Multi-band elastic optical networks (MB-EONs) transmit information in multiple bands to increase the available capacity. However, they suffer from quality of transmission (QoT) degradation caused by the inter-channel stimulated Raman scattering effect, which requires addressing through tailored resource assignment. Additionally, dynamically arriving and departing optical service requests generate spectrum fragmentation (SF), where spectrum resources become scattered into non-continuous chunks and aggravate service blocking ratio (SBR) even when the total available bandwidth is sufficient. To jointly address these challenges, we propose an SF- and QoT-aware algorithm for routing, band, modulation format, and spectrum assignment (RBMSA), along with proactive spectrum defragmentation (SD), referred to asSFQA-defrag. The algorithm considers SF metrics and QoT levels of available channels across multiple candidate paths to ensure that the QoT requirements are met while minimizing the SF. The SD process proactively reorganizes spectrum allocation to reduce fragmentation by consolidating the spectrum gaps, which leads to lower blocking of future requests. TheSFQA-defragalgorithm is evaluated against benchmark algorithms that independently consider either QoT or SF in three reference backbone topologies. The results demonstrate thatSFQA-defragsignificantly reduces the SBR and SF compared to benchmarks, albeit with a slight increase in the average path length.

Multidimensional Voronoi constellations (VCs) have been shown to be more power-efficient than quadrature amplitude modulation (QAM) formats given the same uncoded bit error rate, and also have higher achievable information rates. However, a coded modulation scheme that sustains these gains after forward error correction (FEC) coding is still lacking. This paper designs coded modulation schemes with soft-decision FEC codes for VCs, including bit-interleaved coded modulation (BICM) and multilevel coded modulation (MLCM), together with three bit-to-integer mapping algorithms and log-likelihood ratio calculation algorithms. Simulation results show that VCs can achieve up to 1.84 dB signal-to-noise ratio (SNR) gains over QAM with BICM, and up to 0.99 dB SNR gains over QAM with MLCM for the additive white Gaussian noise channel at the bit error rate of$1.81\times 10^{-3}$, with a low decoding complexity.

Lattices with minimal normalized second moments are designed using a new numerical optimization algorithm. Starting from a random lower-triangular generator matrix and applying stochastic gradient descent, all elements are updated towards the negative gradient, which makes it the most efficient algorithm proposed so far for this purpose. A graphical illustration of the theta series, called theta image, is introduced and shown to be a powerful tool for converting numerical lattice representations into their underlying exact forms. As a proof of concept, optimized lattices are designed in dimensions up to 16. In all dimensions, the algorithm converges to either the previously best known lattice or a better one. The dual of the 15-dimensional laminated lattice is conjectured to be optimal in its dimension and its exact normalized second moment is computed.

The capacity of a discrete-time channel with correlated phase noises is investigated. In particular, the electro-optic frequency comb system is considered, where the phase noise of each subchannel is a combination of two independent Wiener phase-noise sources. Capacity upper and lower bounds are derived for this channel and are compared with lower bounds obtained by numerically evaluating the achievable information rates using quadrature amplitude modulation constellations. Capacity upper and lower bounds are provided for the high signal-to-noise ratio (SNR) regime. The multiplexing gain (pre-log) is shown to beM− 1, whereMrepresents the number of subchannels. A constant gap between the asymptotic upper and lower bounds is observed, which depends on the number of subchannelsM. For the specific case ofM= 2, capacity is characterized up to a term that vanishes as the SNR grows large.

40 years ago, Conway and Sloane proposed using the highly symmetrical Coxeter–Todd latticeK12for quantization, and estimated its second moment. Since then, all published lists identifyK12as the best 12-dimensional lattice quantizer. Surprisingly,K12is not optimal: we construct two new 12-dimensional lattices with lower normalized second moments. The new lattices are obtained by gluing together products of two 6-dimensional lattices.

A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.

In optical fiber communication, due to the random variation of the environment, the state of polarization (SOP) fluctuates randomly with time leading to distortion and performance degradation. The memory-less SOP fluctuations can be regarded as a two-by-two random unitary matrix. In this paper, for what we believe to be the first time, the capacity of the polarization drift channel under an average power constraint with imperfect channel knowledge is characterized. An achievable information rate (AIR) is derived when imperfect channel knowledge is available and is shown to be highly dependent on the channel estimation technique. It is also shown that a tighter lower bound can be achieved when a unitary estimation of the channel is available. However, the conventional estimation algorithms do not guarantee a unitary channel estimation. Therefore, by considering the unitary constraint of the channel, a data-aided channel estimator based on the Kabsch algorithm is proposed, and its performance is numerically evaluated in terms of AIR. Monte Carlo simulations show that Kabsch outperforms the least-square error algorithm. In particular, with complex, Gaussian inputs and eight pilot symbols per block, Kabsch improves the AIR by 0.20 to 0.30 bits/symbol throughout the range of studied signal-to-noise ratios.

Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up version of the coding lattice. In this paper, we design low-complexity VCs with a cubic coding lattice of up to 32 dimensions, in which pseudo-Gray labeling is applied to minimize the bit error rate. The designed VCs have considerable shaping gains of up to 1.03 dB and finer choices of spectral efficiencies in practice compared with conventional VCs. A mutual information estimation method and a log-likelihood approximation method based on importance sampling for very large constellations are proposed and applied to the designed VCs. With error-control coding, the proposed VCs can have higher information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity.

Dynamic reconfigurability in optical and mobile networks can facilitate heterogeneous service provisioning while utilizing minimal resources. This allows cost-efficient service delivery resulting in high revenues for network operators. Deployment of elastic mobile and optical networks is a key driver for enabling reconfigurability in modern networks. Elastic optical networks can be exploited as the fronthaul portion of new generation of mobile networks. Such elastic optical fronthaul networks facilitate joint reconfiguration of flexible radio and optical elements and provide considerable performance improvements. In this paper, we focus on the joint dynamic selection of functional splits and configuration of optical transponders and illustrate that designing a converged network with optical and radio elements improves network power efficiency. A time-averaged stochastic optimization problem is formulated and its solution is derived using a modified version of the Lyapunov drift technique. Simulation results demonstrate that the proposed scheme can reduce the average power consumption by up to 70% compared to a cloud radio access network with a traditional optical fronthaul. Further, the results show that the modified Lyapunov technique can afford stringent fronthaul delays below 250$\mu \text{s}$. We also discuss how future technology upgrades such as increasing the number of radio antenna ports and decreasing the granularity of fiber spectrum grid may influence the results.

End-to-end autoencoder (AE) learning has the potential of exceeding the performance of human-engineered transceivers and encoding schemes, without a priori knowledge of communication-theoretic principles. In this work, we aim to understand to what extent and for which scenarios this claim holds true when comparing with fair benchmarks. Our particular focus is on memoryless multiple-input multiple-output (MIMO) and multi-user (MU) systems. Four case studies are considered: two point-to-point (closed-loop and open-loop MIMO) and two MU scenarios (MIMO broadcast and interference channels). For the point-to-point scenarios, we explain some of the performance gains observed in prior work through the selection of improved baseline schemes that include geometric shaping as well as bit and power allocation. For the MIMO broadcast channel, we demonstrate the feasibility of a novel AE method with centralized learning and decentralized execution. Interestingly, the learned scheme performs close to nonlinear vector-perturbation precoding and significantly outperforms conventional zero-forcing. Lastly, we highlight potential pitfalls when interpreting learned communication schemes. In particular, we show that the AE for the considered interference channel learns to avoid interference, albeit in a rotated reference frame. After de-rotating the learned signal constellation of each user, the resulting scheme corresponds to conventional time sharing with geometric shaping.

In a classical 1983 paper, Conway and Sloane presented fast encoding and decoding algorithms for a special case of Voronoi constellations (VCs), for which the shaping lattice is a scaled copy of the coding lattice. Feng generalized their encoding and decoding methods to arbitrary VCs. Less general algorithms were also proposed by Kurkoski and Ferdinand, respectively, for VCs with some constraints on their coding and shaping lattices. In this work, we design VCs with a cubic coding lattice based on Kurkoski's encoding and decoding algorithms. The designed VCs achieve up to 1.03 dB shaping gains with a lower complexity than Conway and Sloane's scaled VCs. To minimize the bit error rate (BER), pseudo-Gray labeling of constellation points is applied. In uncoded systems, the designed VCs reduce the required SNR by up to 1.1 dB at the same BER, compared with the same VCs using Feng's and Ferdinand's algorithms. In coded systems, the designed VCs are able to achieve lower BER than the scaled VCs at the same SNR. In addition, a Gray penalty estimation method for such VCs of very large size is introduced.

The emerging video services are associated with stringent quality-of-service (QoS) requirements and place high bandwidth demands on the core networks. Edge caching can facilitate the stringent QoS demands while easing the bandwidth requirement from core networks. However, such schemes require on-field caching equipment, in which energy consumption is a function of cache utilization. Designing opportunistic caching strategies for energy efficiency is therefore essential in such schemes. This paper studies the possibilities for achieving high energy efficiency, QoS, and low bandwidth consumption from the core network, in an optically fronthauled fog-enhanced radio access network that implements edge caching. An analytical model for such a network has been derived to measure latency, bandwidth consumption, and cache utilization. It is deduced from the results that low latency (high QoS) and bandwidth consumption can be ensured in such schemes while reducing the energy consumption by up to 93%. The derived model allows to design caching strategies for addressing the trade-off between energy efficiency, QoS, and bandwidth efficiency.

The standard solution for short-haul fiber-optic communications is to deploy noncoherent systems, i.e., to modulate and detect only the light intensity. In such systems, the signal is corrupted with optical noise from amplifiers and with thermal (electrical) noise. The capacity of noncoherent optical links has been studied extensively in the presence of either optical noise or thermal noise. In this paper, for the first time, we characterize the capacity under an average power constraint with both noise sources by establishing upper and lower bounds. In the two extreme cases of zero optical noise or zero thermal noise, we assess our bounds against some well-known results in the literature; improvements in both cases are observed. Next, for amplified fiber-optic systems, we study the trade-off between boosting signal energy (mitigating the effects of thermal noise) and adding optical noise. For a wide spectrum of system parameters and received power levels, we determine the optimal amplification gain. While mostly either no amplification or high-gain amplification is optimal, the best performance is for some parameter intervals achieved at finite gains.

The problem of phase-noise compensation for correlated phase noise in coded multichannel optical transmission is investigated. To that end, a simple multichannel phase-noise model is considered and the maximum a posteriori detector for this model is approximated using two frameworks, namely factor graphs (FGs) combined with the sum-product algorithm (SPA) and a variational Bayesian (VB) inference method. The resulting pilot-aided algorithms perform iterative phase-noise compensation in cooperation with a decoder, using extended Kalman smoothing to estimate the a posteriori phase-noise distribution jointly for all channels. The system model and the proposed algorithms are verified using experimental data obtained from space-division multiplexed multicore-fiber transmission. Through Monte Carlo simulations, the algorithms are further evaluated in terms of phase-noise tolerance for coded transmission. It is observed that they significantly outperform the conventional approach to phase-noise compensation in the optical literature. Moreover, the FG/SPA framework performs similarly or better than the VB framework in terms of phase-noise tolerance of the resulting algorithms, for a slightly higher computational complexity.

A new lower bound on the average reconstruction error variance of multidimensional sampling and reconstruction is presented. It applies to sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic process with constant, isotropically bandlimited spectrum and reconstruction by the best linear interpolator. The lower bound is exact for any lattice at sufficiently high and low sampling rates. The two threshold rates where the error variance deviates from the lower bound gives two optimality criteria for sampling lattices. It is proved that at low rates, near the first threshold, the optimal lattice is the dual of the best sphere-covering lattice, which for the first time establishes a rigorous relation between optimal sampling and optimal sphere covering. A previously known result is confirmed at high rates, near the second threshold, namely, that the optimal lattice is the dual of the best sphere-packing lattice. Numerical results quantify the performance of various lattices for sampling and support the theoretical optimality criteria.

It is well known that matched filtering and sampling (MFS) demodulation together with minimum Euclidean distance (MD) detection constitute the optimal receiver for the additive white Gaussian noise channel. However, for a general nonlinear transmission medium, MFS does not provide sufficient statistics, and, therefore, is suboptimal. Nonetheless, this receiver is widely used in optical systems, where the Kerr nonlinearity is the dominant impairment at high powers. In this paper, we consider a suite of receivers for a two-user channel subject to a type of nonlinear interference that occurs in wavelength-division-multiplexed channels. The asymptotes of the symbol error rate (SER) of the considered receivers at high powers are derived or bounded analytically. Moreover, Monte-Carlo simulations are conducted to evaluate the SER for all the receivers. Our results show that receivers that are based on MFS cannot achieve arbitrary low SERs, whereas the SER goes to zero as the power grows for the optimal receiver. Furthermore, we devise a heuristic demodulator, which together with the MD detector yields a receiver that is simpler than the optimal one and can achieve arbitrary low SERs. The SER performance of the proposed receivers is also evaluated for some single-span fiber-optical channels via split-step Fourier simulations.

A classical problem in digital communications is to evaluate the symbol error probability (SEP) and bit error probability (BEP) of a multidimensional constellation over an additive white Gaussian noise channel. In this paper, we revisit this problem for nonequally likely symbols and study the behavior of the optimal maximum a posteriori (MAP) detector at asymptotically high signal-to-noise ratios. Exact closed-form asymptotic expressions for SEP and BEP for arbitrary constellations and input distributions are presented. The well-known union bound is proven to be asymptotically tight under general conditions. The performance of the practically relevant maximum likelihood (ML) detector is also analyzed. Although the decision regions with MAP detection converge to the ML regions at high signal-to-noise ratios, the ratio between the MAP and ML detectors in terms of both SEP and BEP approaches a constant, which depends on the constellation and a priori probabilities. Necessary and sufficient conditions for asymptotic equivalence between the MAP and ML detectors are also presented.

Matched filtering and sampling, which is known to be the optimal receiver for the linear additive white Gaussian noise channel, is in general suboptimal for a nonlinear medium. Nonetheless, it is commonly used in fiber-optical communication systems with nonlinear distortion. In this paper, a novel demodulation scheme is proposed for a two-user memoryless interference channel, with a type of nonlinear crosstalk that occurs in wavelength-multiplexed optical transmission. We show by simulations that by using this demodulation scheme, unlike matched filtering and sampling, the symbol error rate decreases to zero in the high-power regime.

The Closest Vector Problem (CVP) and the Shortest Vector Problem (SVP) are prime problems in lattice-based cryptanalysis, since they underpin the security of many lattice-based cryptosystems. Despite the importance of these problems, there are only a few CVP-solvers publicly available, and their scalability was never studied. This paper presents a scalable implementation of an enumeration-based CVP-solver for multi-cores, which can be easily adapted to solve the SVP. In particular, it achieves super-linear speedups in some instances on up to 8 cores and almost linear speedups on 16 cores when solving the CVP on a 50-dimensional lattice. Our results show that enumeration-based CVP-solvers can be parallelized as effectively as enumeration-based solvers for the SVP, based on a comparison with a state of the art SVP-solver. In addition, we show that we can optimize the SVP variant of our solver in such a way that it becomes 35%-60% faster than the fastest enumeration-based SVP-solver to date.

We present a comprehensive study of the information rate loss of the max-log approximation for M-ary pulse-amplitude modulation (PAM) in a bit-interleaved coded modulation (BICM) system. It is widely assumed that the calculation of L-values using the max-log approximation leads to an information loss. We prove that this assumption is correct for all M-PAM constellations and labelings with the exception of a symmetric 4-PAM constellation labeled with a Gray code. We also show that for max-log L-values, the BICM generalized mutual information (GMI), which is an achievable rate for a standard BICM decoder, is too pessimistic. In particular, it is proved that the so-called harmonized GMI, which can be seen as the sum of bit-level GMIs, is achievable without any modifications to the decoder. We then study how bit-level channel symmetrization and mixing affect the MI and the GMI for max-log L-values. Our results show that these operations, which are often used when analyzing BICM systems, preserve the GMI. However, this is not necessarily the case when the MI is considered. Necessary and sufficient conditions under which these operations preserve the MI are provided.

There exists a rich flora of channel models for optical fiber channels, which differ not only in the types of transmission scenario they describe but also in the type of analysis they support. In this tutorial paper, we review several channel models used in optical communications, and discuss their suitability for information-theoretic analyses. Key issues are how nonlinearity, channel memory, and multiuser interference are modeled.

Motivated by results in optical communications, where the performance can degrade dramatically if the transmit power is sufficiently increased, the channel capacity is characterized for various kinds of memoryless vector channels. It is proved that for all static point-to-point channels, the channel capacity under an equal-power constraint is a nondecreasing function of power. As a consequence, maximizing the mutual information over all input distributions with a certain power is for such channels equivalent to maximizing it over the larger set of input distributions with upperbounded power. The channel coding theorem is formally proved for an equal-power constraint. For interference channels such as optical wavelength-division multiplexing systems, the primary channel capacity is always nondecreasing with power if all interferers transmit with identical distributions as the primary user. Also, if all input distributions in an interference channel are optimized jointly, then the achievable sum-rate capacity is again nondecreasing. The results generalize to the channel capacity as a function of a wide class of costs, not only power.

In many practical communication systems, one binary encoder/decoder pair is used to communicate over a set of parallel channels. Examples of this setup include multi-carrier transmission, rate-compatible puncturing of turbo-like codes, and bit-interleaved coded modulation (BICM). A bit mapper is commonly employed to determine how the coded bits are allocated to the channels. In this paper, we study spatially coupled low-density parity check codes over parallel channels and optimize the bit mapper using BICM as the driving example. For simplicity, the parallel bit channels that arise in BICM are replaced by independent binary erasure channels (BECs). For two parallel BECs modeled according to a 4-PAM constellation labeled by the binary reflected Gray code, the optimization results show that the decoding threshold can be improved over a uniform random bit mapper, or, alternatively, the spatial chain length of the code can be reduced for a given gap to capacity. It is also shown that for rate-loss free, circular (tail-biting) ensembles, a decoding wave effect can be initiated using only an optimized bit mapper.

In this paper, a new method for power-efficient intersymbol interference-free transmission over the bandlimited intensity-modulation direct-detection channel is proposed. A new time-varying bias signal is added to the transmitted signal to make it nonnegative and provide a more power-efficient transmission than the previously considered constant bias. To exploit the benefits of the new signaling method, Nyquist and root-Nyquist pulses suitable for the use with this kind of bias are designed using two different methods. In the first method, new pulses are obtained by adding Nyquist pulses in the time domain with different combining coefficients, whereas in the second method, the pulses are obtained by the design of their frequency response. Analytical expressions for the asymptotic optical power efficiency and symbol error rate of the proposed schemes are derived and evaluated. At a spectral efficiency of 1 b/s/Hz, using on-off keying modulation and the proposed bias signal and pulses, up to 0.628 dB gains in asymptotic power efficiency can be achieved compared to the previously best known signaling scheme, which is based on squared sinc pulse shaping.

In this paper, we propose a novel detector for single-channel long-haul coherent optical communications, termed stochastic digital backpropagation (SDBP), which takes into account noise from the optical amplifiers in addition to handling deterministic linear and nonlinear impairments. We discuss the design approach behind this detector, which is based on the maximum a posteriori (MAP) principle. As closed-form expressions of the MAP detector are not tractable for coherent optical transmission, we employ the framework of Bayesian graphical models, which allows a numerical evaluation of the proposed detector. Through simulations, we observe that by accounting for nonlinear signal-noise interactions, we achieve a significant improvement in system reach with SDBP over digital backpropagation (DBP) for systems with periodic inline optical dispersion compensation. In uncompensated links with high symbol rates, the performance difference in terms of system reach for SDBP over DBP is small. In the absence of noise, the proposed detector is equivalent to the well-known DBP detector.

Asymptotic expressions of the mutual information between any discrete input and the corresponding output of the scalar additive white Gaussian noise channel are presented in the limit as the signal-to-noise ratio (SNR) tends to infinity. Asymptotic expressions of the symbol-error probability (SEP) and the minimum mean-square error (MMSE) achieved by estimating the channel input given the channel output are also developed. It is shown that for any input distribution, the conditional entropy of the channel input given the output, MMSE, and SEP have an asymptotic behavior proportional to the Gaussian Q-function. The argument of the Q-function depends only on the minimum Euclidean distance (MED) of the constellation and the SNR, and the proportionality constants are functions of the MED and the probabilities of the pairs of constellation points at MED. The developed expressions are then generalized to study the high-SNR behavior of the generalized mutual information (GMI) for bit-interleaved coded modulation (BICM). By means of these asymptotic expressions, the long-standing conjecture that Gray codes are the binary labelings that maximize the BICM-GMI at high SNR is proven. It is further shown that for any equally spaced constellation whose size is a power of two, there always exists an anti-Gray code giving the lowest BICM-GMI at high SNR.

Two decoder structures for coded modulation over the Gaussian channel are studied: 1) the maximum likelihood symbol-wise decoder and 2) the (suboptimal) bit-wise decoder based on the bit-interleaved coded modulation paradigm. We consider a 16-ary quadrature amplitude constellation labeled with a Gray labeling. It is shown that the asymptotic loss in terms of pairwise error probability, for any two codewords caused by the bit-wise decoder, is bounded by 1.25 dB. The analysis also shows that the asymptotic loss is zero for a wide range of linear codes, including all rate-1/2 convolutional codes.

This paper introduces a novel opportunistic transmission strategy for cognitive radios (CRs). The primary user (PU) is assumed to transmit in a time-slotted manner according to a two-state Markov model, and the CR is either sensing, that is, obtaining a causal, noisy observation of a primary user (PU) state, or transmitting, but not both at the same time. In other words, the CR observations of the PU are censored whenever the CR is transmitting. The objective of the CR transmission strategy is to maximize the utilization ratio (UR), i.e., the relative number of the PU-idle slots that are used by the CR, subject to that the interference ratio (IR), i.e., the relative number of the PU-active slots that are used by the CR, is below a certain level. We introduce an a-posteriori LLR-based CR transmission strategy, called CLAPP, and evaluate this strategy in terms of the achievable UR for different PU model parameters and received signal-to-noise ratios (SNRs). The performance of CLAPP is compared with a simple censored energy detection scheme. Simulation results show that CLAPP has 52% gain in UR over the best censored energy detection scheme for a maximum IR level of 10% and an SNR of -2dB.

The asymptotic behavior of the mutual information (MI) at high signal-to-noise ratio (SNR) for discrete constellations over the scalar additive white Gaussian noise channel is studied. Exact asymptotic expressions for the MI for arbitrary one-dimensional constellations and input distributions are presented in the limit as the SNR tends to infinity. Asymptotics of the minimum mean-square error (MMSE) are also developed. It is shown that for any input distribution, the MI and the MMSE have an asymptotic behavior proportional to a Gaussian Q-function, whose argument depends on the minimum Euclidean distance of the constellation and the SNR. Closed-form expressions for the coefficients of these Q-functions are calculated.

An asymptotically optimal trellis-coded modulation (TCM) encoder requires the joint design of the encoder and the binary labeling of the constellation. Since analytical approaches are unknown, the only available solution is to perform an exhaustive search over the encoder and the labeling. For large constellation sizes and/or many encoder states, however, an exhaustive search is unfeasible. Traditional TCM designs overcome this problem by using a labeling that follows the set-partitioning principle and by performing an exhaustive search over the encoders. In this paper we study binary labelings for TCM and show how they can be grouped into classes, which considerably reduces the search space in a joint design. For 8-ary constellations, the number of different binary labelings that must be tested is reduced from 8!=40320 to 240. For the particular case of an 8-ary pulse amplitude modulation constellation, this number is further reduced to 120 and for 8-ary phase shift keying to only 30. An algorithm to generate one labeling in each class is also introduced. Asymptotically optimal TCM encoders are tabulated which are up to 0.3 dB better than the previously best known encoders.

We study the design of amplitude phase-shift keying (APSK) constellations for a coherent fiber-optical communication system where nonlinear phase noise (NLPN) is the main system impairment. APSK constellations can be regarded as a union of phase-shift keying (PSK) signal sets with different amplitude levels. A practical two-stage (TS) detection scheme is analyzed, which performs close to optimal detection for high enough input power. We optimize APSK constellations with 4, 8, and 16 points in terms of symbol error probability (SEP) under TS detection for several combinations of input power and fiber length. For 16 points, performance gains of 3.2 dB can be achieved at a SEP of 10^{-2} compared to 16-QAM by choosing an optimized APSK constellation. We also demonstrate that in the presence of severe nonlinear distortions, it may become beneficial to sacrifice a constellation point or an entire constellation ring to reduce the average SEP. Finally, we discuss the problem of selecting a good binary labeling for the found constellations.

The optimal bit-wise demodulator for M-ary pulse amplitude modulation (PAM) over the additive white Gaussian noise channel is analyzed in terms of uncoded bit-error rate (BER). The BER analysis is based on studying the bit patterns that form a labeling. New closed-form BER expressions for 4-PAM with any labeling are developed. Moreover, closed-form BER expressions for 11 out of 23 possible bit patterns for 8-PAM are presented, which enable us to obtain the BER for 8-PAM with some of the most popular labelings, including the binary reflected Gray code and the natural binary code. Numerical results show that, regardless of the labeling, there is no difference between the optimal demodulator and the symbol-wise demodulator for any BER of practical interest (below 0.1).

The generalized mutual information (GMI) of bit-interleaved coded modulation (BICM) systems, sometimes called the BICM capacity, is investigated at low signal-to-noise ratio (SNR). The combinations of input alphabet, input distribution, and binary labeling that achieve the Shannon limit${- 1.59}\;{\rm dB}$are completely characterized. The main conclusion is that a BICM system with probabilistic shaping achieves the Shannon limit at low SNR if and only if it can be represented as a zero-mean linear projection of a hypercube. Hence, probabilistic shaping offers no extra degrees of freedom to optimize the low-SNR BICM-GMI, in addition to what is provided by geometrical shaping. The analytical conclusions are confirmed by numerical results, which also show that for a fixed input alphabet, probabilistic shaping can improve the BICM-GMI in the low and medium SNR range.

We consider the design of amplitude phase-shift keying (APSK) constellations, targeting their application to coherent fiber-optical communications. Phase compensation is used at the receiver to combat nonlinear phase noise caused by the Kerreffect. We derive the probability density function of the post-compensated observation for multilevel constellations. Optimal APSK constellations in terms of symbol error probability (SEP) are found assuming a two-stage detector. Performance gains of 3:2 dB can be achieved compared to 16-QAM at a SEP of 10-2. We optimize the number of rings, the number of points per ring, as well as the radius distribution of the constellation. For low to moderate nonlinearities, radius optimization only yields minor improvements over an equidistant spacing of rings. In the highly nonlinear regime, however, a smaller SEP can be achieved by “sacrificing” the outer ring of the constellation, in favor of achieving good SEP in the remaining rings.

A novel general ready-to-use bit-error rate (BER) expression for one-dimensional constellations is developed. The BER analysis is performed for bit patterns that form a labeling. The number of patterns for equally spaced M-PAM constellations with different BER is analyzed.

Probabilistic shaping for bit-interleaved coded modulation (BICM) systems at low signal-to-noise ratio (SNR) is investigated. Using known results for BICM systems with a uniform input distribution, the combinations of input alphabet, input distribution, and binary labeling that achieve the Shannon limit -1.59 dB are fully characterized. It is found that a BICM system achieves the Shannon limit at low SNR if and only if it can be represented as a zero-mean linear projection of a hypercube, which is the same condition as for uniform input distributions. Hence, probabilistic shaping offers no extra degrees of freedom to optimize the low-SNR regime of BICM systems, in addition to what is provided by geometrical shaping.

Optimal trellis-coded modulation (TCM) schemes are obtained by jointly designing the convolutional encoder and the binary labeling of the constellation. Unfortunately this approach is infeasible for large encoder memories or constellation sizes. Traditional TCM designs circumvent this problem by using a labeling that follows the set-partitioning principle and by performing an exhaustive search over the encoders. Therefore, traditional TCM schemes are not necessarily optimal. In this paper, we study binary labelings for TCM and show how they can be grouped into classes, which considerably reduces the search space in a joint design. For the particular case of 8-ary modulation the search space for the labelings is reduced from 8! to 240. Using this classification, we formally prove that for any channel it is always possible to design a TCM system based on the binary-reflected Gray code with identical performance to the one proposed by Ungerboeck in 1982. Moreover, the classification is used to tabulate asymptotically optimal TCM schemes.

In this paper we study opportunistic transmission strategies for cognitive radios (CR) in which causal noisy observation from a primary user (PU) state is available. The PU is assumed to be operating in a slotted manner, according to a two-state Markov model. The objective is to maximize utilization ratio (UR), i.e., the relative number of the PU-idle slots that are used by CR, subject to interference ratio (IR), i.e., the relative number of the PU-active slots that are used by the CR, below a certain level. We introduce an a-posteriori LLR-based cognitive transmission strategy and show that this strategy is optimum in the sense of maximizing UR given a certain maximum allowed IR. Two methods for calculating threshold for this strategy in practical situations are presented. One of them performs well in higher SNRs but might have too large IR at low SNRs and low PU activity levels, and the other is proven to never violate the allowed IR at the price of a reduced UR. In addition, an upper-bound for the UR of any CR strategy operating in the presence of Markovian PU is presented. Simulation results have shown a more than 116% improvement in UR at SNR of -3dB and IR level of 10% with PU state estimation. Thus, this opportunistic CR mechanism possesses a high potential in practical scenarios in which there exists no information about true states of PU.

An analytical discrete-time model is introduced for single-wavelength polarization multiplexed nonlinear fiber-optical channels based on the symmetrized split-step Fourier method (SSFM). According to this model, for high enough symbol rates, a fiber-optic link can be described as a linear dispersive channel with additive white Gaussian noise (AWGN) and a complex scaling. The variance of this AWGN noise and the attenuation are computed analytically as a function of input power and channel parameters. The results illustrate a cubic growth of the noise variance with input power. Moreover, the cross effect between the two polarizations and the interaction of amplifier noise and the transmitted signal due to the nonlinear Kerr effect are described. In particular, it is found that the channel noise variance in one polarization is affected twice as much by the transmitted power in that polarization than by the transmitted power in the orthogonal polarization. The effect of pulse shaping is also investigated through numerical simulations. Finally, it is shown that the analytical performance results based on the new model are in close agreement with numerical results obtained using the SSFM for a symbol rate of 28 Gbaud and above.

In this paper, the design and analysis of a new bandwidth-efficient signaling method over the bandlimited intensity-modulated direct-detection (IM/DD) channel is presented. The channel can be modeled as a bandlimited channel with nonnegative input and additive white Gaussian noise (AWGN). Due to the nonnegativity constraint, standard methods for coherent bandlimited channels cannot be applied here. Previously established techniques for the IM/DD channel require bandwidth twice the required bandwidth over the conventional coherent channel. We propose a method to transmit without intersymbol interference in a bandwidth no larger than the bit rate. This is done by combining Nyquist or root-Nyquist pulses with a constant bias and using higher-order modulation formats. In fact, we can transmit with a bandwidth equal to that of coherent transmission. A trade-off between the required average optical power and the bandwidth is investigated. Depending on the bandwidth required, the most power-efficient transmission is obtained by the parametric linear pulse, the so-called "better than Nyquist" pulse, or the root-raised cosine pulse.

We optimize modulation formats for the additive white Gaussian noise channel with nonnegative input, also known as the intensity-modulated direct-detection channel, with and without confining them to a lattice structure. Our optimization criteria are the average electrical, average optical, and peak power. The nonnegative constraint on the input to the channel is translated into a conical constraint in signal space, and modulation formats are designed by sphere packing inside this cone. Some dense packings are found, which yield more power-efficient modulation formats than previously known. For example, at a spectral efficiency of 1.5 bit/s/Hz, the modulation format optimized for average electrical power has a 2.55 dB average electrical power gain over the best known format to achieve a symbol error rate of 10-6. The corresponding gains for formats optimized for average and peak optical power are 1.35 and 1.72 dB, respectively. Using modulation formats optimized for peak power in average-power limited systems results in a smaller power penalty than when using formats optimized for average power in peak-power limited systems. We also evaluate the modulation formats in terms of their mutual information to predict their performance in the presence of capacity-achieving error-correcting codes, and finally show numerically and analytically that the optimal modulation formats for reliable transmission in the wideband regime have only one nonzero point.

Reutilization of the spectrum licensed to services with low occupancy is of great interest for cognitive radios (CRs). To achieve this goal, we introduce a simple hidden Markov model which captures the primary users activity, signal uncertainties, and noise. For evaluating the performance of any CR, two new criteria are presented entitled spectrum utilization ratio (UR) and interference ratio (IR). Based on this model and new measures, a new a- posterior log-likelihood-ratio based CR is designed and implemented. Its performance is compared with standard energy- detection based spectrum-sensing CR. We demonstrate more than 300% increase in UR for up to 1% allowed interference at the SNR of -5dB.

We optimize modulation formats for the additive white Gaussian noise channel with a nonnegative input constraint, also known as the intensity-modulated direct detection channel, with and without confining them to a lattice structure. Our optimization criteria are the average electrical and optical power. The nonnegativity input signal constraint is translated into a conical constraint in signal space, and modulation formats are designed by sphere packing inside this cone. Some remarkably dense packings are found, which yield more power-efficient modulation formats than previously known. For example, at a spectral efficiency of 1 bit/s/Hz, the obtained modulation format offers a 0.86 dB average electrical power gain and 0.43 dB average optical power gain over the previously best known modulation formats to achieve a symbol error rate of 10-6. This modulation turns out to have a lattice-based structure. At a spectral efficiency of 3/2 bits/s/Hz and to achieve a symbol error rate of 10-6, the modulation format obtained for optimizing the average electrical power offers a 0.58 dB average electrical power gain over the best lattice-based modulation and 2.55 dB gain over the best previously known format. However, the modulation format optimized for average optical power offers a 0.46 dB average optical power gain over the best lattice-based modulation and 1.35 dB gain over the best previously known format.

In this paper, the design and analysis of a new bandwidth-efficient signalling method over the bandlimited intensity-modulated direct-detection (IM/DD) channel is proposed. The channel can be modeled as a bandlimited channel with nonnegative input and additive white Gaussian noise. Due to the nonnegativity constraint, the methods previously proposed for conventional bandlimited channels cannot be applied here. We propose a method to transmit without intersymbol interference in a narrower bandwidth compared to previous works, by combining Nyquist pulses with a constant bias. In fact, we can transmit with a bandwidth equal to that of coherent transmission. A trade-off between the required average optical power and the bandwidth is investigated. At low bandwidths, the most power-efficient transmission is obtained by either the parametric linear pulse or the so-called "better than Nyquist" pulse, depending on the exact bandwidth.

Recent results have shown that the performance of bit-interleaved coded modulation (BICM) using convolutional codes in nonfading channels can be greatly improved if the bit-level interleaver takes a trivial form (BICM-T), i.e., if it does not interleave the bits at all. The reported gains reach a few decibels and are obtained using a less complex BICM system. In this paper, we give a formal explanation for these results and show that BICM-T is in fact the combination of a TCM transmitter and a BICM receiver. Analytical bounds that predict the performance of BICM-T are developed and a new type of distance spectrum for the convolutional code is introduced.

We present a closed-form maximum likelihood-based data detection algorithm for long-haul optical channels with dominant nonlinear phase noise induced by self-phase modulation. The closed-form detector is evaluated in terms of symbol error rate as a function of input power, and compared with other sub-optimal detectors as well as a non-parametric detector. We show that the performance of the detector deteriorates for high input power levels yielding an optimal operation region. We also provide insights into the behavior of the detector in the highly nonlinear regime.

In this paper, different aspects of the bit-interleaved coded modulation (BICM) capacity for the Gaussian channel are analyzed. Analytical bounds for the BICM capacity are developed. These bounds suggest that the BICM capacity at high signal-to-noise ration (SNR) is determined by the multiplicity of the minimum Euclidean distance over all the subconstellations generated by the mapper. Based on this observation, we conjecture that for any constellation, the highest BICM capacity at high SNR is always obtained by a Gray code, if one exists. Ready-to-use expressions based on Gauss - Hermite quadratures to compute the coded modulation and BICM capacities for any SNR are also presented. Using these expressions, it is shown that the BICM capacity is in general a nonconvex, nonconcave function of the input bit distribution. For 8PAM and 8PSK, there exist 12 and 7 classes of mappings, respectively, with equivalent high-SNR behavior, of which the best class comprises all Gray codes.

Recent results have shown that the performance of bit-interleaved coded modulation (BICM) using convolutional codes in nonfading channels can be significantly improved when the interleaver takes a trivial form (BICM-T), i.e., when it does not interleave the bits at all. In this paper, we give a formal explanation for these results and show that BICM-T is, in fact, the combination of a TCM transmitter and a BICM receiver. To predict the performance of BICM-T, a new type of distance spectrum for convolutional codes is introduced, analytical bounds based on this spectrum are developed, and asymptotic approximations are presented. It is shown that the free Hamming distance of the code is not the relevant optimization criterion for BICM-T. Asymptotically optimal convolutional codes for different constraint lengths are tabulated and BICM-T is shown to offer asymptotic gains of about 2 dB over traditional BICM designs based on random interleavers. The asymptotic gains over uncoded transmission are found to be the same as those obtained by Ungerboeck's one-dimensional trellis-coded modulation (1D-TCM), and therefore, in nonfading channels, BICM-T is shown to be as good as 1D-TCM.

Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binary code results in a higher capacity than the binary reflected Gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e., constellations that make BICM achieve the Shannon limit -1.59 dB. It is shown that theEb/N0required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC; if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.

An exact expression is derived for the symbol error probability when the vertices of the four-dimensional 24-cell are used for digital transmission in Gaussian noise. Corresponding expressions for other regular convex polytopes are summarized. Numerically stable versions of these error probabilities are also obtained.

Most of the calculations in standard sphere decoders are redundant in the sense that they either calculate quantities that are never used or calculate some quantities more than once. A new method, which is applicable to lattices as well as finite constellations, is proposed to avoid these redundant calculations while still returning the same result. Pseudocode is given to facilitate immediate implementation. Simulations show that the speed gain with the proposed method increases linearly with the lattice dimension. At dimension 60, the new algorithms avoid about 75% of all floating-point operations.

A multilevel coded modulation (MLCM) system in the presence of nonlinear phase noise for fiber optical communication is introduced. The proposed scheme exploits a 16-point ring constellation with nonlinear post compensation of the self phase modulation produced via the Kerr effect. A new set partitioning based on the Ungerboeck approach is introduced to maintain unequal error protection in amplitude and phase direction. The rate allocation for the MLCM component codes for different fiber lengths and transmit powers are done numerically. Simulation results show that the proposed MLCM system provides up to 2 dB gain over a forward error correcting scheme for a block error rates around 10^{-9}, with the same overhead (7%) and complexity.

In this contribution, time varying threshold sequential detectors are employed for energy detection-based spectrum sensing in low- SNR regimes. Sequential detection is proven to be faster (on average) than any other multi-sample detector for a set of given probabilities of detection and false-alarm. In this report, exact performance of a sequential detector for spectrum sensing is analyzed using the direct method. The theoretical results presented herein are verified with Monte-Carlo simulations. It is shown that for a SNR of -10dB, among tests with Wald and triangular thresholds with similar probabilities of mis-detection and false- alarm, triangular performs 54% faster in terms of maximum detection time (90 percentile).

We present a new rate allocation scheme for multilevel coded modulation based on the minimization of the total block error rate (BLER). The proposed method uses affine code components and hard decision multistage decoding. Exhaustive search for the rate allocation which minimizes the total BLER justifies the near-optimum performance of the introduced method in moderate to high SNRs. Compared to previous approaches this new rate allocation scheme can improve the performance of the system by 1 dB at BLER = 10-6for 16-QAM with Ungerboeck set partitioning. Interestingly, our results indicate that the optimum rate allocation is a function of the SNR. Finally, the performance of some specific codes are evaluated by simulation and union bounds to verify the theoretical results.

In this paper we formally analyze the interleaver and code design for QAM-based BICM transmissions using the binary reflected Gray code. We develop analytical bounds on the bit error rate and we use them to predict the performance of BICM when unequal error protection (UEP) is introduced by the constellation labeling. Based on these bounds the optimum design of interleaver and code is found, and numerical results for representative configurations are presented. When the new design is used, the improvements may reach 2 dB, and they are obtained without any increase on the transceiver's complexity. We also introduce the concept of generalized optimum distance spectrum convolutional codes, which are the optimum codes for QAM-based BICM transmissions.

In the above titled paper (ibid., vol. 54, no. 12, pp. 5447-5455, Dec. 08), there are three errors that are corrected here.

In this paper we present a general methodology for the interleaver and code design for QAM-based BICM transmissions. We develop analytical bounds on the bit error rate and we use them to predict the performance of BICM when unequal error protection (UEP) is introduced by the constellation labeling. Based on these bounds, the optimum design of interleaver and code is presented. The improvements obtained reached 2 dB for the analyzed cases, and are obtained without complexity increase. Although previous works noted the influence of the interleaver design and the UEP, to the best of our knowledge, this paper is the first to analyze formally this problem for BICM transmissions.

In this paper the issue of improving the performance of iterative decoders based on sub-optimal calculation of the messages exchanged during iterations (L-values) is addressed. It is well known in the literature that a simple-yet very effective-way to improve the performance of suboptimal iterative decoders is based on applying a scaling factor to the L-values. In this paper, starting with a theoretical model based on the so-called consistency condition of a random variable, we propose a methodology for correcting the L-values that relies only on the distribution of the soft information exchanged in the iterative process. This methodology gives a clear explanation of why the well-known linear scaling factor provides a very good performance. Additionally, the proposed methodology allows us to avoid the exhaustive search required otherwise. Numerical simulations show that for turbo codes the scaling factors found closely follow the optimum values, which translates to a close-to-optimal BER performance. Moreover, for LDPC codes, the proposed methodology produces a better BER performance compared with the known method in the literature.

In this tutorial paper we analyze the capacity of bit-interleaved coded modulation (BICM) with quadrature amplitude modulation (QAM) constellations, and we pay special attention to different bit-to-symbol labeling strategies. The relation between the BICM capacity and the capacity of other CM schemes such as trellis coded modulation (TCM) and multilevel codes (MLC) is analyzed. Motivated by the fact that for BICM with some particular labelings, the same Eb/No maps to more than one BICM capacity value, we study the relation between the capacity and Eb/No. In particular, we present some analytical results on this relation, and we also give an intuitive explanation for the somehow contradictory behavior of these curves.

The problem of finding the optimal labeling (bit-to-symbol mapping) of multilevel coherent phase shift keying (PSK), pulse amplitude modulation (PAM), and quadrature amplitude modulation (QAM) constellations with respect to minimizing the bit-error probability (BEP) over a Gaussian channel is addressed. We show that using the binary reflected Gray code (BRGC) to label the signal constellation results in the lowest possible BEP for high enough signal energy-to-noise ratios and analyze what is "high enough" in this sense. It turns out that the BRGC is optimal for PSK and PAM systems whenever the target BEP is at most a few percent, which covers most systems of practical interest. New and simple closed-form expressions are presented for the BEP of PSK, PAM, and QAM using the BRGC

The generalization of the sampling theorem to multidimensional signals is considered, with or without bandwidth constraints. The signal is modeled as a stationary random process and sampled on a lattice. Exact expressions for the mean-square error of the best linear interpolator are given in the frequency domain. Moreover, asymptotic expansions are derived for the average mean-square error when the sampling rate tends to zero and infinity, respectively. This makes it possible to determine the optimal lattices for sampling. In the low-rate sampling case, or equivalently for rough processes, the optimal lattice is the one which solves the packing problem, whereas in the high-rate sampling case, or equivalently for smooth processes, the optimal lattice is the one which solves the dual packing problem. In addition, the best linear interpolation is compared with ideal low-pass filtering (cardinal interpolation).

This paper is concerned with the problem of selecting a binary labeling for the signal constellation in an M-PSK communication system. A good starting point is labelings having the Gray property, but this is not altogether enough, since the number of distinct Gray labelings that result in different bit error probability grows rapidly with increasing constellation size. By introducing a recursive Gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings (not only Gray), that will give the lowest possible average probability of bit errors. Under certain assumptions on the channel, the answer is that the labeling originally proposed by Gray, the binary reflected Gray code, is the optimal labeling for M-PSK systems, which has, surprisingly, never been proved before.

This paper concerns the problem of selecting a binary labeling for the signal constellation in M-PSK, M-PAM, and M-QAM communication systems. Gray labelings are discussed and the original work by Frank Gray is analyzed. As is noted, the number of distinct Gray labelings that result in different bit-error probability grows rapidly with increasing constellation size. By introducing a recursive Gray labeling construction method called expansion, the paper answers the natural question of what labeling, among all possible constellation labelings, will give the lowest possible average probability of bit errors for the considered constellations. Under certain assumptions on the channel, the answer is that the labeling proposed by Gray, the binary reflected Gray code, is the optimal labeling for all three constellations, which has, surprisingly, never been proved before.

The problem of calculating the average bit-error probability (BEP) of coherent M-ary phase-shift keying (PSK) over a Gaussian channel has been studied previously in the literature. A solution to the problem for systems using a binary reflected Gray code (BRGC) to map bits to symbols was first presented by P.J. Lee (see ibid., vol.COM-34, p.488-91, 1986). We show that the results obtained by Lee are incorrect for M/spl ges/16. We show that the reason for this is an invalid assumption that the bit-error rate (BER) is independent of the transmitted symbols, an assumption which has also propagated to textbooks. We give a new expression for the BER of M-PSK systems using the BRGC and compare this with Lee's results.

In this semitutorial paper, a comprehensive survey of closest point search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closest point search algorithm, based on the Schnorr-Euchner (1995) variation of the Pohst (1981) method, is implemented. Given an arbitrary point x /spl isin/ /spl Ropf//sup m/ and a generator matrix for a lattice /spl Lambda/, the algorithm computes the point of /spl Lambda/ that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan (1983, 1987) algorithm and an experimental comparison with the Pohst (1981) algorithm and its variants, such as the Viterbo-Boutros (see ibid. vol.45, p.1639-42, 1999) decoder. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, computing the Voronoi (1908)-relevant vectors, and finding a Korkine-Zolotareff (1873) reduced basis.

Let A(n, d) denote the maximum possible number of codewords in an (n, d) binary code. We establish four new bounds on A(n, d), namely, A(21, 4)/spl les/43689, A(22, 4)/spl les/87378, A(22, 6)/spl les/6941, and A(23, 4)/spl les/173491. Furthermore, using previous upper bounds on the size of constant-weight binary codes, we reapply known methods to generate a table of bounds on A(n, d) for all n/spl les/28. This table extends the range of parameters compared with previously known tables.

Let A(n,d,w) denote the maximum possible number of codewords in an (n,d,w) constant-weight binary code. We improve upon the best known upper bounds on A(n,d,w) in numerous instances for n/spl les/24 and d/spl les/12, which is the parameter range of existing tables. Most improvements occur for d=8, 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n/spl les/28 and d/spl les/14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n,d,w) by means of mapping constant-weight codes into Euclidean space. This approach produces, among other results, a bound on A(n,d,w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doubly-constant-weight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doubly-bounded-weight codes, which may be thought of as a generalization of the doubly-constant-weight codes. Subsequently, a class of Euclidean-space codes, called zonal codes, is introduced, and a bound on the size of such codes is established. This is used to derive bounds for doubly-bounded-weight codes, which are in turn used to derive bounds on A(n,d,w). We also develop a universal method to establish constraints that augment the Delsarte inequalities for constant-weight codes, used in the linear programming bound. In addition, we present a detailed survey of known upper bounds for constant-weight codes, and sharpen these bounds in several cases. All these bounds, along with all known dependencies among them, are then combined in a coherent framework that is amenable to analysis by computer. This improves the bounds on A(n,d,w) even further for a large number of instances of n, d, and w.

Soft-decision decoding of block codes is regarded as the geometrical problem of identifying the Voronoi region within which a given input vector lies. A measure, called the neighbor ratio, is proposed to characterize how many facets a Voronoi region has. Theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes. An asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1/2 or not. For rates below this threshold, all pairs of codewords tend to share a Voronoi facet; for higher rates, a relatively small fraction of them do.

A training algorithm for the design of lattices for vector quantization is presented. The algorithm uses a steepest descent method to adjust a generator matrix, in the search for a lattice whose Voronoi regions have minimal normalized second moment. The numerical elements of the found generator matrices are interpreted and translated into exact values. Experiments show that the algorithm is stable, in the sense that several independent runs reach equivalent lattices. The obtained lattices reach as low second moments as the best previously reported lattices, or even lower. Specifically, we report lattices in nine and ten dimensions with normalized second moments of 0.0716 and 0.0708, respectively, and nonlattice tessellations in seven and nine dimensions with 0.0727 and 0.0711, which improves on previously known values. The new nine- and ten-dimensional lattices suggest that Conway and Sloane's (1993) conjecture on the duality between the optimal lattices for packing and quantization might be false. A discussion of the application of lattices in vector quantizer design for various sources, uniform and nonuniform, is included.

The Voronoi regions of a block code govern many aspects of the code's performance on a Gaussian channel, and they are fundamental instruments in, for example, error probability analysis and soft-decision decoding. The article presents an efficient method for finding the boundaries of the Voronoi regions for an arbitrary binary linear block code. Two theoretical results together lead to the Voronoi regions. First, it is shown that the question of the Voronoi neighborship can be reduced into testing a simpler relation, called the Gabriel neighborship. Second, a fast method of recognising Gabriel neighbors is proposed. These results are finally employed to describe the Voronoi regions for the Golay codes and several BCH codes, including Hamming codes.

We show that the channel distortion for maximum-entropy encoders, due to noise on a binary-symmetric channel, is minimized if the vector quantizer can be expressed as a linear transform of a hypercube. The index assignment problem is regarded as a problem of linearizing the vector quantizer. We define classes of index assignments with related properties, within which the best index assignment is found by sorting, not searching. Two powerful algorithms for assigning indices to the codevectors of nonredundant coding systems are presented. One algorithm finds the optimal solution in terms of linearity, whereas the other finds a very good, but suboptimal, solution in a very short time.