Sum Product Algorithm

This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.

The performance of several existing and partly new algorithms for positioning of sensor node based on distance estimate is compared when the distance estimates are obtained from a measurement campaign. The distance estimates are based on time-of-arrival measurements done by ultrawideband devices in an indoor office environment. Two different positioning techniques are compared: statistical and geometrical. In statistical category, distributed weighted-multidimensional scaling (dwMDS), least squares, and sum ...

Approximate inference in large and densely connected graphical models is a challenging but highly relevant problem. Belief propagation, as a method for performing approximate inference in loopy graphs, has shown empirical success in many applications. However, convergence of belief propagation can only be guaranteed for simple graphs. Whether belief propagation converges depends strongly on the applied message update scheme, and specialized schemes can be highly beneficial. Yet, residual belief propagation is the only established method utilizing this fact to improve convergence properties. In experiments, we observe that residual belief propagation fails to converge if local oscillations occur and the same sequence of messages is repeatedly updated. To overcome this issue, we propose two novel message update schemes. In the first scheme we add noise to oscillating messages. In the second scheme we apply weight decay to gradually reduce the influence of these messages and consequently enforce convergence. Furthermore, in contrast to previous work, we consider the correctness of the obtained marginals and observe significant performance improvements when applying the proposed message update schemes to various Ising models with binary random variables.

We study iterative receivers for joint decoding and channel-state estimation for transmission on block-fading channels of root-LDPC-coded signals. Root-LDPC codes are known to be most performant codes for block-fading channels, as their spacial "root" structure allows to get the full-diversity property. This property ensures a good error decoding performance of root LDPC codes, especially in contrast with the performance standard LDPC codes (having the maximum diversity equal to 1). However, as any channel code, root-LDPC codes also suffer from the diversity loss when the channel state information is not known at the receiver. In this work we propose a joint channel estimation-decoding scheme for root-LDPC codes that helps to overcome this problem and still to have the full-diversity. V 0 . 1 1 Joint decoding and channel state estimation block-fading channels EB

This paper investigates hardware cyber-security risks associated with channel decoders, which are commonly acquired as a black box in semiconductor industry. It is shown that channel decoders are potentially attractive targets for hardware cyber-security attacks and can be easily embedded with malicious blocks. Several attack scenarios are considered in this work and suitable methods for mitigating the risks are proposed. These methods are based on randomizing the inputs of the channel decoder to obstruct the communications between attackers and the malicious blocks, ideally without changing the decoding performance.

In satellites, nonlinear amplifiers used near saturation severely distort the transmitted signal and cause difficulties in its reception. Nevertheless, the nonlinearities introduced by memoryless bandpass amplifiers preserve the symmetries of the M-ary quadrature amplitude modulation (M-QAM) constellation. In this paper, a cluster-based sequence equalizer (CBSE) that takes advantage of these symmetries is presented. The proposed equalizer exhibits enhanced performance compared to other techniques, including the conventional linear transversal equalizer, Volterra equalizers, and RBF network equalizers. Moreover, this gain in performance is obtained at a substantially lower computational cost.

This paper introduces a class of structured lowdensity parity-check (LDPC) codes whose parity check matrices are arrays of permutation matrices. The permutation matrices are obtained from Latin squares and form a finite field under some matrix operations. They are chosen so that the Tanner graphs do not contain subgraphs harmful to iterative decoding algorithms. The construction of column-weight-three codes is presented. Although the codes are optimized for the Gallager A/B algorithm over the binary symmetric channel (BSC), their error performance is very good on the additive white Gaussian noise channel (AWGNC) as well.

In this paper, we investigate the sequence estimation problem of faster-than-Nyquist (FTN) signaling as a promising approach for increasing spectral efficiency (SE) in future communication systems. In doing so, we exploit the concept of Gaussian separability and propose two probabilistic data association (PDA) algorithms with polynomial time complexity to detect binary phase-shift keying (BPSK) FTN signaling. Simulation results show that the proposed PDA algorithm outperforms the recently proposed SSSSE and SSSgbKSE algorithms for all SE values with a modest increase in complexity. The PDA algorithm approaches the performance of the semidefinite relaxation (SDRSE) algorithm for SE values of 0.96 bits/sec/Hz, and it is within the 0.5 dB signal-to-noise ratio (SNR) penalty at SE values of 1.10 bits/sec/Hz for the fixed values of β = 0.3.

We investigate faster-than-Nyquist modulation based on short finite pulses over the AWGN channel. We consider several pulse shapes and compare their in- formation rates for several system setups. We com- pare the effect of increasing the alphabet size ver- sus of increasing the signaling rate. The outcome is that for these pulses the FTN symbol rate is of greater importance than the alphabet size. Finally we test some concatenated coding schemes where faster- than-Nyquist modulation constitutes the innermost encoder; the outcome is very good.

All analog circuits are affected by device mismatch, slight errors in the physical characteristics of analog devices. In many applications, mismatch reduces precision, making power and complexity advantages irrelevant. In this paper, we examine the effects of mismatch for decoders based on Gilbert multipliers and the sum-product algorithm.

Iterative decoders, including Turbo decoders, provide near-optimal error protection for various communication channels and storage media. CMOS analog implementations of these decoders offer dramatic savings in complexity and power consumption, compared to digital architectures. Conventional CMOS analog decoders must have supply voltage greater than 1 V. A new low-voltage architecture is proposed which reduces the required supply voltage by at least 0.4 V. It is shown that the low-voltage architecture can be used to implement the general sum-product algorithm. The low-voltage analog architecture is then useful for implementing Turbo and low-density parity check decoders. The low-voltage architecture introduces new requirements for signal normalization, which are discussed. Measured results for two fabricated low-voltage analog decoders are also presented.

Efficient implementations of the sum-product algorithm (SPA) for decoding low-density parity-check (LDPC) codes using loglikelihood ratios (LLR) as messages between symbol and parity-check nodes are presented. Various reduced-complexity derivatives of the LLR-SPA are proposed. Both serial and parallel implementations are investigated, leading to trellis and tree topologies, respectively. Furthermore, by exploiting the inherent robustness of LLRs, it is shown, via simulations, that coarse quantization tables are sufficient to implement complex core operations with negligible or no loss in performance. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate design point in high-speed applications from a performance, latency, and computational complexity perspective.

Rapid variation of channel coefficients is one of the most challenging problems in wireless communication. To provide and keep communication in desired quality, channel coefficients should be estimated continuously. This can be made by using pilot symbols between data blocks which are known by the transmitter and receiver. The channel coefficients between pilot symbols can be estimated by interpolation but this method has a disadvantage in fast fading channels since the channel coefficient estimates have better quality around the pilot blocks than away from them. To solve this problem, we propose to extend the pilot block by making use of the soft information produced by channel decoder. We track the channel estimates in time by the LMS algorithm bidirectionally so that we can estimate coefficients more accurately by interpolation . We also introduce a new interleaver which divides the bits into subregions based on their proximity to pilot blocks and permutes them within their own r...

In this paper, by applying the concept of linear prediction, which is widely used for fading channels, to phaseuncertain communications, we generalize existing linear predictive detection algorithms for transmission over channels with phase noise and frequency offset. This approach leads to the derivation of detection algorithms, which are referred to as phasor linear predictive (pLP), for trellis-based maximum a posteriori (MAP) sequence detection (based on the Viterbi algorithm) and MAP symbol detection: trellis-based (using the forwardbackward algorithm) and graph-based (using the sum-product algorithm). The effectiveness of the proposed pLP detection algorithms is evaluated for several communication schemes. The derived algorithms outperform previously appeared finitememory detection solutions in terms of robustness against fast channel dynamics. Moreover, the proposed detection strategy lends itself to attractive extensions to adaptive schemes.

Global Navigation Satellite Systems (GNSS) are a widely used technology for positioning and navigation. GNSS positioning relies on pseudorange measurements from satellites to receivers. A pseudorange is the apparent distance between two agents deduced from the time-of-flight of a signal sent from one agent to the other. Because of the lack of synchronization between the agents' clocks, it is a biased version of their distance. This paper introduces a new rigidity theory adapted to pseudorange measurements. The peculiarity of pseudoranges is that they are asymmetrical measurements. Therefore, unlike other usual rigidities, the graphs of pseudorange frameworks are directed. In this paper, pseudorange rigidity is proved to be a generic property of the underlying undirected graph of constraints. The main result is a characterization of rigid pseudorange graphs as combinations of rigid distance graphs and connected graphs. This new theory is adapted for GNSS. It provides new insights into the minimum number of satellites needed to locate a receiver, and is applied to the localization of GNSS cooperative networks of receivers. The interests of asymmetrical constraints in the context of formation control are also discussed.

In this paper, we examine code-aided synchronization in the presence of carrier frequency uncertainties and phase noise. As code-aided synchronization can only achieve high estimation accuracy if the initial parameter offset is sufficiently small, we employ a data-aided coarse synchronization unit comprising a feed-forward frequency estimator and a LMMSE estimator for pre-compensating frequency offsets and phase noise. Furthermore, we make use of known iterative synchronization techniques for frequency and phase offsets. We here extend the concept of iterative synchronization to iterative phase noise compensation and we show analytically under which circumstances our approach can achieve a worthwhile improvement in terms of estimation accuracy.

In this paper, we examine code-aided synchronization in the presence of carrier frequency uncertainties and phase noise. As code-aided synchronization can only achieve high estimation accuracy if the initial parameter offset is sufficiently small, we employ a data-aided coarse synchronization unit comprising a feed-forward frequency estimator and a LMMSE estimator for pre-compensating frequency offsets and phase noise. Furthermore, we make use of known iterative synchronization techniques for frequency and phase offsets. We here extend the concept of iterative synchronization to iterative phase noise compensation and we show analytically under which circumstances our approach can achieve a worthwhile improvement in terms of estimation accuracy.

This work investigates reliable wireless broadcast with asynchronous data access. A wireless broadcast system based on LT-codes, a realization of the recently introduced Fountain codes, is introduced. We review the traditional problem formalization for Fountain codes operating on erasure channels and generalize the framework to arbitrary types of channels coining the terminology and the concept of information collecting. The sum-product algorithm in order to exploit soft-information available at the wireless receiver for decoding of LT codes is applied. Simulation results on the AWGN, as well as the fading channel, show remarkable performance gains compared to traditional erasure decoding.

Resumo-Um esquema de codificação de faixa eficiente para receptores iterativos foi investigado. Canais não-coerentes M-APSK (do inglês, M-ary Amplitude Phase Shift Keying) foram considerados. O esquema propõe uma nova abordagem para um receptor iterativo baseado em grafos-fatores. A abordagem considera o uso de codificação diferencial não binária. Uma comparação de desempenho do esquema para códigos LDPC (do inglês, Low-Density Parity-Check) com comprimentos curtos será descrita. Palavras-Chave-Canais não coerentes, constelações M-APSK, decodificação iterativa, grafos-fatores.

The error floor phenomenon, associated with iterative decoders, is one of the most significant limitations to the applications of low-density parity-check (LDPC) codes. A variety of techniques from code design to decoder implementation have been proposed to address the error floor problem, among which post-processors have shown to be both effective and implementation-friendly. In this work, we take the inspiration from simulated annealing to generalize the post-processor design using three methods: quenching, extended heating, and focused heating, each of which targets a different error structure. The resulting post-processor is demonstrated to lower the error floors by two orders of magnitude for two structured code examples, a (2209, 1978) array LDPC code, and a (1944, 1620) LDPC code used by the IEEE 802.11n standard. The post-processor can be integrated to a belief-propagation decoder with minimal overhead. The post-processor design is equally applicable to other structured LDPC codes.

Recently, a novel method for developing filtering algorithms, based on the interconnection of two Bayesian filters and called double Bayesian filtering, has been proposed. In this manuscript we show that the same conceptual approach can be exploited to devise a new smoothing method, called double Bayesian smoothing. A double Bayesian smoother combines a double Bayesian filter, employed in its forward pass, with the interconnection of two backward information filters used in its backward pass. As a specific application of our general method, a detailed derivation of double Bayesian smoothing algorithms for conditionally linear Gaussian systems is illustrated. Numerical results for two specific dynamic systems evidence that these algorithms can achieve a better complexity-accuracy tradeoff and tracking capability than other smoothing techniques recently appeared in the literature. Pasquale Di Viesti† Giorgio M. Vitetta† Emilio Sirignano† [email protected] giorgio.vitetta@un...

In this manuscript, a general method for deriving filtering algorithms that involve a network of interconnected Bayesian filters is proposed. This method is based on the idea that the processing accomplished inside each of the Bayesian filters and the interactions between them can be represented as message passing algorithms over a proper graphical model. The usefulness of our method is exemplified by developing new filtering techniques, based on the interconnection of a particle filter and an extended Kalman filter, for conditionally linear Gaussian systems. Numerical results for two specific dynamic systems evidence that the devised algorithms can achieve a better complexity-accuracy tradeoff than marginalized particle filtering and multiple particle filtering. Giorgio M. Vitetta† Pasquale Di Viesti† [email protected] [email protected] Emilio Sirignano† Francesco Montorsi [email protected] [email protected] †Dept. of Engineering ”Enzo Ferra...

In this manuscript, a general method for deriving filtering algorithms that involve a network of interconnected Bayesian filters is proposed. This method is based on the idea that the processing accomplished inside each of the Bayesian filters and the interactions between them can be represented as message passing algorithms over a proper graphical model. The usefulness of our method is exemplified by developing new filtering techniques, based on the interconnection of a particle filter and an extended Kalman filter, for conditionally linear Gaussian systems. Numerical results for two specific dynamic systems evidence that the devised algorithms can achieve a better complexity-accuracy tradeoff than marginalized particle filtering and multiple particle filtering.

In this manuscript the fixed-lag smoothing problem for conditionally linear Gaussian state-space models is investigated from a factor graph perspective. More specifically, after formulating Bayesian smoothing for an arbitrary state-space model as forward-backward message passing over a factor graph, we focus on the above mentioned class of models and derive two novel particle smoothers for it. Both the proposed techniques are based on the well known two-filter smoothing approach and employ marginalized particle filtering in their forward pass. However, on the one hand, the first smoothing technique can only be employed to improve the accuracy of state estimates with respect to that achieved by forward filtering. On the other hand, the second method, that belongs to the class of Rao-Blackwellized particle smoothers, provides also a point mass approximation of the so called joint smoothing distribution. Finally, our smoothing algorithms are compared, in terms of estimation accuracy and computational requirements, with a Rao-Blackwellized particle smoother recently proposed by Lindsten et al. in [20].

Using a finite field approach, a novel algebraic construction of low-density parity-check (LDPC) convolutional codes with fast encoding property is proposed. According to the matrices of quasi-cyclic (QC) codes constructed based on the multiplicative groups of finite fields and the algebraic property that a binary circulant matrix is isomorphic to a finite ring, we first generate a polynomial-form parity-check matrix of an LDPC convolutional code under a given rate over a given finite field. Then some related modifications are made upon the original polynomial-form matrix to obtain the new one with fast encoding property. Simulation results show that the proposed LDPC convolutional codes have good performance with the iterative belief propagation decoding algorithm.

Stopping criteria for the iterative decoding of lowdensity parity-check codes are considered. For a successful decoding task an inherent stopping criterion is used: the fulfillment of all parity-check constraints. For an unsuccessful task the decoder usually completes a preset maximum number of iterations. Proper iteration control is required to save energy and time on unnecessary decoder operation when processing undecodable blocks. In this paper we propose an iteration control policy that is driven by the combination of two decision metrics. One is the number of satisfied parity-check constraints and the second one is provided by a specific message computation kernel: the Self-Corrected Min-Sum decoding algorithm. Our results show that this hybrid control policy offers superior performance in terms of energy efficiency compared to previously proposed techniques. In addition we show empirically how stopping criteria should be tuned as a function of false alarm and missed detection rates.

In modern communication systems, high-order modulation schemes have been widely employed to improve the spectral efficiency. In this context, soft decision methods are usually preferred, due to their superior performance over hard decisions. Soft decoders use metrics such as the log-likelihood ratios (LLRs) to measure the reliability of their decisions. Typically, the complexity of the exact channel LLR calculation increases exponentially with the modulation order of the system. The most well-known LLR approximation, the Max-Log-MAP symbol-to-bit demapper, still demands a large number of operations. In this paper, we present a novel simplified scheme that greatly reduces the complexity of the LLR demapper at high-order PSK and APSK modulations compared to the Max-Log-MAP soft-demapper. Moreover, it is shown that the proposed method has a negligible overall system performance loss compared to the same method.

We compare the performance of short-length linear binary codes on the binary erasure channel and the binaryinput Gaussian channel. We use a universal decoder that can decode any linear binary block code: Gaussian-elimination based Maximum-Likelihood decoder on the erasure channel and probabilistic Ordered Statistics Decoder on the Gaussian channel. As such we compare codes and not decoders. The word error rate versus the channel parameter is found for LDPC, Reed-Muller, Polar, and BCH codes at length 256 bits. BCH codes outperform other codes in absence of cyclic redundancy check. Under joint decoding, the concatenation of a cyclic redundancy check makes all codes perform very close to optimal lower bounds.

This paper proposes a decoding algorithm for nonbinary low-density parity-check (NB-LDPC) codes, aiming to improve the error rate performance for NAND flash memory. Several NB-LDPC decoding methods for NAND flash memory have been studied. Some approaches rely on hard decisions, and these are relatively simple but do not have a good error rate performance. Others are based on soft decisions that require multiple reads for each flash memory cell, leading to significant memory throughput degradation. To improve the error rate performance without suffering performance degradation owing to multiple reads, an iterative pseudo-soft-reliability-based decoding algorithm is proposed. Using Galois field addition to calculate the Hamming distance at the initialization, the proposed algorithm not only improves the error rate performance but also reduces the average number of iterations compared with those of conventional hard-decision-based decoding algorithms. INDEX TERMS Error correction codes, Hamming distance, hard decision, iterative hard-reliability-based majority-logic decoding algorithm, NAND flash memory, nonbinary low-density parity-check codes.

We develop a robust multiuser detector for a Frequency Division Multiplexing (FDM) system where each user employs a binary continuous phase modulation (CPM) generated through a low-cost transmitter, thus characterized by a significant modulation index uncertainty, and sent over a channel affected by phase noise. In this FDM system the spectral efficiency can be increased by reducing the spacing between two adjacent channels. The proposed receiver is designed by adopting a simplified representation of a binary CPM signal with the principal component of its Laurent decomposition and is obtained by using the framework based on factor graphs and the sum-product algorithm. This detector can be used for iterative detection/decoding of a coded scheme in which each user employs a binary CPM modulator serially concatenated with an outer encoder through a pseudo-random interleaver. It does not require an explicit estimation of the modulation index nor of the channel phase and is very robust to large uncertainties of the nominal value of the modulation index.

The paper proposes a numerically stable recursive algorithm for the exact computation of the linear-chain conditional random field gradient. It operates as a forward algorithm over the log-domain expectation semiring and has the purpose of enhancing memory efficiency when applied to long observation sequences. Unlike the traditional algorithm based on the forward-backward recursions, the memory complexity of our algorithm does not depend on the sequence length. The experiments on real data show that it can be useful for the problems which deal with long sequences.

We consider the decoding of LDPC codes in presence of non-Gaussian noise, especially a set of ǫ-mixture models. For each of these models, the optimal LLRs are presented. We study the performance degradation due to the use of incorrect LLR in presence of a given noise model. Without modifying the existing LDPC decoder, we propose robust initial LLR which require minimum knowledge about the underlying noise model and are computationally less complex. Since BER simulations are computationally heavy, we use density evolution to compare the thresholds of different LLRs.

This paper considers two recently-proposed receivers, Tikh and DCT. Both receivers are computationallyefficient, iterative and designed to be robust against phase noise on the local oscillators of digital bandpass communication systems. The presented results build on our prior research. We discuss the initialization of the DCT receiver, explore reducing the computational complexity by simplifying the receiver scheduling and study the effect of a small frequency offset. Coded PSK signaling and additive white Gaussian noise are assumed.

The sum-product algorithm (belief/probability propagation) can be naturally mapped into analog transistor circuits. These circuits enable the construction of analog-VLSI decoders for turbo codes, low-density parity-check codes, and similar codes.

In this manuscript the fixed-lag smoothing problem for conditionally linear Gaussian state-space models is investigated from a factor graph perspective. More specifically, after formulating Bayesian smoothing for an arbitrary state-space model as forward-backward message passing over a factor graph, we focus on the above mentioned class of models and derive two novel particle smoothers for it. Both the proposed techniques are based on the well known two-filter smoothing approach and employ marginalized particle filtering in their forward pass. However, on the one hand, the first smoothing technique can only be employed to improve the accuracy of state estimates with respect to that achieved by forward filtering. On the other hand, the second method, that belongs to the class of Rao-Blackwellized particle smoothers, provides also a point mass approximation of the so called joint smoothing distribution. Finally, our smoothing algorithms are compared, in terms of estimation accuracy and computational requirements, with a Rao-Blackwellized particle smoother recently proposed by Lindsten et al. in [20].

This paper presents a Relaxed Half-Stochastic (RHS) low-density parity-check (LDPC) decoding algorithm that uses some elements of the sum-product algorithm (SPA) in its variable nodes, but maintains the low-complexity interleaver and check node structures characteristic of stochastic decoders. The algorithm relies on the principle of successive relaxation to convert binary stochastic streams to a log-likelihood ratio (LLR) representation. Simulations of a (2048, 1723) RS-LDPC code show that the RHS algorithm can outperform 100-iterations floating-point SPA decoding. We describe approaches for lowcomplexity implementation of the RHS algorithm. Furthermore, we show how the stochastic nature of the belief representation can be exploited to lower the error floor.

This article surveys recent development on Euclidean interpoint distances (IPDs). IPDs find applications in many scientific fields and are the building blocks of several multivariate techniques such as comparison of distributions, clustering, classification, and multidimensional scaling. In this article, we explore IPDs, discuss their properties and applications, and present their distributions for several families, including the multivariate normal, multivariate Bernoulli, multivariate power series, and the unified hypergeometric distributions. We consider two groups of observations in R d and present a simultaneous plot of the empirical cumulative distribution functions of the within and between IPDs to visualize and examine the equality of the underlying distribution functions of the observations.

Message-passing algorithms have emerged as powerful techniques for approximate inference in graphical models. When these algorithms converge, they can be shown to find local (or sometimes even global) optima of variational formulations to the inference problem. But many of the most popular algorithms are not guaranteed to converge. This has lead to recent interest in convergent message-passing algorithms. In this paper, we present a unified view of convergent message-passing algorithms. We present a simple derivation of an abstract algorithm, tree-consistency bound optimization (TCBO) that is provably convergent in both its sum and max product forms. We then show that many of the existing convergent algorithms are instances of our TCBO algorithm, and obtain novel convergent algorithms "for free" by exchanging maximizations and summations in existing algorithms. In particular, we show that Wainwright's non-convergent sum-product algorithm for tree based variational bounds, is actually convergent with the right update order for the case where trees are monotonic chains.

This paper presents an efficient algorithm for finding the dominant trapping sets of a low-density parity-check (LDPC) code. The algorithm can be used to estimate the error floor of LDPC codes or to be used as a tool to design LDPC codes with low error floors. For regular codes, the algorithm is initiated with a set of short cycles as the input. For irregular codes, in addition to short cycles, variable nodes with low degree and cycles with low approximate cycle extrinsic message degree (ACE) are also used as the initial inputs. The initial inputs are then expanded recursively to dominant trapping sets of increasing size. At the core of the algorithm lies the analysis of the graphical structure of dominant trapping sets and the relationship of such structures to short cycles, low-degree variable nodes and cycles with low ACE. The algorithm is universal in the sense that it can be used for an arbitrary graph and that it can be tailored to find a variety of graphical objects, such as absorbing sets and Zyablov-Pinsker (ZP) trapping sets, known to dominate the performance of LDPC codes in the error floor region over different channels and for different iterative decoding algorithms. Simulation results on several LDPC codes demonstrate the accuracy and efficiency of the proposed algorithm. In particular, the algorithm is significantly faster than the existing search algorithms for dominant trapping sets.

This paper presents five methods for constructing nonbinary LDPC codes based on finite geometries. These methods result in five classes of nonbinary LDPC codes, one class of cyclic LDPC codes, three classes of quasi-cyclic LDPC codes and one class of structured regular LDPC codes. Experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the AWGN channel and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates with either algebraic hard-decision decoding or Kötter-Vardy algebraic soft-decision decoding at the expense of a larger decoding computational complexity.

The increasing number of internet users in recent years has put a significant strain on communication systems. In response, new mobile network generations are released every decade. Currently, research efforts are focused on developing the 6th generation and beyond. This thesis explores strategies to enhance system reliability, improve energy efficiency, reduce latency, and create overall better communication systems. To achieve these goals, in this thesis, we incorporate the simultaneous wireless information and power transfer (SWIPT) / reconfigurable intelligent surface (RIS) into the low-density parity check (LDPC)-coded cooperative communications. The main contributions of this thesis are summarized as follows:
(1) We investigate the LDPC-coded cooperation with the SWIPT technology for energy harvesting on the relays. There are multiple sources, multiple relays, and one destination in the considered systems. Sources encode the messages using the LDPC codes and transmit them in a broadcast manner to the destination and relay. My proposed work uses orthogonal frequency division multiplexing (OFDM) to transmit the codewords. The relays decode the message, and if it is correct they re-encode and transmit it to the destination. The SWIPT technology is also applied on the relay to harvest the energy from the radio frequency (RF) signals, this harvested energy is used for transmitting the signals and also for multiplexing the signals. At the destination, the min-sum decoding algorithm is used to decode the codewords and recover the original information bits. Theoretical analysis and simulation results demonstrate the superiorities of the proposed LDPC-coded cooperation with the SWIPT technology. Also, we compare the performances of the proposed system under different conditions like various decoding iterations, different channels, and various energy efficiencies.
(2) We investigate the RIS-aided LDPC-coded cooperative communications. We make use of the RIS architecture to replace the traditional relays. RIS, also known as intelligent reflection surface (IRS), is an artificial structure consisting of passive radio elements, each of which could adjust the reflection of the incident electromagnetic waves with unnatural properties. LDPC codes are used on the source to encode the messages, and the codewords are then transmitted to the destination and RIS. The RIS only reflects the message to the destination. At the destination, joint iterative decoding algorithms are performed to recover the original information bits. Theoretical analysis and simulation results also demonstrate the superiorities of the proposed RIS-aided LDPC-coded cooperative communications. Also, we compare the performances of the proposed system under different conditions like various decoding iterations, and various numbers of RIS elements. Furthermore, we compare the performance of the two proposed systems. It demonstrates that the RIS-aided cooperative communication system outperforms the relay-aided system.

KEYWORDS: LDPC, Encoding, Decoding, SWIPT, RIS