In time entanglement-based quantum key distribution (TE-QKD), Alice and Bob extract the raw key bits from the arrival times of entangled photon pairs. Each entangled pair can contribute to multiple key bits depending on how precisely Alice and Bob can measure the photon arrival times. Thus, TE-QKD can potentially increase the secret key rate compared to typical QKD implementations, which extract up to a bit per photon. Because of entanglement, the times of photon arrivals at Alice’s and Bob’s detectors and, thus, their raw keys should be identical. However, practical photon detectors suffer from time jitter errors. These errors cause discrepancies between Alice’s and Bob’s raw keys. Therefore, Alice must send information to Bob through the public channel to reconcile their raw keys. The amount of data sent for reconciliation represents a loss, rendering secret keys shorter than the raw keys. We compute the secret key rates possible in systems with detector jitter errors and show that they are much higher than those achievable in polarization entanglement-based QKD. We then construct codes for information reconciliation to approach these rates. We demonstrate that short and moderate-length standard error-correcting codes represent excellent information reconciliation choices, making TE-QKD a promising technology.

This paper investigates the decoding of a remarkable set of lattices: We treat in a unified framework the Leech lattice in dimension 24, the Nebe lattice in dimension 72, and the Barnes-Wall lattices. A new interesting lattice, named$L_{3\cdot 24}$, is constructed as a simple application of the single parity check on the Leech lattice. The common aspect of these lattices is that they can be obtained via a single parity check or via the$k$-ing construction. We exploit these constructions to introduce a new efficient paradigm for decoding. This leads to efficient list decoders and quasi-optimal decoders on the Gaussian channel. Both theoretical and practical performance (point error probability and complexity) of the new decoders are provided.

We characterize the complexity of the lattice decoding problem from a neural network perspective. The notion of Voronoi-reduced basis is introduced to restrict the space of solutions to a binary set. On the one hand, this problem is shown to be equivalent to computing a continuous piecewise linear (CPWL) function restricted to the fundamental parallelotope. On the other hand, it is known that any function computed by a ReLU feed-forward neural network is CPWL. As a result, we count the number of affine pieces in the CPWL decoding function to characterize the complexity of the decoding problem. It is exponential in the space dimension$n$, which induces shallow neural networks of exponential size. For structured lattices we show that folding, a technique equivalent to using a deep neural network, enables to reduce this complexity from exponential in$n$to polynomial in$n$. Regarding unstructured MIMO lattices, in contrary to dense lattices many pieces in the CPWL decoding function can be neglected for quasi-optimal decoding on the Gaussian channel. This makes the decoding problem easier and it explains why shallow neural networks of reasonable size are more efficient with this category of lattices (in low to moderate dimensions).

We present new efficient recursive decoders for the Barnes-Wall lattices based on their squaring construction. The analysis of the new decoders reveals a quasi-quadratic complexity in the lattice dimension. The error rate is shown to be close to the universal lower bound in dimensions 64 and 128.

Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in ℝn, known as the closest vector problem (CVP), becomes a classification problem in the fundamental parallelotope with a piecewise linear function defining the boundary. Theoretical results are obtained by studying root lattices. We show how the number of pieces in the boundary function reduces dramatically with folding, from exponential to linear. This translates into a two-layer ReLU neural network requiring a number of neurons growing exponentially in n to solve the CVP, whereas this complexity becomes polynomial in n for a deep ReLU neural network.

We prove that the ensemble of random Generalized Low-Density (GLD) lattices can attain the Poltyrev limit for an alphabet size increasing polylogarithmically with the lattice dimension. Our main theorem imposes no constraints on the normalized minimum distance of the code associated to the lattice ensemble, any asymptotically good code is suitable. This is a great improvement with respect to the first theorem on Poltyrev goodness of GLD lattices (2015). Our new bound is based on a new method referred to as the buckets approach where we employ the asymptotics of the restricted compositions of the Hamming weight. The new bound has applications in many coding areas beyond the specific lattice ensemble considered in this paper.

Constellation shaping is necessary to approach channel capacity for information rates above 1 bit/dim. Probabilistic shaping shows a small gap to capacity, however a complex distribution matcher is required to modify the source distribution. Spherical shaping of lattice constellations also reduces the gap to capacity, but practical Voronoi shaping is feasible in small dimensions only. In this paper, our codebook is a real geometrically non-uniform Gaussian-like constellation. We prove that this discrete codebook achieves channel capacity when the number of points goes to infinity. Then we build a special mapping to interface between non-binary low-density codes and the codebook, allowing the code alphabet size to be equal to the square root of the codebook size. Excellent performance is shown with fast-encoding and practical iterative probabilistic decoding, e.g. 0.7 dB gap to capacity at 6 bits/s/Hz with a code defined over the ring Z/8Z.

This paper deals with Low-Density Construction-A (LDA) lattices, which are obtained via Construction A from non-binary low-density parity-check codes. More precisely, a proof is provided that Voronoi constellations of LDA lattices achieve capacity of the AWGN channel under lattice encoding and decoding for every signal-to-noise ratio greater than 1. This is obtained after showing the same result for more general Construction-A lattice constellations. The theoretical analysis is carried out in a way that allows to describe how the prime number underlying Construction A behaves as a function of the lattice dimension. Moreover, no dithering is required in the transmission scheme, simplifying some previous solutions of the problem. Remarkably, capacity is achievable with LDA lattice codes whose parity-check matrices have constant row and column Hamming weights. Some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound for the minimum Euclidean distance of LDA lattices in our ensemble.

We generalize probabilistic amplitude shaping (PAS) with binary codes [1] to the case of non-binary codes defined over prime finite fields. Firstly, we introduce probabilistic shaping via time sharing where shaping applies to information symbols only. Then, we design circular quadrature amplitude modulations (CQAM) that allow to directly generalize PAS to prime finite fields with full shaping.

The problem of communicating over the additive white Gaussian noise (AWGN) channel with lattice codes is addressed in this paper. Theoretically, Voronoi constellations have proved to yield very powerful lattice codes when the fine/coding lattice is AWGN-good and the coarse/shaping lattice has an optimal shaping gain. However, achieving Shannon capacity with these premises and practically implementable encoding algorithms is in general not an easy task. In this paper, a new way to encode and demap Construction-A Voronoi lattice codes is presented. As a meaningful application of this scheme, the second part of the paper is focused on Leech constellations of lowdensity Construction-A (LDA) lattices: LDA Voronoi lattice codes are presented whose numerically measured waterfall region is situated at less than 0.8 dB from Shannon capacity. These LDA lattice codes are based on dual-diagonal nonbinary low-density parity-check codes. With this choice, encoding, iterative decoding, and demapping have all linear complexity in the block length.

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size M. Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is o(Mℵ), for a given differential evolution parameter ℵ, where Mℵitself is small with respect to the huge cardinality of the coloring ensemble. Stopping sets of a product code are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size (d + 1)2, where d is the minimum Hamming distance of the MDS component code. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both the block and the independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.

We consider the problem of physical-layer network coding when the channel exhibits block fading. Specifically, we focus on the use of lattice codes in a compute-and-forward framework for realizing physical-layer network coding. We construct a novel lattice ensemble called the root-Low-Density Construction-A (root-LDA) ensemble which uses Construction A with root-low-density parity check (LDPC) codes. Using extensive simulations, we show that the proposed lattice codes exhibit full diversity when used over the block fading channels. In addition, their performance is comparable to the performance of LDA lattice codes optimized by the progressive edge growth algorithm over the additive white Gaussian noise AWGN channel. This suggests that root-LDA lattice codes provide a robust solution to the problem of implementing physical layer network coding over fading channels.

Stopping sets for MDS-based product codes under iterative row-column algebraic decoding are analyzed in this paper. A union bound to the performance of iterative decoding is established for the independent symbol erasure channel. This bound is tight at low and very low error rates. We also proved that the performance of iterative decoding reaches the performance of Maximum-Likelihood decoding at vanishing channel erasure probability. Numerical results are shown for product codes at different coding rates.

We propose new rate-flexible low-density parity-check (LDPC) coding schemes for secrecy over a compound channel with L parallel links. These codes, called anti-root LDPC codes, have good performance at both finite and asymptotic code length while all links are jointly encoded. Firstly, an algebraic security scheme is developed based on the anti-root LDPC ensemble and a source splitter. Secondly, an information theoretic security scheme is built from the same splitter with the adjunction of a random sequence. Then, we present a new diversity-security tradeoff for channels exhibiting block fading or block erasure. Finally, we describe anti-root LDPC ensembles with higher diversity or security orders to attain the aforementioned tradeoff.

In this paper we analyze nonanticipative (delayless) transmission of source symbols with memory over channels with memory (with and without feedback). We employ duality of {source, channel} pairs with respect to {distortion function, transmission cost} pairs to show achievability of nonanticipative transmission in terms of excess distortion probability. We apply the method to the Binary Markov source with Hamming distortion function and the Binary Unit Memory channel with transmission cost, with the joint-design operating optimally and in real-time, with and without feedback encoding and decoding.

Nonergodic fading is a useful model for various wireless communication channels in both indoor and outdoor environments. With this model, a codeword is divided into multiple blocks such that fading is constant within a block and independent across blocks. Building on Poltyrev's work on infinite lattice constellations for the Gaussian channel, we derive a Poltyrev outage limit for lattice constellations transmitted over a block-faded channel. We prove that the diversity order of this Poltyrev outage limit is equal to the number of degrees of freedom in the channel. An important application is in decoding low-density lattice codes. With block fading, the presence of an outage may dramatically increase the runtime of both sphere and iterative decoders. Using our newly defined Poltyrev outage limit, decoding is not performed whenever an outage is declared, which drastically reduces the overall decoding time.

The recently discovered family of generalized low-density (GLD) lattices brings new mathematical challenges to coding theorists and practitioners. Given the excellent performance of integer GLD lattices in high dimensions and motivated by the simple lattice structure used for fast iterative decoding, this paper is a first attempt to analyze GLD lattices for asymptotically large dimensions. Firstly, we describe non-binary GLD codes and show their asymptotic goodness in terms of minimum Hamming distance. Secondly, we consider a GLD lattice ensemble built via Construction A from non-binary GLD codes, and analyze their goodness with respect to Poltyrev limit on the Gaussian channel. Finally, at large dimensions and using a large code alphabet, we prove that infinite GLD lattice constellations attain Poltyrev capacity limit under maximum likelihood decoding.

Sparse ISI channels have been considered for applications in underwater acoustic, wireless multipath, aeronautical, and satellite systems. They are characterized by having a long gain vector with few significant taps. BCJR-based simulation methods used when the channel memory is low become quickly impractical since the number of states grows exponentially large with it. A novel techniques is proposed in this paper based on simple information theoretical considerations and counting arguments, which is applied to the i.i.d. binary antipodal input case. Simulation results are reported illustrating how the information rate can be determined for either short ISI channels and long sparse ISI channels with 50-symbol memory.

In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of the nested lattice codebooks over Eisenstein integers surpass those of lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity.

In this work, we propose phase precoding for the compute-and-forward (CoF) protocol. We derive the phase precoded computation rate and show that it is greater than the original computation rate of CoF protocol without precoder. To maximize the phase precoded computation rate, we need to `jointly' find the optimum phase precoding matrix and the corresponding network equation coefficients. This is a mixed integer programming problem where the optimum precoders should be obtained at the transmitters and the network equation coefficients have to be computed at the relays. To solve this problem, we introduce phase precoded CoF with partial feedback. It is a quantized precoding system where the relay jointly computes both a quasi-optimal precoder from a finite codebook and the corresponding network equations. The index of the obtained phase precoder within the codebook will then be fedback to the transmitters. A “deep hole phase precoder” is presented as an example of such a scheme. We further simulate our scheme with a lattice code carved out of the Gosset lattice and show that significant coding gains can be obtained in terms of equation error performance.

We propose the construction of a new family of lattice sphere packings. Given a small-dimensional lattice, we start by building a first lattice in a large dimension by the direct sum of the small lattice. Then, the coordinates of the first large lattice are permuted to yield a second large-dimensional lattice. Finally, our generalized low-density (GLD) lattice is the intersection of the first and the second lattice. We restrict our construction in this paper to integer lattices. GLD lattices are the result of mixing classical lattice theory with modern coding theory. They are potential candidates not only for channel coding as coded modulations, but also for physical-layer network coding and for secure digital communications.

A new method for spatial coupling of low-density parity-check ensembles is proposed. The method is inspired from overlapped layered coding. Edges of local ensembles and those defining the spatial coupling are separately built. The new method allows the construction of non-uniform coupling chains with near-Shannon spatially-varying thresholds under iterative decoding. A direct application of non-uniform spatial coupling is unequal error protection of information.

This work investigates polar coding for block-fading channels. We show that polarization does occur at infinity for three types of channel multiplexers. Nevertheless, the polarization process is not unique, as it is shaped by the choice of the multiplexer. The fading-plane approach is used to study the outage behavior of polar coding at a fixed transmission rate. Two types of multiplexers are shown to provide full diversity at finite and infinite code length.

We address the problem of transmission of information over the AWGN channel using lattices. In particular, we will deal with previously introduced LDA lattices which are obtained by Construction A from LDPC codes over the finite field Fp. We will show how to build a particular ensemble of LDA lattices related to bipartite graphs with good expansion properties. We investigate the quality of this family under lattice decoding and show that a random member in it can be reliably decoded for any value of the channel noise variance up to Poltyrev limit. Values of p and the parameters for which optimal performance is guaranteed under lattice decoding are in accordance with the optimal parameters found experimentally under iterative decoding.

The outage probability limit is a fundamental and achievable lower bound on the word error rate of coded communication systems affected by fading. This limit is mainly determined by two parameters: the diversity order and the coding gain. With linear precoding, full diversity on a block fading channel can be achieved without error-correcting code. However, the effect of precoding on the coding gain is not well known, mainly due to the complicated expression of the outage probability. Using a geometric approach, this paper establishes simple upper bounds on the outage probability, the minimization of which yields to precoding matrices that achieve very good performance. For discrete alphabets, it is shown that the combination of constellation expansion and precoding is sufficient to closely approach the minimum possible outage achieved by an i.i.d. Gaussian input distribution, thus essentially maximizing the coding gain.

Multiple antennas are used to increase reliability and bit rate for a given bandwidth. For a fixed transmission rate, discrete input alphabets and no channel state information at the transmitter, optimal space-time codes (STCs) achieving both gains (full rate and full diversity) are well known. However, the complexity of maximum likelihood decoding increases exponentially with the number of space and time dimensions of the STC. Despite praiseworthy efforts to reduce the decoding complexity, optimal STCs are not used in practice in the case of more than two transmit antennas because of complexity reasons. It is generally accepted that reducing the dimension of the STC leads to a reduced diversity order. We show that this is not true for coded communication, assuming the presence of an outer error-correcting code with any coding rate. We propose a new class of full-rate full-diversity STCs; more specifically, time-varying space-only codes. This new class is referred to as EMI codes. Full diversity is proven in terms of outage probability, for the case where the number of receive antennas is larger than or equal to the number of transmit antennas, and is numerically verified in terms of outage and word error probability using LDPC codes.

We describe a new family of integer lattices built from construction A and non-binary LDPC codes. An iterative message-passing algorithm suitable for decoding in high dimensions is proposed. This family of lattices, referred to as LDA lattices, follows the recent transition of Euclidean codes from their classical theory to their modern approach as announced by the pioneering work of Loeliger (1997), Erez, Litsyn, and Zamir (2004-2005). Besides their excellent performance near the capacity limit, LDA lattice construction is conceptually simpler than previously proposed lattices based on multiple nested binary codes and LDA decoding is less complex than real-valued message passing.

In wireless communications, the block fading (BF) channel is an important channel model. A key quality indicator in coded transmission is the word error rate (WER), which is the fraction of packets that cannot be decoded correctly at the receiver. We study Low-Density Parity-Check (LDPC) coded modulation with precoding, with the aim to minimize the WER on BF channels without channel state information at the transmitter. In the literature, it was not yet known how to optimize the system parameters for this channel model, mainly due to the fading gain distribution. One of the existing approaches to combining coding and modulation is bit-interleaved coded modulation with iterative decoding (BICM-ID). This work uses precoding to optimize BICM-ID with LDPC codes for BF channels, and is a continuation of previous work that used precoding to minimize the outage probability limit. We present the selection of the precoding matrix, the mapping function and the error-correcting code yielding a WER that closely approaches this minimum outage probability. Using a geometric approach, the off-line system optimization effort for the BF channel is limited to at most B+1 times the effort for Gaussian channels, where B is the number of blocks in the BF channel.

We study precoding for the outage probability minimization of block fading (BF) channels and BF relay channels. Recently, an upper bound on the outage probability with precoding was established for BF channels, but only for high instantaneous SNR. This upper bound is much easier to minimize than the actual outage probability, so that optimal precoding matrices can be determined without much computational effort. Here, we provide a proof for the upper bound on the outage probability at low instantaneous SNR. Next, the structure of the precoding matrix is simplified so that it can be easily constructed for an arbitrary number of blocks in the BF channel. Finally, we apply this technique to cooperative communications.

We study Low-Density Parity-Check (LDPC) codes with iterative decoding on block-fading (BF) Relay Channels. We consider two users that employ coded cooperation, a variant of decode-and-forward with a smaller outage probability than the latter. An outage probability analysis for discrete constellations shows that full diversity can be achieved only when the coding rate does not exceed a maximum value that depends on the level of cooperation. We derive a new code structure by extending the previously published full-diversity root-LDPC code, designed for the BF point-to-point channel, to exhibit a rate-compatibility property which is necessary for coded cooperation. We estimate the asymptotic performance through a new density evolution analysis and the word error rate performance is determined for finite length codes. We show that our code construction exhibits near-outage limit performance for all block lengths and for a range of coding rates up to 0.5, which is the highest possible coding rate for two cooperating users.

Expectation-maximization (EM) based iterative algorithms are investigated in order to estimate the impulse response of a frequency-selective multipath channel in a coded OFDM system. Two ways of choosing the EM complete data are compared: a complete data built from observations and transmitted symbols (CL-EM) and a complete data chosen by decomposing noise and observation components (NCD-EM). Both CL-EM and NCD-EM algorithms are derived for a coded OFDM system. The rate of convergence of both EM algorithms is theoretically determined. It is found that the rate of convergence of CL-EM is independent from the number of channel taps at high signal-to-noise ratio (SNR), while that of NCD-EM varies with the number of taps. It is shown that CL-EM converges in a few iterations. Furthermore, considering the complexity per iteration, CL-EM has a lower complexity than its counterpart. We also establish a Cramer-Rao bound (CRB) for coded OFDM transmission. Simulation results show that CL-EM has a good performance-complexity trade-off and it achieves the CRB.

The outage probability limit is a fundamental lower bound on the word error rate of coded communications systems. It is mainly determined by two parameters: the diversity order and the coding gain. With linear precoding, the maximum achievable coding rate yielding full-diversity can exceed the upper limit given by the standard Singleton bound. However, the effect of precoding on the coding gain is not well understood. This paper analyzes linear precoding from an information theoretical point of view and tries to optimize the coding gain. For discrete constellations, it is shown that constellation expansion together with one optimized precoding parameter is sufficient to approach the best outage achieved by a Gaussian alphabet, thus maximizing the coding gain.

We study irregular binary turbo codes over nonergodic block-fading channels. We first propose an extension of channel multiplexers initially designed for regular turbo codes. We then show that, using these multiplexers, irregular turbo codes that exhibit a small decoding threshold over the ergodic Gaussian-noise channel perform very close to the outage probability on block-fading channels, from both density evolution and finite-length perspectives.

We design powerful low-density parity-check (LDPC) codes with iterative decoding for the block-fading channel. We first study the case of maximum-likelihood decoding, and show that the design criterion is rather straightforward. Since optimal constructions for maximum-likelihood decoding do not perform well under iterative decoding, we introduce a new family of full-diversity LDPC codes that exhibit near-outage-limit performance under iterative decoding for all block-lengths. This family competes favorably with multiplexed parallel turbo codes for nonergodic channels.

In this paper, we consider the problem of coding for the half-duplex nonorthogonal amplify-and-forward (NAF) cooperative channel where the transmitter to relay and the interrelay links are highly reliable. We derive bounds on the diversity order of the NAF protocol that are achieved by a distributed space-time bit-interleaved coded modulation (D-ST-BICM) scheme under iterative APP detection and decoding. These bounds lead to the design of space-time precoders that ensure maximum diversity order and high coding gains. The word error rate performance of D-ST-BICM are also compared to outage probability limits.

Our motivation is the design of space-time coding which is optimal under both maximum likelihood and iterative decoding. We describe the construction of new full-rate space-time codes with non-vanishing determinant that satisfy the genie conditions for iterative probabilistic decoding. The problem combining the genie conditions and the rank criterion is rewritten in terms of a quadratic form. The construction over ¿[i] (the cubic lattice) yields a family of codes defined by Pythagorean triples. The space-time code built over ¿[i] and involving the quaternion algebra (i,5/¿(i)) is referred to as the Aladdin-Pythagoras code. The construction over ¿[j] (the hexagonal lattice) also yields a full-rate non-vanishing determinant code that is suitable for iterative decoding on multiple antenna channels.

Reconciliation is an essential part of any secret-key agreement protocol and hence of a quantum key distribution (QKD) protocol, where two legitimate parties are given correlated data and want to agree on a common string in the presence of an adversary, while revealing a minimum amount of information. In this paper, we show that for discrete-variable QKD protocols, this problem can be advantageously solved with low density parity check (LDPC) codes optimized for the binary symmetric channel (BSC). In particular, we demonstrate that our method leads to a significant improvement of the achievable secret key rate, with respect to earlier interactive reconciliation methods used in QKD.

Cooperative communications is a well known technique to yield transmit diversity in a multi-user environment. Network coding can increase the spectral efficiency in networks. These two techniques can be combined to achieve a double diversity order for a maximum coding rate Rc= 2/3 on the Multiple Access Relay Channel (MARC), where two sources share a common relay in their transmission to the destination. However, codes have to be carefully designed to obtain the intrinsic channel diversity. Up till now, no full-diversity capacity-achieving code for the MARC at a coding rate Rc= 2/3 has been published. We present a strategy to produce excellent low-density parity-check (LDPC) codes with rate 2/3, i.e., exhibiting full-diversity and operating close to the outage probability limit. Simulation of the word error rate performance of the new proposed family of LDPC codes for the MARC confirms the full-diversity.

Joint channel estimation and decoding using belief propagation on factor graphs requires the quantization of probability densities since continuous parameters are involved. We propose to replace these densities by standard messages where the channel estimate is accurately modeled as a Gaussian mixture over multipath channel. Upward messages include symbol extrinsic information and downward messages carry mean values and variances for the Gaussian modeled channel estimate. Such unquantized message propagation leads to a complexity reduction and a performance improvement. Over multipath channel, the proposed belief propagation almost achieves the performance of iterative APP equalizer and outperforms MMSE equalizer.

Joint channel estimation and decoding using belief propagation on factor graphs requires the quantization of probability densities since continuous parameters are involved. We propose to replace these densities by standard messages where the channel estimate is accurately modeled as a Gaussian mixture. Upward messages include symbol extrinsic information and downward messages carry a mean and a variance for the Gaussian modeled channel estimate. Such unquantized message propagation leads to a complexity reduction and a performance improvement. For QAM modulated symbols, the proposed belief propagation almost achieves the performance of expectation-maximization under good initialization and surpasses it under bad initialization.

We propose an original technique for the design of convolutional Tanner structures that are full diversity under iterative decoding. The code design is based on the analysis of the local trellis neighborhood and is suitable for transmission over wireless non-ergodic channels. This new technique enables us to split the giant convolutional checknode into multiple smaller checknodes which is a means to mimic the standard analysis of LDPC codes under iterative message passing decoding.

A new graph-based construction of generalized low density codes (GLD-Tanner) with binary BCH constituents is described. The proposed family of GLD codes is optimal on block erasure channels and quasi-optimal on block fading channels. Optimality is considered in the outage probability sense. A classical GLD code for ergodic channels (e.g., the AWGN channel, the i.i.d. Rayleigh fading channel, and the i.i.d. binary erasure channel) is built by connecting bitnodes and subcode nodes via a unique random edge permutation. In the proposed construction of full-diversity GLD codes (referred to as root GLD), bitnodes are divided into 4 classes, subcodes are divided into 2 classes, and finally both sides of the Tanner graph are linked via 4 random edge permutations. The study focuses on non-ergodic channels with two states and can be easily extended to channels with 3 states or more.

We propose a method for extracting an errorless secret key in a continuous-variable quantum key distribution protocol, which is based on Gaussian modulation of coherent states and homodyne detection. The crucial feature is an eight-dimensional reconciliation method, relying on the algebraic properties of octonions. By using this coding scheme with an appropriate signal-to-noise ratio, the distance for secure continuous-variable quantum key distribution can be significantly extended.

We show how to build full-diversity product codes under both iterative encoding and decoding over non-ergodic channels, in presence of block erasure and block fading. The concept of a rootcheck or a root subcode is introduced by generalizing the same principle recently invented for low-density parity-check codes. We also describe some channel related graphical properties of the new family of product codes, a family referred to as root product codes.

In order to increase the performance of joint channel estimation and decoding through belief propagation on factor graphs, we approximate the distribution of channel estimate in the factor graph as a mixture of Gaussian distributions. The result is a continuous downward and upward message propagation in the factor graph instead of discrete probability distributions. Using continuous downward messages, the computation complexity of belief propagation is reduced without performance degradation. With both continuous upward and downward messages, belief propagation almost achieves the same performance as expectation-maximization under good initialization and outperforms it under bad initialization.

This study proposes an efficient way of interference mitigation for ultra-wideband energy detection. A receiver for pulse position modulation systems is investigated. The inter-slot (i.e., intra-symbol) and inter-symbol interferences are studied and a probabilistic equalizer is derived. This energy equalizer is embedded into the loop of an iterative channel decoder. Computer simulations are performed on the channel models from the IEEE 802.15.3a task group.

The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading multiple-input multiple-output (MIMO) channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upper bound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders, called dispersive nucleo algebraic (DNA) precoders are proposed. The proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension. With turbo codes, they exhibit a frame error rate which does not increase with frame length.

We investigate a space-time coding scheme with four or more transmit antennas based on a matrix-Alamouti scheme in a quasi-static fading environment. The receiver consists of an Alamouti combiner followed by an iterative joint detection and decoding process. A priori probabilities fed back from the decoder are used to both get more reliable estimates of the transmitted symbols and efficiently remove inter-block interference. Interblock interference is removed using techniques known in multiuser communication. The theoretical bounds of this scheme are established through the study of its outage probability.

In this paper, we investigate reliable error rate estimation techniques for MAC layer adaptive mechanisms. In particular, we propose and analyze a novel error rate estimator based on soft output information available as output of receivers with turbo principle. Contrary to previous works on the subject, we relax the assumption of perfect knowledge of signal-noise-to-ratio (SNR) at the receiver, and we analyze the impact of a SNR estimation error on the error rate estimate. We show that, differently to previous techniques, the proposed estimation method is insensitive to such SNR estimation error. Our analytical and simulation results validate the conclusion.

An accurate approximation for the conditional error probability on quasi-static multiple-input multiple-output (MIMO) antenna channels is proposed. For a fixed channel matrix, it is possible to accurately predict the performance of quadrature amplitude modulations (QAM) transmitted over the MIMO channel in presence of additive white Gaussian noise. The tight approximation is based on a simple Union bound for the point error probability in the n-dimensional real space. Instead of making an exhaustive evaluation of all pairwise error probabilities (intractable in many cases), a Pohst or a Schnorr-Euchner lattice enumeration is used to limit the local theta series inside a finite radius sphere. The local theta series is derived from the original lattice theta series and the point position within the finite multidimensional QAM constellation. In particular, we take into account the number of constellation facets (hyperplanes) that are crossing the sphere center. As a direct application to the accurate approximation for the conditional error probability, we describe a new adaptive QAM modulation for quasi-static multiple antenna channels

We consider channel decoding of redundant data with context based memory with the recently proposed class of nonsystematic split-LDPC codes. Methods based on denoising techniques and context estimation from universal source coding are used to extract redundancy between decoding iterations of split-LDPC codes. Simulation results show that the use of nonsystematic split-LDPC codes gains in most cases on standard systematic LDPC codes even without prior knowledge of the source statistics. Different context extraction methods are considered, including a novel approach that is used to capture different types of context based redundancy.

A new approximation for the conditional error probability on quasi-static multiple antenna (MIMO) channels is proposed. For a fixed channel matrix, it is possible to predict the performance of quadrature-amplitude modulations (QAM) transmitted over the MIMO channel in presence of additive white Gaussian noise (AWGN). The tight approximation is based on a simple union bound for the point error probability in the n-dimensional real space. A Pohst or a Schnorr-Euchner lattice enumeration is used to limit the local Theta series inside a finite radius sphere. As applications to this approximation, we describe a new adaptive QAM modulation and a new antenna selection criterion

Nonsystematic channel codes are superior to systematic codes in the presence of source redundancy. We study the performance of split-LDPC codes (we recently proposed), which are based on splitting redundant data bits into coded bits. We propose a novel method to build extrinsic information transfer (EXIT) chart to approximate the thresholds of such codes. EXIT charts provide a fast and close to accurate prediction of the thresholds of split-LDPC codes for nonuniform sources. The thresholds approximated by fast EXIT chart analysis are very close to those obtained by density evolution (DE) analysis that we recently proposed for split-LDPC codes. The EXIT chart analysis can thus be used to efficiently search for good split-LDPC codes. Simulations verify good performance close to the approximate thresholds predicted by the EXIT charts

The paper studies symbol-by-symbol maximum a posteriori (MAP) decoding algorithms for non binary codes over an extension field GF(q). This decoding rule minimizes the probability of symbol error over a time-discrete memory less channel by employing the dual code. It is shown that these algorithms meet all requirements needed for iterative decoding as the output of the decoder can be split into three independent estimates: soft channel value, a priori term and extrinsic value. It represents a better form of coding for the q-ary LDPC codes, which have been shown to outperform binary LDPC codes and Reed-Solomon codes on the AWGN channel and it gives us a new fast and reduced-complexity decoding algorithm. The complexity of this rule varies inversely with code rate, making the technique particularly attractive for high rate codes. Finally, we see that complexity is reduced by using the dual code, and the algorithm is accelerated by using the fast Hadamard transform (FHT). Examples are given for both single parity check (non iterative case) and LDPC (iterative case) non binary codes

In this correspondence, we present an interleaving scheme that yields quasi-cyclic turbo codes. We prove that randomly chosen members of this family yield with probability almost 1 turbo codes with asymptotically optimum minimum distance, i.e., growing as a logarithm of the interleaver size. These interleavers are also very practical in terms of memory requirements and their decoding error probabilities for small block lengths compare favorably with previous interleaving schemes.

We consider channel estimation in multiple-input multiple-output (MIMO) systems using iterative detection at the receiver. Space-time bit-interleaved coded modulation (BICM) and soft-input soft-output maximum a posteriori (MAP) symbol detection and decoding are considered. Channel coefficients are updated at each iteration of the detector using a semi-blind estimation approach based on the expectation maximization (EM) algorithm. We first show that a "classical" and non-optimized EM implementation, as already proposed in some previous works, gives a biased estimate of the channel coefficients. We then try to optimize the EM implementation and propose a modification to it that provides an unbiased channel estimate and leads to a better convergence of the iterative detector. We show that considerable improvement in the receiver performance can be obtained by using our proposed modified unbiased (MU) EM algorithm, especially for large number of transmit antennas and short training sequences. We also show that when MIMO signal detection is strongly asymmetric in the sense of too few receive antennas, the EM-based channel estimation may be of little interest. Moreover, we consider a simple semi-blind estimation scheme, based on hard decisions on reliable decoded data bits, and compare its performance with the EM based estimation methods

We consider channel re-estimation based on the EM (expectation maximization) algorithm in an iterative MIMO detector based on parallel interference cancellation, also known under the name of turbo-BLAST. While providing an appropriate formulation of EM, we show that a previously proposed EM implementation gives a biased estimate, and propose modifications in the EM implementation in order to obtain an unbiased estimate. Testing the proposed algorithms by simulations, we show that the achieved improvement is considerable for important diversity orders.

Messages coded and transmitted over a channel may contain some redundancy, which is usually not utilized by standard channel decoding techniques. Non-systematic codes have a potential for significant advantage over systematic codes if a method is found to utilize such redundancy. We propose a novel general encoder/decoder structure for non-systematic low-density parity-check (LDPC) codes, that can be used in several different configurations for efficiently exploiting redundancy in decoding of redundant data sequences. Simulation results for one configuration of this method demonstrate clear performance gains over standard systematic LDPC codes when a decoder utilizes the source redundancy for decoding of redundant data sequences. These gains increase as the non-uniformity of the source increases, and also with use of some of the other configurations of the new general method

This paper deals with packet error rate (PER) prediction and its impact on adaptive modulation and coding (AMC) for an OFDM turbo coded transmission on a multipath channel. We compare four link quality metrics (LQM), including the signal to noise ratio (SNR), the capacity, and two LQM which we designed to exploit the soft outputs of the turbo-decoder. Since error prediction is used to perform AMC, we assess the throughput performance of a realistic system employing each of the four LQM. The comparison shows that while SNR performs poorly and capacity remains suboptimum, the two metrics exploiting soft outputs approach the throughput which would be achieved by perfect PER prediction

In this paper, we extend the iterative demapping technique to observation separation (OS), which is an iterative channel estimation and symbol detection procedure. We consider a bit interleaved coded modulation with iterative decoding (BICM-ID) on a multipath channel with coded packet oriented transmission system with 16QAM. The channels considered are the time-invariant multipath and the channel estimation corresponds to the pseudo-inverse type. We propose here to study the influence of mapping over the performance of turbo equalization. Turbo equalization is a receiving process performing iteratively equalization and channel decoding; soft information generated by each receiving function is improved through the iterations until the optimum behavior of the system reached. In the studied system the turbo equalization and the channel estimation part is based on our previous work. In the context of bit interleaved coded modulation (BICM), the iterative demapping is an iterative process between the soft demapping device and the channel decoding. We analyze here the improvement of mapping optimization for turbo equalization and we proposed an optimized binary mapping of 16-QAM constellation in order to increase coding gain, which can be used for some practically important cases. It is demonstrated that owning to a carefully designed mapping, different to the classical Gray mapping, bit error rate performance is improved.

Multidimensional binary mappings for bit-interleaved coded modulations (BICMs) on ergodic multiple-antenna channels with iterative decoding are presented. After derivation of a closed-form expression for the pairwise error probability under ideal maximum-likelihood (ML) decoding, the design criterion for mapping optimization is established from the ML performance of the ideally interleaved channel. It coincides with the figure of merit derived from the genie condition when the iterative receiver converges to perfect a priori information. Multidimensional mapping constructions that exhibit high signal-to-noise ratio (SNR) gains without increasing the complexity of the a posteriori probability (APP) detection are proposed. They allow for a reduced decoding complexity as they achieve near turbo code performance with a single convolutional code.

This paper presents the study of interleavers that both use a reduced amount of random choice and moderate algebraic structure. The interleaver is essentially chosen at random among a family that produces quasicyclic turbo codes. A typical interleaver produces a turbo code with minimum distance log N. For moderate lengths these interleavers turn out to be also quite practical, comparing favorably with S-random interleavers in a number of instances.

We present a linear preceding solution to achieve full diversity with iteratively decoded bit-interleaved coded modulation on multiple antenna channels while minimizing the detection complexity.

Soft output detection for signals transmitted on linear channels is investigated. A particular emphasis is made for signal detection on multiple antenna channels. The a posteriori information at the detector output is evaluated from a shifted spherical list of point candidates. The spherical list is centered on the maximum likelihood point, which has the great advantage of stabilizing the list size. Thus, the sphere radius is selected in order to control the list size and to cope with the boundaries of the finite multiple antenna constellation. Our new soft output sphere decoder is then applied to the computation of constrained channel capacity and to the iterative detection of a coded transmission. For example, we achieved a signal-to-noise ratio at 1.25 dB from capacity limit on a 4/spl times/4 MIMO channel with 16-QAM modulation and a 4-state rate 1/2 parallel turbo code.

The paper aims at comparing the respective benefits of STBICM (space-time bit-interleaved coded modulation) and space-time trellis codes (STTC) for a MIMO block fading multipath channel. Giving a general framework for the calculation of the outage probability, we try to point out the best transmit scheme that approaches the closest the outage probability, assuming the most efficient receiver (i.e., disregarding any complexity aspect). We conclude that iteratively decoded STBICM should be used in preference to STTC.

A new joint detection method based on sphere packing lattice decoding is presented in this paper. The algorithm is suitable for both synchronous and asynchronous multiple access direct-sequence code-division multiple-access (DS-CDMA) systems, and it may jointly detect up to 64 users with a reasonable complexity. The detection complexity is independent of the modulation size and large M-PAM or M-QAM constellations can be used. Furthermore, a theoretical gain analysis is performed in which the multiple-access system performance is derived from the lattice parameters.

We describe a robust time and burst synchronization strategy for data over cable transmission systems. The strategy includes coarse and fine timing and involves a total of 7 steps. Analytical analysis of false alarm and detection error probabilities is also accomplished. We study the influence of a parameter (called contrast threshold) on the performance and we take into account frequency and sampling time errors. It is shown that a successful synchronization is guaranteed by fixing the contrast threshold between 3 and 4 dB while operating at moderate signal-to-noise ratios.

A generic model of bit-interleaved coded modulation (BICM) on a multipath Rayleigh fading multiple-input multiple-output (MIMO) channel is derived. A practical low-complexity trellis-based receiver performing iteratively channel estimation, multilayer coded data detection and channel decoding is introduced. The inner multilayer data detector, employing reduced-state techniques together with generalized per-survivor processing, presents two-fold advantages. It enables one to cope with severe channel intersymbol interference and allows the use of more transmit antennas than receive antennas. Simulations show that our approach can dramatically improve the downlink performance of time division multiple access (TDMA) systems with high order modulation, such as EDGE, keeping a reasonable complexity at the receiver side.

We present a framework for iterative multiuser joint decoding of code-division multiple-access (CDMA) signals, based on the factor-graph representation and on the sum-product algorithm. In this framework, known parallel and serial, hard and soft interference cancellation algorithms are derived in a unified way. The asymptotic performance of these algorithms in the limit of large code block length can be rigorously analyzed by using density evolution. We show that, for random spreading in the large-system limit, density evolution is considerably simplified. Moreover, by making a Gaussian approximation of the decoder soft output, we show that the behavior of iterative multiuser joint decoding is approximately characterized by the stable fixed points of a simple one-dimensional nonlinear dynamical system.

We derive a low-complexity receiver scheme for joint multiuser decoding and parameter estimation of code division multiple access signals. The resulting receiver processes the users serially and iteratively and makes use of soft-in soft-out single-user decoders, of soft interference cancellation and of expectation-maximization parameter estimation as the main building blocks. Computer simulations show that the proposed receiver achieves near single-user performance at very high channel load (number of users per chip) and outperforms conventional schemes with similar complexity.

We establish a simple relation between high-diversity multidimensional rotations obtained from totally complex cyclotomic fields and the discrete Fourier transform. The diversity distribution of an Hadamard-like random rotation is derived analytically. It is shown that a random multidimensional rotation exhibits an excellent diversity distribution and can be combined to quadrature amplitude modulation (QAM) constellations to combat channel fading. We also describe a mean-square error (MSE) universal lattice decoder suitable for large dimensions up to 1024. The MSE criterion treats the lattice structure as intersymbol interference. The universal decoder is applied to both Gaussian and Rayleigh fading channels to decode dense lattice sphere packings and rotated cubic constellations, respectively.

We build a class of pseudo-random error correcting codes, called generalized low density codes (GLD), from the intersection of two interleaved block codes. GLD code performance approaches the channel capacity limit and the GLD decoder is based on simple and fast SISO (soft input-soft output) decoders of smaller block codes. GLD codes are a special case of Tanner codes and a generalization of Gallager's LDPC codes. It is also proved by an ensemble performance argument that these codes are asymptotically good in the sense of the minimum distance criterion. The flexibility in selecting the parameters of GLD codes makes them suitable for small and large block length forward error correcting schemes.

We present a maximum-likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver. The decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point. By judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment.

The increasing need for high data-rate transmissions over time- or frequency-selective fading channels has drawn attention to modulation schemes with high spectral efficiency such as QAM. With the aim of increasing the "diversity order" of the signal set we consider multidimensional rotated QAM constellations. Very high diversity orders can be achieved and this results in an almost Gaussian performance over the fading channel, This multidimensional modulation scheme is essentially uncoded and enables one to trade diversity for system complexity, at no power or bandwidth expense.

Recent work on lattices matched to the Rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency. We present a new family of lattice constellations, based on complex algebraic number fields, which have good performance on Rayleigh fading channels. Some of these lattices also present a reasonable packing density and thus may be used at the same time over a Gaussian channel. Conversely, we show that particular versions of the best lattice packings (D/sub 4/, E/sub 6/, E/sub 8/, K/sub 12/, /spl Lambda//sub 16/, /spl Lambda//sub 24/), constructed from totally complex algebraic cyclotomic fields, present better performance over the Rayleigh fading channel. The practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links. Some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results.