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The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of Kötter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if μ erasures and δ deviations occur, then errors of rank t can always be corrected provided that 2t ≤ d - 1 + μ + δ, where d is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application, where n packets of length M over Fq are transmitted, the complexity of the decoding algorithm is given by O(dM) operations in an extension field Fqn. [ABSTRACT FROM AUTHOR]

The problem of error-control in random linear network coding is considered. A "noncoherent" or "channel oblivious" model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vectorspace preserving, information transmission is modeled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum-distance decoder for this metric achieves correct decoding if the dimension of the space V∩U is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style "list-1" minimum-distance decoding algorithm is provided. [ABSTRACT FROM AUTHOR]

This paper demonstrates that the transmission distance of an ON-OFF-keyed high-speed pseudolinear optical communication system can be dramatically improved by means of runlength-limited coding. The key idea is to impose constraints on transmitted sequences such that the minimum interpulse spacing increases, and the average transmission power decreases. Both effects help suppress intrachannel four-wave mixing, major nonlinear penalty in pseudolinear long-haul links. Several coding schemes with different properties are designed for various constraints and compared against a prototypical reference system. Finally, numerical simulations are performed for a benchmark system employing differential phase-shift keying. The results obtained indicate that the runlength-limited coding approach can be considered a legitimate alternative to this more advanced modulation format. [ABSTRACT FROM AUTHOR]

Two design techniques are proposed for high-throughput low-density parity-check (LDPC) decoders. A broadcasting technique mitigates routing congestion by reducing the total global wirelength. An interlacing technique increases the decoder throughput by processing two consecutive frames simultaneously. The brief discusses how these techniques can be used for both fully parallel and partially parallel LDPC decoders. For fully parallel decoders with code lengths in the range of a few thousand bits, the half-broadcasting technique reduces the total global wirelength by about 26% without any hardware overhead. The block interlacing scheme is applied to the design of two fully parallel decoders, increasing the throughput by 60% and 71% at the cost of 5.5 % and 9.5 % gate count overhead, respectively. [ABSTRACT FROM AUTHOR]

A partial ordering on general finite-state Markov channels is given, which orders the channels in terms of probability of symbol error under iterative estimation decoding of a low-density parity-check (LDPC) code. This result is intended to mitigate the complexity of characterizing the performance of general finite-state Markov channels, which is difficult due to the large parameter space of this class of channel. An analysis tool, originally developed for the Gilbert-Elliott channel, is extended and generalized to general finite-state Markov channels. In doing so, an operator is introduced for combining finite-state Markov channels to create channels with larger state alphabets, which are then subject to the partial ordering. As a result, the probability of symbol error performance of finite-state Markov channels with different numbers of states and wide ranges of parameters can be directly compared. Several examples illustrating the use of the techniques are provided, focusing on binary finite-state Markov channels and Gaussian finite-state Markov channels. Furthermore, this result is used to order Gilbert-Elliott channels with different marginal state probabilities, which was left as an open problem by previous work. [ABSTRACT FROM AUTHOR]

In a system utilizing multiuser diversity, regular feedback of channel-quality predictions to the base station is required for each user. Typically, the measure of channel quality must be quantized at each mobile station before it can be sent back. In this paper, we present two distributed scalar quantization schemes that optimize two different performance criteria: a) the minimization of the probability Pe of incorrectly identifying the user with the best channel quality and b) maximization of the resulting throughput R. For a typical Rayleigh-fading system with 30 users per sector, numerical optimization results show that the Pe and R realized by the uniform quantization strategy with 16 quantization levels for each user can be achieved by only three quantization levels using the two proposed strategies. A practical approximation of the proposed schemes is studied and is shown to provide near-optimal performance for both performance criteria as the number of quantization levels becomes large. [ABSTRACT FROM AUTHOR]

This paper presents a reduced-complexity approximate density evolution (DE) scheme for low-density parity-check (LDPC) codes in channels with memory in the form of a hidden Markov chain. This approximation is used to design degree sequences representing some of the best known LDPC code ensembles for the Gilbert-Elliott channel, and example optimizations are also given for other Markov channels. The problem of approximating the channel estimation is addressed by obtaining a specially constructed message-passing schedule in which the channel messages all approach their stable densities. It is shown that this new schedule is much easier to approximate than the standard schedule, but has the same ultimate performance in the limits of long block length and many decoding iterations. This result is extended to show that all message-passing schedules that satisfy mild conditions will have the same threshold under density evolution. [ABSTRACT FROM AUTHOR]

This paper considers a class of iterative message- passing decoders for low-density parity-check codes in which the decoder can choose its decoding rule from a set of decoding algorithms at each iteration. Each available decoding algorithm may have a different per-iteration computation time and performance. With an appropriate choice of algorithm at each iteration, overall decoding latency can be reduced significantly, compared with standard decoding methods. Such a decoder is called a gear-shift decoder because it changes its decoding rule (shifts gears) in order to guarantee both convergence and maximum decoding speed (minimum decoding latency). Using extrinsic information transfer charts, the problem of finding the optimum (minimum decoding latency) gear-shift decoder is formulated as a computationally tractable dynamic program. The optimum gear-shift decoder is proved to have a decoding threshold equal to or better than the best decoding threshold among those of the available algorithms. In addition to speeding up software decoder implementations, gear-shift decoding can be applied to optimize a pipelined hardware decoder, minimizing hardware cost for a given decoder throughput. [ABSTRACT FROM AUTHOR]

Efficient soft-decision decoding of Reed-Solomon (RS) codes is made possible by the Koetter-Vardy (KV) algorithm which consists of a front-end to the interpolation-based Guruswami-Sudan (GS) list decoding algorithm. This paper approaches the soft-decision KV algorithm from the point of view of a communications systems designer who wants to know what benefits the algorithm can give, and how the extra complexity introduced by soft decoding can be managed at the systems level. We show how to reduce the computational complexity and memory requirements of the soft-decision front-end. Applications to wireless communications over Rayleigh fading channels and magnetic recording channels are proposed. For a high-rate RS(255,239) code, 2-3 dB of soft-decision gain is possible over a Rayleigh fading channel using 16-quadrature amplitude modulation. For shorter codes and at lower rates, the gain can be as large as 9 dB. To lower the complexity of decoding on the systems level, the redecoding architecture is explored, which uses only the appropriate amount of complexity to decode each packet. An error-detection criterion based on the properties of the KV decoder is proposed for the redecoding architecture. Queueing analysis verifies the practicality of the redecoding architecture by showing that only a modestly sized RAM buffer is required. [ABSTRACT FROM AUTHOR]

Density evolution analysis of low-density parity-check (LDPC) codes in memoryless channels is extended to the Gilbert-Elliott (GE) channel, which is a special case of a large class of channels with hidden Markov memory. In a procedure referred to as estimation decoding, the sum-product algorithm (SPA) is used to perform LDPC decoding jointly with channel-state detection. Density evolution results show (and simulation results confirm) that such decoders provide a significantly enlarged region of successful decoding within the GE parameter space, compared with decoders that do not exploit the channel memory. By considering a variety of ways in which a GE channel may be degraded, it is shown how knowledge of the decoding behavior at a single point of the GE parameter space may be extended to a larger region within the space, thereby mitigating the large complexity needed in using density evolution to explore the parameter space point-by-point. Using the GE channel as a straightforward example, we conclude that analysis of estimation decoding for LDPC codes is feasible in channels with memory, and that such analysis shows large potential gains. [ABSTRACT FROM AUTHOR]

We consider a class of message-passing decoders for low-density parity-check (LDPC) codes whose messages are binary valued. We prove that if the channel is symmetric and all codewords are equally likely to be transmitted, an optimum decoding rule (in the sense of minimizing message error rate) should satisfy certain symmetry and isotropy conditions. Using this result, we prove that Gallager's Algorithm B achieves the optimum decoding threshold among all binary message-passing decoding algorithms for regular codes. For irregular codes, we argue that when the nodes of the message-passing decoder do not exploit knowledge of their decoding neighborhood, optimality of Gallager's Algorithm B is preserved. We also consider the problem of designing irregular LDPC codes and find a bound on the achievable rates with Gallager's Algorithm B. Using this bound, we study the case of low error-rate channels and analytically find good degree distributions for them. [ABSTRACT FROM AUTHOR]

A conditionally Gaussian channel is a vector channel in which the channel output, given the channel input, has a Gaussian distribution with (well-behaved) input-dependent mean and covariance. We study the capacity-achieving probability measure for conditionally Gaussian channels subject to bounded-input constraints and average cost constraints. Many practical communication systems, including additive Gaussian noise channels, certain optical channels, fading channels, and interference channels rail within this framework. Subject to bounded-input constraint (and average cost constraints), we show that the channel capacity is achievable and we derive a necessary and sufficient condition for a probability measure to be capacity achieving. Under certain conditions, the capacity-achieving mea- sure is proved to be discrete. [ABSTRACT FROM AUTHOR]

We introduce the concept of convolutional factor graphs, which represent convolutional factorizations of multivariate functions, just as conventional (multiplicative) factor graphs represent multiplicative factorizations. Convolutional and multiplicative factor graphs arise as natural Fourier transform duals. In coding theory applications, algebraic duality of group codes is essentially an instance of Fourier transform duality. Convolutional factor graphs arise when a code is represented as a sum of subcodes, just as conventional multiplicative factor graphs arise when a code is represented as an intersection of supercodes. With auxiliary variables, convolutional factor graphs give rise to "syndrome realizations" of codes, just as multiplicative factor graphs with auxiliary variables give rise to "state realizations." We introduce normal and coenormal extensions of a multivariate function, which essentially allow a given function to be represented with either a multiplicative or a convolutional factorization, as is convenient. We use these function extensions to derive a number of duality relationships among the corresponding factor graphs, and use these relationships to obtain the duality properties of Forney graphs as a special case. [ABSTRACT FROM AUTHOR]

The highly stochastic nature of wireless environments makes it desirable to monitor link loss rates in wireless sensor networks. In a wireless sensor network, link loss monitoring is particularly supported by the data aggregation communication paradigm of network traffic: the data collecting node can infer link loss rates on all links in the network by exploiting whether packets from various sensors are received, and there is no need to actively inject probing packets for inference purposes. In this paper, we present a low complexity algorithmic framework for link loss monitoring based on the recent modeling and computational methodology of factor graphs. The proposed algorithm iteratively updates the estimates of link losses upon receiving (or detecting the loss of) recently sent packets by the sensors. The algorithm exhibits good performance and scalability, and can be easily adapted to different statistical models of networking scenarios. In particular, due to its low complexity, the algorithm is particularly suitable as a long-term monitoring facility. [ABSTRACT FROM AUTHOR]

We introduce a new one-dimensional (1-D) analysis of low-density parity-check (LDPC) codes on additive white Gaussian noise channels which is significantly more accurate than similar 1-1) methods. Our method assumes a Gaussian distribution in message-passing decoding only for messages from variable nodes to check nodes. Compared to existing work, which makes a Gaussian assumption both for messages from check nodes and from variable nodes, our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence. Similar to previous work, the problem of designing irregular LDPC codes it-duces to a linear programming problem. However, our method al- lows irregular code design in a wider range of rates without any limit on the maximum variable-node degree. We use our method to design irregular LDPC codes with rates greater than 1/4 that perform within a few hundredths of a decibel from the Shannon limit. The designed codes perform almost as well as codes designed by density evolution. [ABSTRACT FROM AUTHOR]

We apply irregular low-density parity-check (LDPC) codes to the design of multilevel coded quadrature amplitude modulation (QAM) schemes for application in discrete multitone systems in frequency-selective channels. A combined Gray/Unger-boeck scheme is used to label each QAM constellation. The Gray-labeled bits are protected using an irregular LDPC code with iterative soft-decision decoding, while other bits are protected using a high-rate Reed-Solomon code with hard-decision decoding (or are left uncoded). The rate of the LDPC code is selected by analyzing the capacity of the channel seen by the Gray-labeled bits and is made adaptive by selective concatenation with an inner repetition code. Using a practical bit-loading algorithm, we apply this coding scheme to an ensemble of frequency-selective channels with Gaussian noise. Over a large number of channel realizations, this coding scheme provides an average effective coding gain of more than 7.5 dB at a bit-error rate of 10-7 and a block length of approximately 10-5 b. This represents a gap of approximately 2.3 dB from the Shannon limit of the additive white Gaussian noise channel, which could be closed to within 0.8-1.2 dB using constellation shaping. [ABSTRACT FROM AUTHOR]

Presents an optical intensity-modulated direct detection channels. Presentation of a structured signal space model for optical intensity channels; Definition of lattice codes satisfying channel constraints; Computation of coding and shaping gain relative to baseline; Applicability of bandwidth-efficient schemes for high data-rate applications.

Describes a sequential decoder for linear block codes that performs maximum-likelihood soft-decision decoding. Description of a metric priority measure; Analysis on the performance of sequential decoders that use the variable bias term (VBT) metric; Results of simulating the sequential decoder with the VBT metric, the Fano metric and the zero bias-term metric.

Discusses the intractability of permuting a block code to minimize trellis complexity. Difficulty in finding a coordinate permutation of a given code to minimize trellis complexity; Problem of finding a coordinate permutation that minimizes the number of vertices at a given depth in the minimal trellis for a binary linear block code.

Presents a proof of a conjecture of McEliece regarding the expansion index of minimal trellis. Trellis complexity; Linear block codes; Rectangular codes; Vitebi decoding.

Demonstrates that the family of maximal fixed-cost codes, with codeword costs defined in a right-cancellative semigroup, are rectangular. Admission of biproper trellis presentations; Minimal trellises; Lattice codes; Constant-weight subcodes.

Highlights a study which proposed an alternative to the transmission systems based on quasi phase shift key (QPSK) with unequal error protection (UEP). Details on QPSK; Information on UEP; Method used to conduct the study; Results of the study; Conclusion based on the study.

The Covid-19 pandemic showed forcefully the fundamental importance broadband data communication and the internet has in our society. Optical communications forms the undisputable backbone of this critical infrastructure, and it is supported by an interdisciplinary research community striving to improve and develop it further. Since the first 'Roadmap of optical communications' was published in 2016, the field has seen significant progress in all areas, and time is ripe for an update of the research status. The optical communications area has become increasingly diverse, covering research in fundamental physics and materials science, high-speed electronics and photonics, signal processing and coding, and communication systems and networks. This roadmap describes state-of-the-art and future outlooks in the optical communications field. The article is divided into 20 sections on selected areas, each written by a leading expert in that area. The sections are thematically grouped into four parts with 4–6 sections each, covering, respectively, hardware, algorithms, networks and systems. Each section describes the current status, the future challenges, and development needed to meet said challenges in their area. As a whole, this roadmap provides a comprehensive and unprecedented overview of the contemporary optical communications research, and should be essential reading for researchers at any level active in this field. [ABSTRACT FROM AUTHOR]

In this paper, the outage performance of wireless networks with unstructured topologies is investigated. The network reliability perspective of graph theory is used to obtain the network outage polynomial of generalized wireless networks for both uncorrelated and correlated wireless channels. A relationship is established between the max-flow min-cut theorem and key communication performance indicators. The diversity order is equal to the size of the minimum cut-set between source and destination, and the coding gain is the number of cut-sets with size equal to the minimum cut. An ergodic capacity analysis of arbitrary network topologies based on the network outage polynomial is also presented. Numerical results are used to illustrate the technical definitions and verify the derivations. [ABSTRACT FROM AUTHOR]

The recipients of the 2018 IEEE Information Theory Society Paper Award are Mansoor I. Yousefi and Frank R. Kschischang for the three-part paper “Information Transmission Using the Nonlinear Fourier Transform, I, II, III” which appeared in the IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 4312–4328 (Part I), pp. 4329–4345 (Part II), and pp. 4346–4369 (Part III), July 2014. [ABSTRACT FROM AUTHOR]

Resistive memories store information in a crossbar arrangement of two-terminal devices that can be programmed to patterns of high or low resistance. While extremely compact, this technology suffers from the “sneak-path” problem: certain information patterns cannot be recovered, as multiple low resistances in parallel make a high resistance indistinguishable from a low resistance. In this paper, a multi-layer device is considered, and the number of bits it can store is derived exactly and asymptotic bounds are developed. The information density of a series of isolated arrays with extreme aspect ratios is derived in the single- and multi-layer cases with and without peripheral selection circuitry. This density is shown to be non-zero in the limit, unlike that of the arrays with moderate aspect ratios previously considered. A simple encoding scheme that achieves capacity asymptotically is presented. [ABSTRACT FROM AUTHOR]

Recent decades have witnessed steady improvements in our ability to harness the information-carrying capability of optical fibres. Will this process continue, or will progress eventually stall? Information theory predicts that all channels have a limited capacity depending on the available transmission resources, and thus it is inevitable that the pace of improvements will slow. However, information theory also provides insights into how transmission resources should, in principle, best be exploited, and thus may serve as a guide for where to look for better ways to squeeze more out of a precious resource. This tutorial paper reviews the basic concepts of information theory and their application in fibre-optic communications. [ABSTRACT FROM AUTHOR]

Over a finite field F q m , the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew polynomials to the familiar concept of linear independence for vector spaces. This relation allows for the definition of a representable matroid called the F q m [ x ; σ ] -matroid, with rank function that makes it a metric space. Specific submatroids of this matroid are individually bijectively isometric to the projective geometry of F q m equipped with the subspace metric. This isometry allows one to use the F q m [ x ; σ ] -matroid in a matroidal network coding application. [ABSTRACT FROM AUTHOR]