool in the analysis of multiuser networks. Despite considerable recent progress, the capacity region of the Gaussian $K$ -user interference channel is still unknown in general, in part due to the challenges associated with alignment on the signal scale using lattice codes. This paper develops a new framework for lattice interference alignment, based on the compute-and-forward approach. Within this framework, each receiver decodes by first recovering two or more linear combinations of the transmitted codewords with integer-valued coefficients and then solving these linear combinations for its desired codeword. For the special case of symmetric channel gains, this framework is used to derive the approximate sum capacity of the Gaussian interference channel, up to an explicitly defined outage set of the channel gains. The key contributions are the capacity lower bounds for the weak through strong interference regimes, where each receiver should jointly decode its own codeword along with part of the interfering codewords. As part of the analysis, it is shown that decoding $K$ linear combinations of the codewords can approach the sum capacity of the $K$ -user Gaussian multiple-access channel up to a gap of no more than ${K}/{2}\log{K}$ bits.

Wiretap lattice codes from number fields with no small norm elements

We consider the problem of designing reliable and confidential lattice codes, known as wiretap lattice codes, for fast fading channels. We identify as code design criterion for finite lattice constellations a finite sum of inverse of algebraic norms, in fact the analogous for finite sums of the known criterion for infinite constellations, which motivates us to study totally real number fields with few elements of small norms, over which ideal lattices can be built to provide wiretap lattice codes. More precisely, we first narrow down our search to Abelian number fields known to provide reliable lattice codes, among which we look for no small inert primes and large regulators.