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The growing number of Internet of Thing (IoT) and Ultra-Reliable Low Latency Communications (URLCC) use cases in next generation communication networks calls for the development of efficient Forward Error Correction (FEC) mechanisms. These use cases usually imply using short to mid-sized information blocks and requires low-complexity and/or fast decoding procedures. This paper investigates the joint learning of short to mid block-length coding schemes and associated Belief-Propagation (BP) like decoders using Machine Learning (ML) techniques. An interpretable auto-encoder (AE) architecture is proposed, ensuring scalability to block sizes currently challenging for ML-based linear block code design approaches. By optimizing a coding scheme w.r.t. the targeted decoder, the proposed system offers a good complexity/performance trade-off compared to various codes from literature with length up to 128 bits. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of hybrid coupled serially concatenated codes (HC-SCC), in which coupling is achieved by re-encoding partial outputs from both the outer and the inner encoders. Firstly, we derive the exact density evolution (DE) equations over the binary erasure channels (BECs) for the HC-SCCs, which can be used to analyze the impact of the coupling ratios and the coupling memories on the performances of the HC-SCCs. Secondly, we present a genie-aided lower bound for the proposed HC-SCCs to estimate their error-floors. Thirdly, we use the DE equations to design randomly punctured HC-SCCs, including rate-compatible HC-SCCs. The analytical and the numerical results show that: 1) for comparable coupling memories, the iterative decoding thresholds of the proposed HC-SCC ensembles are better than those of the recently introduced spatially coupled serially concatenated code (SC-SCC) ensembles and the partially parity-coupled turbo code (PPC-TC) ensembles; 2) with outer coupling memory five and inner coupling memory five, the iterative decoding thresholds of the rate-compatible HC-SCCs over the BECs are 0.0003 away from the corresponding channel capacities for a wide range of coding rates; 3) simulation results over the additive white Gaussian noise (AWGN) channels and the BECs are provided to confirm the performance advantages of the HC-SCCs. Particularly, the simulation results show that the HC-SCCs admit lower error-floor than the SC-SCCs and the rate-compatible HC-SCCs perform better than the PPC-TCs, which are consistent with the results of DE analysis. [ABSTRACT FROM AUTHOR]

Matrix multiplication is a fundamental building block in various distributed computing algorithms. In order to multiply large matrices, it is common practice to distribute the computation into multiple tasks running on different nodes. In order to tolerate stragglers among such nodes, various coding schemes have been proposed by adding additional coded tasks. However, most existing coding schemes for matrix multiplication are constructed for only one matrix multiplication, while batch matrix multiplication is common in large-scale distributed computing workloads. In this paper, we propose Rook Coding (RC), a novel polynomial-based coding framework for computing the multiplication of $n$ pairs of matrices in batch. Designed to achieve lower encoding time in practice, we construct RC as polynomials of much simpler forms than existing coding schemes for batch matrix multiplication, achieving a recovery threshold of $O(n^{\log _{2} ~3})$. Compared to existing coding schemes, RC achieves a lower encoding complexity in practice, because of its simpler forms in the encoding polynomials. Through extensive experiments, we show that RC can save the time of the whole job thanks to its low overhead of encoding. [ABSTRACT FROM AUTHOR]

This paper proposes a new probabilistic amplitude shaping (PAS) approach for concatenated two-level multi-level coding (MLC). The proposed system is based on a concatenated forward error correction (FEC) scheme where outer codes are serially concatenated with inner two-level MLC. This concatenated two-level MLC scheme has recently been shown to have a potential for achieving better performance-complexity trade-offs than the conventional bit-interleaved coded modulation (BICM). Meanwhile, PAS has recently been demonstrated to offer remarkable performance gains as well as rate adaptivity. However, the majority of existing works on PAS assume the use of the binary reflected Gray code as a bit-labeling, and its application to coded modulation schemes with other bit-labelings, such as two-level MLC, may not be straightforward. In this paper, we devise a bit-labeling scheme and propose a new PAS structure for an efficient integration of PAS with two-level MLC systems. More specifically, we propose to generate signed amplitude symbols with the distribution matcher (DM) for maximizing both coding and shaping gains achieved by two-level MLC and PAS, respectively, while the conventional PAS generates unsigned amplitude symbols. It is demonstrated by simulation results that, with 256QAM and inner polar codes, the proposed two-level MLC with PAS simultaneously offers 75% reduction in the number of required inner encoding and soft-decision (SD) decoding operations for given outer and inner FEC code lengths, and up to 0.3 dB performance gain over the conventional PAS scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose an optimization method for protograph-based spatially coupled low-density parity-check (SC-LDPC) codes under window decoding (WD). Previous works on constructing SC-LDPC codes for WD typically focused on optimizing asymptotic performance metrics such as the WD threshold. However, in this paper, it is observed that the WD threshold is not an appropriate metric to sufficiently explain the finite-length behavior of SC-LDPC codes under WD. Thus, we propose a new performance metric, called the window mean parameter, based on a scaling analysis to capture the WD performance more accurately and formulate a code optimization algorithm that optimizes the proposed performance metric. Since the proposed metric depends on the window size, the optimization algorithm can provide a code family of SC-LDPC codes optimized for various target window sizes. Simulation results confirm that the improvement in the proposed metric leads to a finite-length performance improvement, resulting in one to two orders of the frame error rate gain over the conventional SC-LDPC codes for a wide range of window sizes. Furthermore, we investigate structural characteristics of the proposed codes to provide a supplementary explanation for the performance improvement, which also promotes a better understanding of SC-LDPC codes for WD. [ABSTRACT FROM AUTHOR]

Sparse code multiple access (SCMA) is a promising non-orthogonal multiple access technique to support massive connectivity for future wireless Internet of Things (IoT) networks. As the main feature of SCMA, modulation and spread spectrum is embedded into codebook mapping, offering significant codebook shaping gains to mitigate inter-cell interference. Maximizing the constellation-constrained average mutual information (AMI) is an effective way for SCMA codebook optimization. However, deriving a closed-form expression of the AMI is analytically intractable, while it is computationally costly to estimate the AMI by numerical methods. To address this challenge, this paper first derives a lower bound of the AMI with a closed-form expression. On this basis, we propose a novel codebook optimization method referred to as joint bare bones particle swarm optimization (JBBPSO) through maximizing the AMI lower bound. The proposed low-complexity method jointly optimizes the mother codebook including basic constellation and other non-zero-dimensional constellations, and the rotation angles of multiple users. Numerical results show that our proposed optimized codebooks outperform the state-of-the-art SCMA codebooks in terms of both the lower bound and the error performance. [ABSTRACT FROM AUTHOR]

The prefixed Varshamov-Tenengolts (VT) codes, which are subsets of VT codes with predetermined prefixes, can be used for error correction over segmented edit channels. In this paper, we investigate the construction and analysis of this class of codes. First, we derive upper bounds on the size of zero-error codes for segmented edit channels with segment-by-segment decoding. Second, we establish a one-to-one correspondence between prefixed VT codes and Levenshtein codes. Based on this relation, we can obtain explicit formulas on the size of prefixed VT codes via the existing results on the size of Levenshtein codes. Third, we construct a new zero-error prefixed VT code and show that the size of the constructed code is strictly larger than that of the existing prefixed VT code for the segmented deletion channel. Finally, an efficient systematic encoding method of prefixed VT codes is proposed for the segmented edit channels. [ABSTRACT FROM AUTHOR]

Conventional design methods of polar-coded modulation schemes aim to minimize the block error rate (BLER) under successive cancellation (SC) decoding. However, codes designed by conventional methods are not competitive under successive cancellation list (SCL) decoding. This paper presents a new design method based on the BLER upper bound under maximum-likelihood (ML) decoding (ML-BLER upper bound). The ML-BLER upper bound depends on the weight enumerating function (WEF) of the polar-coded modulation scheme over the squared Euclidean distance. In this paper, the polar-coded modulation is randomized by the concept of interleaved polar (i-polar) codes, and the WEF averaged over the ensemble of the polar-coded modulation schemes can be derived. Three polar-coded modulation schemes are considered, i.e., the bit-interleaved polar-coded modulation with a single interleaver (BIPCM-SI), the bit-interleaved polar-coded modulation with multiple interleavers (BIPCM-MI), and the multi-level polar-coded modulation (MLPCM). A new bit channel selection algorithm for polar-coded modulation schemes is proposed, which takes the polarization effect and the ML-BLER upper bound as design criteria. Design examples show that, under SCL decoding, the polar-coded modulation schemes (without CRC) with the proposed channel selection algorithm outperform those with conventional algorithms and are competitive as compared to the state-of-the-art 5G LDPC codes. [ABSTRACT FROM AUTHOR]

Many systems possess the flexibility to serve requests in more than one way, such as distributed storage systems that store multiple copies of the data. In such systems, the latency of serving the requests may potentially be reduced by sending redundant requests: a request may be sent to more servers than needed and deemed served when the requisite number of servers complete service. Such a mechanism trades off the possibility of faster execution of the request with the increase in the load on the system. Several recent works empirically evaluate the latency performance of redundant requests in diverse settings. In this paper, we perform an analytical study of the latency performance of redundant requests, with the primary goals of characterizing under what scenarios sending redundant requests will help (and under what scenarios it will not), and of designing optimal redundant-requesting policies. We show that when service times are i.i.d. memoryless or “heavier,” and when the additional copies of already-completed jobs can be removed instantly, maximally scheduling redundant requests achieves the optimal average latency. On the other hand, when service times are i.i.d. “lighter” or when service times are memoryless and removal of jobs is not instantaneous, then not having any redundancy in the requests is optimal under high loads. Our results are applicable to arbitrary arrival processes. [ABSTRACT FROM AUTHOR]

An important ingredient of the future 5G systems will be ultra-reliable low-latency communication (URLLC). A way to offer URLLC without intervention in the baseband/PHY layer design is to use interface diversity and integrate multiple communication interfaces, each interface based on a different technology. In this paper, we propose to use coding to seamlessly distribute coded payload and redundancy data across multiple available communication interfaces. We formulate an optimization problem to find the payload allocation weights that maximize the reliability at specific target latency values. In order to estimate the performance in terms of latency and reliability of such an integrated communication system, we propose an analysis framework that combines traditional reliability models with technology-specific latency probability distributions. Our model is capable to account for failure correlation among interfaces/technologies. By considering different scenarios, we find that the optimized strategies can in some cases significantly outperform strategies based on $k$ -out-of- $n$ erasure codes, where the latter do not account for the characteristics of the different interfaces. The model has been validated through simulation and is supported by experimental results. [ABSTRACT FROM PUBLISHER]

This paper formulates the polar-code construction problem for the successive-cancellation list (SCL) decoder as a maze-traversing game, which can be solved by reinforcement-learning techniques. The proposed method provides a novel technique for polar-code construction that no longer depends on sorting and selecting bit-channels by reliability, as in most current algorithms. Instead, this technique decides whether the input bits should be frozen in a purely sequential manner. The equivalence of optimizing the polar-code construction for the SCL decoder under this technique and maximizing the expected reward of traversing a maze is drawn. Simulation results show that the standard polar-code constructions that are designed for the successive-cancellation decoder are no longer optimal for the SCL decoder with respect to the frame error rate (FER). In contrast, the proposed game-based construction method finds code constructions that have similar or lower FER for various code lengths and various list sizes of the SCL decoder, compared to the state-of-the-art construction methods. The advantage of the game-based constructions over the standard constructions increases with the channel signal-to-noise ratio and the list size of SCL decoding. Moreover, the learning is highly efficient in terms of the number of required training samples and computational operations. [ABSTRACT FROM AUTHOR]

In this paper, we present a sequential decoding metric function, which leads to significantly improved computational complexity while maintaining the superiority of polarization-adjusted convolutional (PAC) codes’ error-correction performance. With the proposed metric function, the PAC codes’ decoding computational complexity is comparable to the computational complexity of sequential decoding of conventional convolutional codes (CCs). Moreover, simulation results show an improvement in the error-correction performance of low rate PAC codes when using the proposed metric function. Simulation results also show that using the proposed metric, the upper bound on the PAC codes’ computational complexity has a Pareto distribution. To reduce the worst-case latency of PAC sequential decoder, we limit the number of searches performed by sequential decoder. The results show that for PAC codes of length 128, search-limited sequential decoding can achieve an error-correction performance close to the error-correction performance of polar codes with successive cancellation list decoding with list size 64 and CRC length 11 with considerably less computational complexity. [ABSTRACT FROM AUTHOR]

The problem of computing the Hamming weight distribution of linear codes is considered in this paper. A novel method to enumerate all codewords up to a certain Hamming weight for binary linear block codes in general and in particular for the punctured, shortened, precoded polar codes is introduced. The proposed approach performs a recursive decomposition of the codes using construction X4 that is typically used to combine codes of different lengths. This allows to enumerate the low-weight codewords of the overall code as combinations of the low-weight codewords of the component codes. Numerical results show that the proposed approach can efficiently compute the exact partial weight distribution of the 5G New Radio punctured/shortened polar codes with CRC11 and pure polar codes. In the former and latter cases, the low-weight codeword number is up to 106 and 108, respectively. Besides, randomly punctured and shortened polar codes and randomly precoded polar codes are also considered. To the best of the authors’ knowledge, this is the first method able to solve these problems. [ABSTRACT FROM AUTHOR]

Binary quasi-reversible BCH codes whose defining set contains consecutive elements from negative to positive integers have received considerable attention in recent years due to their efficient decoding suitable for a wide range of applications. This paper combines the concept of the weight and quasi-reversible structures to introduce two subclasses of BCH codes: odd-like/even-like quasi-reversible BCH codes. The step-by-step decoding of these codes is developed as follows: First, the weight evaluation of a received polynomial is able to judge whether the number of errors is odd or even, which helps to simplify the decoding processes. Second, based on Chiò’s pivotal condensation process which can be easily implemented in a parallel computing architecture, the determinant calculation of the band matrix instead of Peterson’s matrix in column-echelon form is faster. Third, a newly proposed non-monic error-locator polynomial is sparser than the conventional ones. As a consequence, the theoretical analysis and experimental results validate potential benefits in requiring fewer finite field additions and multiplications used in the decoding of binary odd-like/even-like quasi-reversible BCH codes up to half the minimum distance when compared with the narrow-sense BCH codes with small error-correcting capability. [ABSTRACT FROM AUTHOR]

The differential chaos shift keying with code index modulation (CIM-DCSK) using Walsh codes can increase the data rate, but error rate performance degrades over practical multipath channel with high delay spread. To tackle this drawback, this paper proposes a parallel CIM-DCSK (PT-CIM-DCSK), where the first time slot is used to transmit the chaotic reference sequence, and the second slot is used to transmit the multiple sequences carrying data bits. In particular, multiple $q$ chaotic sequences are added for data transmission in parallel, and each sequence is modulated by CIM-DCSK with $m_{c}$ bits. By exploiting the quasi-orthogonal characteristics of permuted chaotic signals, the interference between $q$ sequences can be remarkably suppressed. In this way, given the data rate, PT-CIM-DCSK can achieve a larger length of Walsh segment than the CIM-DCSK, since its parallel transmission compensates the rate. In addition, the error performance of the proposed scheme over the multipath Rayleigh fading channel is theoretically analyzed. The theoretical and consistent simulation results demonstrate that the PT-CIM-DCSK can obtain significant performance gains over the conventional CIM-DCSK, and the gain can be up to 5 dB in multipath fading channel with high delay. This is because the larger Walsh segment of PT-CIM-DCSK provides the enhanced robustness against multipath effect. The performance superiority of the proposed scheme is further verified in practical ultra-wideband (UWB) wireless industrial scenarios, which suggests promising for low-cost and low-complexity wireless communication applications. [ABSTRACT FROM AUTHOR]

Tiled diagonal zipper codes (TDZCs) are shown to be a class of competitive forward error correction codes in modern optical fiber communication systems. This paper introduces the multichain tiled diagonal zipper codes (MC-TDZCs) in which multiple TDZC chains are coupled together in a cyclic manner. Firstly, we present the encoding and decoding algorithms of the proposed MC-TDZCs. Secondly, we analyze the graphical structures of the minimum stall patterns of the MC-TDZCs using directed graphs. Thirdly, we derive the multiplicities of these minimum stall patterns. The analytical results show that the size of the minimum-sized stall patterns of the proposed MC-TDZCs is larger than that of the underlying TDZC counterpart. Finally, we present simulation results to show the performance advantages of the proposed MC-TDZCs under the constraint of equal storage size for decoding. Particularly, a rate 0.94 MC-TDZC with four chains shows more than four orders of magnitude improvement in bit error rates compared to its underlying TDZC counterpart. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified adaptive belief propagation (ABP) algorithm, which is referred to as the random ABP (R-ABP) algorithm, for decoding short channel codes adopted in the fifth generation (5G) New Radio (NR) wireless systems. Based on the cyclic redundancy check (CRC) concatenation structure of 5G channel codes, we can take partial or even all CRC bits into iterative R-ABP decoding to improve the error correction performance, while the error detection capability can still be guaranteed by the remaining CRC bits and a proposed threshold-based acceptance criterion. We call this improved R-ABP decoding the threshold-and-CRC-aided R-ABP (TCA-R-ABP) decoding. The simulation results show that our proposed TCA-R-ABP algorithm outperforms the state-of-the-art decoding algorithms for many 5G low-density parity-check (LDPC) and polar codes. Moreover, the proposed TCA-R-ABP algorithm leads to a unified decoder for LDPC and polar codes, which has the potential to be a lower-complexity decoder over the combination of the belief propagation (BP) and CRC-aided successive cancellation list (CA-SCL) decoders. [ABSTRACT FROM AUTHOR]

Online fountain codes (OFCs) have many advantages, such as low overhead, online feedback and optimal encoding, due to the feedback of the instantaneous decoding state. This paper analyzes the characteristics of underwater acoustic networks (UANs) as well as the issues of existing OFCs applied in UANs. Aiming at these issues, two optimization objectives of OFCs are put forward for UANs. In addition, a recursive OFC with limited feedback (ROFC-LF) is presented for UANs. The ROFC-LF reduces the consumption of bandwidth and energy caused by the transmission of useless coding packets. Through limited feedback, the problem of low channel utilization in UANs with half-duplex communication is solved. Furthermore, a data transmission mechanism based on the ROFC-LF for UANs is presented. The theoretical analysis and simulation results show that the proposed transmission mechanism based on the ROFC-LF scheme outperforms the existing OFC schemes in terms of overhead, computational complexity, coding efficiency and energy consumption. Consequently, the ROFC-LF is suitable for UANs with constrained resources. [ABSTRACT FROM AUTHOR]

In entropy coding, universal coding of integers (UCI) is a binary universal prefix code, such that the ratio of the expected codeword length to $\max \{1, H(P)\}$ is less than or equal to a constant expansion factor $K_{\mathcal {C}}$ for any probability distribution $P$ , where $H(P)$ is the Shannon entropy of $P$. $K_{\mathcal {C}}^{*}$ is the infimum of the set of expansion factors. The optimal UCI is defined as a class of UCI possessing the smallest $K_{\mathcal {C}}^{*}$. Based on prior research, the range of $K_{\mathcal {C}}^{*}$ for the optimal UCI is $2\leq K_{\mathcal {C}}^{*}\leq 2.75$. Currently, the code constructions achieve $K_{\mathcal {C}}=2.75$ for UCI and $K_{\mathcal {C}}=3.5$ for asymptotically optimal UCI. In this paper, we construct a class of UCI, termed $\iota $ code, to achieve $K_{\mathcal {C}}=2.5$. This further narrows the range of $K_{\mathcal {C}}^{*}$ to $2\leq K_{\mathcal {C}}^{*}\leq 2.5$. Next, a family of asymptotically optimal UCIs is presented, where their expansion factor infinitely approaches 2.5. Then, tighter upper bounds of the expansion factors are derived for several classic UCIs. Finally, we prove that the length of the first codeword of optimal UCI is one. [ABSTRACT FROM AUTHOR]

A joint energy and correlation detection aided orthogonal frequency division multiplexing differential chaos shift keying (JECD-OFDM-DCSK) system is presented in this paper, where the reference and information-bearing signals of JECD-OFDM-DCSK are superimposed in the same time slot so that the phase offset between the reference and information-bearing signals is addressed. In JECD-OFDM-DCSK, only some subcarriers are activated and additional information bits are transmitted by the indices of these activated subcarriers, thereby improving the data rate while reducing the energy consumption. The energy efficiency, peak to average power ratio (PAPR) performance and system complexity of JECD-OFDM-DCSK are analyzed and then compared to other systems. The comparison results show JECD-OFDM-DCSK can obtain higher energy efficiency and better PAPR performance compared to its competitors at the cost of higher system complexity. Furthermore, the bit error rate (BER) expressions of JECD-OFDM-DCSK are derived in additive white Gaussian noise (AWGN) and multipath fading channels. Simulation results show that the BER performance of JECD-OFDM-DCSK outperforms that of other OFDM-based DCSK systems. Finally, JECD-OFDM-DCSK performs much better than other systems over the underwater acoustic (UWA) channel, which validates the robustness of JECD-OFDM-DCSK. [ABSTRACT FROM AUTHOR]

The performance of distributed computing is restricted by the slowest worker nodes, known as stragglers, in the system. Coded computation has emerged as an efficient technique to mitigate the straggler effects in distributed computing. Most existing works only considered the computation straggler for single-hop networks. However, in multi-hop networks, the straggler effects will occur not only on worker nodes but also on relay nodes. In this paper, we consider a heterogeneous multi-hop network. The nodes in the network are heterogeneous, i.e., their computation capacities and transmission capacities are different. We propose a hierarchical coding scheme for such a network. Firstly, we reorganize it into a hierarchical network containing multiple layers. Each layer in the network consists of several groups. Then, a new hierarchical coding scheme is proposed, where coding is applied to each group to mitigate the stragglers. By taking both the computation time and transmission time into consideration, the overall task completion time is derived. To improve the performance of the network, heterogeneous hierarchical coded computation (HHCC) algorithm is proposed to provide an asymptotically optimal task allocation strategy. Compared with existing uniform uncoded, load balanced uncoded, and heterogeneous coded matrix multiplication schemes, HHCC has significant improvement. [ABSTRACT FROM AUTHOR]

Reconfigurable intelligent surface (RIS) has emerged as an important transmission technology for numerous applications in Internet of Things (IoT) systems. Thus, in this paper, we investigate the application of RIS in energy harvesting (EH) based unmanned aerial vehicle (UAV) communication network with finite block length (BL) codes, where a rotary wing type flying UAV communicates with the multiple single antenna IoT users with the aid of multiple RISs mounted on several skyscraper buildings. To transmit the signal to a particular IoT user, the UAV selects an RIS on the basis of either UAV-RIS (i.e., partial) or UAV-RIS-IoT (i.e., full) channel state information (CSI) and then transmits the signal through the selected RIS along with the direct link transmission. In particular, we derive (i) the expression for probability of RIS selection, (ii) the statistical distribution of instantaneously received information signal-to-noise ratio (SNR) at the IoT user. Based on the derived statistics, we analyze the performance of the considered system under finite BL codes in terms of the average outage probability, average block error rate (ABLER) and goodput averaged over entire flying duration. Moreover, the BLER performance with finite BL codes is also compared with the infinite BL codes scenario. Additionally, we also investigate the impact of various channel and system parameters like imperfect CSI, number of RISs and the number of reflecting elements at each RIS, location of IoT users, variable altitude of the UAV, and the severity of channel fading of UAV-RIS link on the system performance. Furthermore, we have obtained the optimum UAV location in each time slot which minimizes the ABLER per time slot over all the users in the network. The analytical results are corroborated with Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as permute-and-add network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their degree which is the number of permutations applied on each incoming vector to compute an outgoing vector. Constructions of permute-and-add network codes for multicast networks are known. In this paper, we provide a new framework based on group algebras to design permute-and-add network codes for arbitrary (not necessarily multicast) networks. Our framework allows the use of any finite group of permutations (including circular shifts, proposed in prior works) and admits a trade-off between coding rate and the degree of the code. Further, our technique permits elegant recovery and generalizations of the key results on permute-and-add network codes known in the literature. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs) are important for distributed storage systems due to their efficient repairing ability of the failed storage nodes. A $q$ -ary optimal $(n,k,r)$ -LRC is an $[n,k,d]$ linear code over $\mathbb {F}_{q}$ such that every code symbol has locality $r$ , and the minimum distance attains the well-known Singleton-like bound. In this paper, we study the maximal code length, code constructions and weight distributions of $q$ -ary optimal LRCs with locality 2 and distance 5, which are of both practical and theoretical interest. Firstly, it is proved that when the code dimension is even or odd, corresponding maximal code lengths of such $q$ -ary optimal LRCs are $3 \cdot \lfloor \frac {q+1}{3} \rfloor $ and $3 \cdot \left \lfloor{ \frac {q-1}{3} }\right \rfloor +5$ , respectively. Up to the equivalence of linear codes, we propose constructions of all the possible $q$ -ary optimal LRCs with locality 2, distance 5 and maximal code length. Then, by characterizing the weight type hierarchy of codewords, we show that the weight distribution of any $q$ -ary optimal LRC with locality 2, distance 5 and even code dimension can be uniquely determined and explicit expression of the weight distribution is given. Moreover, it is shown that all $q$ -ary optimal LRCs with locality 2, distance 5 and even code dimension are maximally recoverable. [ABSTRACT FROM AUTHOR]

This paper presents density evolution (DE) techniques based on reciprocal channel approximation (RCA) for general bit-interleaved coded modulation (BICM) systems. The original RCA-based DE (RCA-DE) technique is implemented under the assumption of binary-input additive white Gaussian noise channels (BI-AWGNCs). In BICM systems, however, $M$ -ary modulation schemes are generally used, and they can be regarded as to be a parallel transmission of $\log _{2} M$ bits. Thus, $\log _{2} M$ separate bit-level channels, which are not Gaussian equivalent, need to be considered. In order to extend the conventional RCA-DE technique to BICM systems, we first establish a model of protograph BICM ensembles. Two methods are developed based on the bit error rate (BER) and the BICM capacity in order to find a corresponding BI-AWGNC that models each of the bit-level channels. Using these methods, we implement the protograph RCA-DE technique for BICM systems and show that it achieves an accurate estimation. As a practical application of the proposed RCA-DE method, we design bit interleavers in order to achieve better BICM performance in the 3GPP New Radio (NR) LDPC coding system. Numerical results show that performance gain of up to 0.3 dB is consistently achieved over a wide range of parameter values. [ABSTRACT FROM AUTHOR]

A capacity result involves two parts: achievability and converse. The achievability proof is usually non-constructive and only the existence of capacity-achieving codes is shown invoking probabilistic techniques. Recently, capacity-achieving codes have been found for several channels demonstrating that such codes can actually be constructed algorithmically. To this end, each construction is designed for a pre-specified channel so that the corresponding algorithm is specifically tailored to it. This paper addresses the general question of whether or not it is possible to find algorithms that can construct capacity-achieving codes for a whole class of channels. To do so, the concept of Turing machines is used which provides the fundamental performance limits of digital computers and therewith fully specifies which tasks are algorithmically feasible in principle. It is shown that there exists no Turing machine that is able to construct capacity-achieving codes for a whole class of channels, where the channel realization from this class is given as an input to the Turing machine. It is further shown that such an algorithmic construction remains impossible when the optimality condition is dropped and codes only need to achieve a fraction of the capacity. Finally, implications on channel-aware transmission, link adaptation, and cross-layer optimization are discussed. [ABSTRACT FROM AUTHOR]

This paper proposes the algebraic soft decoding (ASD) for one-point elliptic codes, where the interpolation problem is solved from the perspective of module basis reduction. In ASD, the interpolation polynomial $\mathcal {Q}(x, y, z)$ is the minimum candidate of a Gröbner basis. Based on a multiplicity matrix, an interpolation ideal can be defined. With the decoding output list size, an equivalent interpolation module can be led to. By further defining the set of interpolation points, a sequence of modules from the elliptic curve coordinate ring can be obtained. Based on the Lagrange interpolation functions over elliptic function field, a basis of the interpolation module can be constructed. The desired Gröbner basis that contains $\mathcal {Q}$ can be determined by reducing the module basis. Re-encoding transform (ReT) is further introduced to reduce the basis reduction complexity. It is also shown that the interpolation can be facilitated by assessing the degree of the Lagrange interpolation polynomials. The decoding complexity is analyzed, which is verified by numerical results. That shows the advantage of this interpolation technique over the conventional Kötter’s interpolation. The ASD performance of elliptic codes is also presented. [ABSTRACT FROM AUTHOR]

In any transmission scheme, security against eavesdroppers is of importance. Contrary to classical encryption, security directly at the physical layer can be achieved by applying suited coding and modulation schemes. In this paper, coded modulation schemes, i.e., the use of higher-order constellations, is studied for generating both security against a wiretapper and reliability for the legitimate user. Specifically, a multilevel coding (MLC) scheme is considered. The influence of the constellation, in particular its cardinality and its dimensionality, on the gained security is assessed. As LSB component code, a practical secure coding scheme based on punctured LDPC codes is used. The security level for the employed LDPC code ensembles within MLC constructions is examined with an adapted density evolution technique. Results from numerical simulations cover the theoretical considerations and match with the derived asymptotic behavior. The results are compared to the situation when straightforwardly applying a binary code according to the BICM approach. [ABSTRACT FROM AUTHOR]

Some nonlinear codes, such as Kerdock and Preparata codes, can be represented as binary images under the Gray map of linear codes over rings. This paper introduces MAP decoding of Kerdock and Preparata codes by working with their quaternary representation (linear codes over $\mathbb {Z}_{4}$) with the complexity of $\mathcal {O}(N^{2}\log _{2} N)$ , where N is the code length in $\mathbb {Z}_{4}$. A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of $\mathcal {O}(N\log _{2} N)$ that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel transmission scheme, called sparse layered MIMO (SL-MIMO), that combines non-orthogonal transmission and singular value decomposition (SVD) precoding. Non-orthogonality in SL-MIMO allows re-using of the eigen-channels which improves the spectral efficiency and error rate performance of the system through enhancing the coding gain and diversity gain. We also present a low-complexity message-passing (MP) detector for the proposed SL-MIMO system which performs quite close to maximum likelihood (ML). The joint moment generating function (MGF) of the ordered eigenvalues is calculated and used to derive a closed-form upper bound on the average word error probability (AWEP) of the SL-MIMO system, and this derived expression is then used to analyze the diversity gain of the system. We use our analytical results to design sub-optimal codebooks to minimize the error rate of the SL-MIMO system. Simulation results in $4\times 4$ and $6\times 6$ multiple-input multiple-output (MIMO) systems with 4-ary, 16-ary, and 64-ary constellations show that our proposed SL-MIMO scheme outperforms competing approaches such as X- and Y-codes in terms of system error rate performance. SL-MIMO has 5.6 dB advantage compared to X-codes and 4.7 dB advantage compared to Y-codes in $6\times 6$ MIMO system with a 64-ary constellation. [ABSTRACT FROM AUTHOR]

Fractional repetition (FR) codes are a class of repair efficient erasure codes that can recover a failed storage node with both optimal repair bandwidth and complexity. In this paper, we study the minimum distance of FR codes, which is the smallest number of nodes whose failure leads to the unrecoverable loss of the stored file. We derive a new upper bound on the minimum distance of FR codes, which is tighter than the Singleton bound and a Singleton-like bound that takes locality into account. Based on regular graphs and combinatorial designs, several families of FR codes with optimal minimum distance are obtained. [ABSTRACT FROM AUTHOR]

In this paper, a method for joint source-channel coding (JSCC) based on concatenated spatially coupled low-density parity-check (SC-LDPC) codes is investigated. A construction consisting of two SC-LDPC codes is proposed: one for source coding and the other for channel coding, with a joint belief propagation-based decoder. Also, a novel windowed decoding (WD) scheme is presented with significantly reduced latency and complexity requirements. The asymptotic behavior for various graph node degrees is analyzed using a protograph-based Extrinsic Information Transfer (EXIT) chart analysis for both LDPC block codes with block decoding and for SC-LDPC codes with the WD scheme, showing robust performance for concatenated SC-LDPC codes. Simulation results show a notable performance improvement compared to existing state-of-the-art JSCC schemes based on LDPC codes with comparable latency and complexity constraints. [ABSTRACT FROM AUTHOR]

The $T$ -Private Information Retrieval ($T$ -PIR) problem is that a user wishes to retrieve a single record from $N$ servers, without revealing the identity of the desired record even if the number of colluding servers up to $T$. Here every server stores all $M$ records and each record can be viewed as a vector over the $q$ -ary finite field $\mathbb {F}_{q}$. To reduce the field size, a capacity-achieving $T$ -PIR scheme based on some MDS array codes for all possible $N>T,M\geq 2$ was constructed by Xu and Zhang in 2019. A key idea in that scheme is to construct some MDS array codes in query phase, whose generator matrices satisfy the recovery property (see (C1) in , page 6), i.e., there exist some specific columns in each generator matrix such that the number of them is equal to the rank of such matrix. We call the set of each column index of those columns in each generator matrix as the column index set. However, that scheme didn’t give an explicit selection of the column index set satisfying the recovery property. In this paper, let $N=d(2t-1)$ , $N>T=dt>1, M\geq 3$ and $d\geq 1$ , under the constraint of $2^{t-1}\geq N$ , we present a novel method to construct capacity-achieving $T$ -PIR schemes over the binary field with an explicit selection of the column index set. Moreover, our scheme has sub-packetization $N(2t-1)^{M-2}$ , which is optimal for all linear capacity-achieving $T$ -PIR schemes determined by Zhang and Xu in 2019. The key to our method is that all locators and column multipliers are precisely selected such that a Generalized Reed-Solomon (GRS) code defined by them has a zero-dimensional subfield sub-code. Compared with all the known capacity-achieving $T$ -PIR schemes for the same non-trivial parameters, ours is the first explicit capacity-achieving $T$ -PIR scheme over the binary field. [ABSTRACT FROM AUTHOR]

The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures based on the Plotkin construction. The decoding algorithm can be applied with slight modifications to Reed-Muller or eBCH codes, that both achieve the capacity of erasure channels, although the list size needed for good performance grows too fast to make the decoding practical even for moderate block lengths. The key ingredient for proving the capacity-achieving property of Reed-Muller and eBCH codes is their group of symmetries. It can be plugged into the concept of Plotkin decomposition to design various permutation decoding algorithms. Although such techniques allow to outperform the straightforward polar-like decoding, the complexity stays impractical. In this paper, we show that although invariance under a large automorphism group is valuable in a theoretical sense, it also ensures that the list size needed for good performance grows exponentially. We further establish the bounds that arise if we sacrifice some of the symmetries. Although the theoretical analysis of the list decoding algorithm remains an open problem, our result provides an insight into the factors that impact the decoding complexity. [ABSTRACT FROM AUTHOR]

In this paper, a simplified successive cancellation (SC) decoding algorithm for polar codes on insertion/deletion error channels is proposed. First, the SC decoding is designed to decode polar codes on insertion/deletion channels and the joint weight distribution is derived to measure the occurrence probability of different scenarios. Some scenarios with small occurrence probability can be pruned to obtain lower decoding complexity with negligible performance loss. Inspired by this, a fixed pruning strategy (FPS) is proposed to reduce the decoding complexity, which can prune as many scenarios as possible with the given performance requirement. By exploiting the periodicity of the joint weight distribution, the upper bound of the block error rate of the pruned SC decoding is derived. Furthermore, according to the convergence of the upper bound, a dynamic self-adjusting pruning strategy is designed to further reduce the decoding complexity and improve the flexibility of the pruning algorithm. Simulation results show that the decoding complexity of the proposed pruning-based decoding algorithms is significantly reduced compared to the state-of-the-art scenario simplified SC decoding algorithm. [ABSTRACT FROM AUTHOR]

Inspired by excellent precision of carrier phase positioning, this paper presents a new carrier phase positioning technique for 5G new radio cellular networks with a focus on clock synchronization and integer ambiguity resolution. A carrier-phase based clock offset estimation method is first proposed to achieve precise clock synchronization among base stations, and proved to achieve the Cramér–Rao Lower Bound (CRLB) asymptotically. A fusion method is developed to fuse the estimated positions of a mobile station (MS) based on time-difference-of-arrival, with the estimated position changes based on the temporal changes of carrier phase measurements. While circumventing the integer ambiguities of the carrier phase measurements, the fusion method provides quality interim estimates of the MS positions, at which the measurements can be linearized to resolve the integer ambiguities. As a result, precise MS positions can be obtained based on the disambiguated carrier phase measurements. Numerical simulations show that the proposed carrier phase positioning can achieve a centimeter-level accuracy in wireless cellular networks. [ABSTRACT FROM AUTHOR]

In this paper, we propose two novel modulation concepts based on a simple maximum distance separable (MDS) code and show that these concepts can achieve better error performance than index modulation (IM) and related schemes. In the first concept, we use amplitude and phase levels to form a simple MDS code, whereas, in the second one, in-phase and quadrature components of codeword elements are used to construct the MDS code. We depict practical schemes for using the proposed concepts with orthogonal frequency division multiplexing (OFDM). We analyze the performance in terms of the minimum Euclidean distance and bit error rate. We also show that the proposed techniques exhibit desirable properties such as efficient low-complexity detection, very simple bits-to-symbols, and symbols-to-bits mappings, and a better error performance when compared to the OFDM-IM and related schemes. More importantly, contrary to the vast majority of IM studies that focus on showing the superiority of the IM techniques against conventional modulation techniques, we show that modulation concepts based on a well-known MDS code can achieve better error performance than the IM and related schemes while exhibiting a structure as simple as these schemes. [ABSTRACT FROM AUTHOR]

High-rate product codes (PCs) and staircase codes (SCs) are ubiquitous codes in high-speed optical communication achieving near-capacity performance on the binary symmetric channel. Their success is mostly due to very efficient iterative decoding algorithms that require very little complexity. In this paper, we extend the density evolution (DE) analysis for PCs and SCs to a channel with ternary output and ternary message passing, where the third symbol marks an erasure. We investigate the performance of a standard error-and-erasure decoder and of its simplification using DE. The proposed analysis can be used to find component code configurations and quantizer levels for the channel output. We also show how the use of even-weight BCH subcodes as component codes can improve the decoding performance at high rates. The DE results are verified by Monte-Carlo simulations, which show that additional coding gains of up to 0.6dB are possible by ternary decoding, at only a small additional increase in complexity compared to traditional binary message passing. [ABSTRACT FROM AUTHOR]

Online fountain codes (OFC) have attracted much attention for their good intermediate performance, which is important for receivers with low-complexity requirement. However, low-complexity receivers generally have limited buffer size to store coded symbols that have not been fully decoded yet, as well as limited power budget for feedback transmissions. In this paper, we propose improved transmission schemes for online fountain codes to reduce the buffer occupancy and feedback transmissions. Firstly, we analyze the relationship between buffer occupancy and overhead as well as the relationship between recovery rate and overhead for online fountain codes. Motivated by the analysis, we propose the weighted online fountain codes (WOFC) which can adapt to various buffer sizes by adjusting the weight to control the probability that a coded symbol can be fully processed immediately, and analyze its performance. Then we further propose weighted online fountain codes with low feedback (WOFC-LF), which utilize the proposed analysis to estimate the recovery rate, and reduce feedback transmissions. Simulation results verify the effectiveness of the analysis for both OFC and WOFC, and demonstrate the superior performance of WOFC-LF with limited buffer size and feedback transmissions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a special class of unit memory convolutional codes (UMCCs), called block oriented UMCCs (BOUMCCs). Distinguished from conventional UMCCs, which usually have small constraint lengths, the BOUMCCs have relatively large constraint lengths. We conduct the performance analysis by assuming a first-order Markov model, which indicates that the performance of the BOUMCCs depends critically on both the error propagation and the sub-frame error rate of the first layer. The error propagation can be alleviated by the use of partial superposition, which is specified by a superposition matrix with a fraction of columns being nulled. Given a superposition fraction, we propose a tree growing and pruning algorithm (TGPA) with a tunable sliding window, which provides a convenient way to trade off the decoding delay and the performance. We also present a structured construction and show by simulation that there is no performance degradation compared with random construction. Numerical results also show that, by taking the TBCCs as basic codes, the performance of BOUMCCs with TGPA is comparable to that of other short codes but with a more flexible construction or a lower complexity. [ABSTRACT FROM AUTHOR]

Multi-level memory cells are used in non-volatile memories to increase the storage density. Using multi-level cells, however, imposes lower read and write speeds, limiting their usability with high-performing applications. In this work we study the tradeoff between storage density and write/read speeds using codes. The contributions are codes that give high-performance write and read processes with minimal reduction in storage density. We describe the codes, give a detailed analytical treatment of their information rate and speed, provide encoding/decoding algorithms, and compare them with more basic access schemes and upper bounds. Using performance coding enables accessing the memory with variable access speeds, thus creating heterogenous storage devices serving a variety of applications with improved efficiency. [ABSTRACT FROM PUBLISHER]

A novel recursive list decoding (RLD) algorithm for Reed-Muller (RM) codes based on successive permutations (SP) of the codeword is presented. A low-complexity SP scheme applied to a subset of the symmetry group of RM codes is first proposed to carefully select a good codeword permutation on the fly. Then, the proposed SP technique is integrated into an improved RLD algorithm that initializes different decoding paths with random codeword permutations, which are sampled from the full symmetry group of RM codes. Finally, efficient latency and complexity reduction schemes are introduced that virtually preserve the error-correction performance of the proposed decoder. Simulation results demonstrate that at the target frame error rate of 10−3 for the RM code of length 256 with 163 information bits, the proposed decoder reduces 6% of the computational complexity and 22% of the decoding latency of the state-of-the-art semi-parallel simplified successive-cancellation decoder with fast Hadamard transform (SSC-FHT) that uses 96 permutations from the full symmetry group of RM codes, while relatively maintaining the error-correction performance and memory consumption of the semi-parallel permuted SSC-FHT decoder. [ABSTRACT FROM AUTHOR]

For high-rate linear systematic maximum distance separable (MDS) codes, most early constructions could initially optimally repair all the systematic nodes but not all the parity nodes. Fortunately, this issue was first solved by Li et al. in (IEEE Trans. Inform. Theory, 64(9), 6257-6267, 2018), where a transformation that can convert any nonbinary MDS array code into another one with desired properties was proposed. However, the transformation does not work for binary MDS array codes. In this paper, we address this issue by proposing another generic transformation that can convert any $[n, k]$ binary MDS array code into a new one, which endows any $r=n-k\ge 2$ chosen nodes with optimal repair bandwidth and optimal rebuilding access properties, and at the same time, preserves the normalized repair bandwidth/rebuilding access for the remaining $k$ nodes under some conditions. As two immediate applications, we show that 1) by applying the transformation multiple times, any binary MDS array code can be converted into one with optimal rebuilding access for all nodes, 2) any binary MDS array code with optimal repair bandwidth or optimal rebuilding access for the systematic nodes can be converted into one with the corresponding optimality property for all nodes. [ABSTRACT FROM AUTHOR]

This paper presents an enhanced permeable layer mechanism useful for highly robust packetized multimedia transmission. Packet header recovery at various protocol layers using MAP estimation is the cornerstone of the proposed solution. The inherently available intra-layer and inter-layer header correlation proves to be very effective in selecting a reduced set of possible header configurations for further processing. The best candidate is then obtained through soft decoding of CRC protected data and CRC redundancy information itself. Simulation results for WiFi transmission using DBPSK modulated signals over AWGN channels show a substantial (4 to 12 dB) link budget improvement over classical hard decision procedures. We also introduce a sub-optimal and hardware realizable version of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The length-restricted code-construction problem arises when using prefix codes for large messages and also for search-tree depth minimization. A problem instance comprises an ordered set of input probabilities which in this paper are assumed, without loss of generality, to be a set of unnormalized integer frequencies {f1, f2,…, fn}, and a maximum codeword length L bits. The package-merge algorithm of Larmore and Hirschberg constructs a minimum-redundancy length-restricted code in O(nL) time. Here we present an algorithm which computes a minimum-redundancy length-restricted code in O((H - L + 1)n) time, by starting with a minimum-redundancy (Huffman) code with a maximum codeword length of H, and then refining it to meet the length limit L. The new algorithm is suited to problems where H - L is small; and when H - L > L - [log n], the new algorithm outperforms all previous methods. Experimental results confirm the behavior of the new algorithm. [ABSTRACT FROM AUTHOR]

Considering cache enabled networks, optimal content placement minimizing the total cost of communication in such networks is studied, leading to a surprising fundamental 0–1 law for non-redundant cache placement strategies, where the total cache sizes associated with each file does not exceed the file size. In other words, for such strategies, we prove that any non-redundant cache placement strategy can be transformed, with no additional cost, to a strategy in which at every node, each file is either cached completely or not cached at all. Moreover, we obtain a sufficient condition under which the optimal cache placement strategy is in fact non-redundant. This result together with the 0–1 law reveals that situations exist, where optimal content placement is achieved just by uncoded placement of whole files in caches. [ABSTRACT FROM PUBLISHER]

Massive multiple access schemes, capable of serving a large number of uncoordinated devices while fulfilling reliability and latency constraints, are proposed. The schemes belong to the class of grant-free coded random access protocols and are tailored to massive multiple input multiple output (MIMO) base station processing. High reliability is obtained owing to an intra-frame spatial coupling effect, triggered by a simple device access protocol combined with acknowledgements (ACKs) from the base station. To provide system design guidelines, analytical bounds on error floor and latency are also derived. The proposed schemes are particularly interesting to address the challenges of massive machine-type communications in the framework of next generation massive multiple access systems. [ABSTRACT FROM AUTHOR]

This article introduces a novel communication paradigm for the unsourced, uncoordinated Gaussian multiple access problem. The major components of the envisioned framework are as follows. The encoded bits of every message are partitioned into two groups. The first portion is transmitted using a compressive sensing scheme, whereas the second set of bits is conveyed using a multi-user coding scheme. The compressive sensing portion is key in sidestepping some of the challenges posed by the unsourced aspect of the problem. The information afforded by the compressive sensing is employed to create a sparse random multi-access graph conducive to joint decoding. This construction leverages the lessons learned from traditional IDMA into creating low-complexity schemes for the unsourced setting, while also accounting for inherent randomness. Under joint message-passing decoding, the proposed scheme offers good performance at a low computational complexity. Findings are supported by numerical simulations, and results are compared to existing alternatives. [ABSTRACT FROM AUTHOR]

A new class of spatially-coupled turbo-like codes (SC-TCs), dubbed generalized spatially coupled parallel concatenated codes (GSC-PCCs), is introduced. These codes are constructed by applying spatial coupling on parallel concatenated codes (PCCs) with a fraction of information bits repeated $q$ times. GSC-PCCs can be seen as a generalization of the original spatially-coupled parallel concatenated codes proposed by Moloudi et al., 2017. To characterize the asymptotic performance of GSC-PCCs, we derive the corresponding density evolution equations and compute their decoding thresholds. The threshold saturation effect is observed and proven. Most importantly, we rigorously prove that the rate- $R$ GSC-PCC ensemble with 2-state convolutional component codes achieves at least a fraction $1-\frac {R}{R+q}$ of the capacity of the binary erasure channel (BEC) for repetition factor $q\geq 2$ and this multiplicative gap vanishes as $q$ tends to infinity. To the best of our knowledge, this is the first class of SC-TCs that are proven to be capacity-achieving. Further, the connection between the strength of the component codes, the decoding thresholds of GSC-PCCs, and the repetition factor is established. The superiority of the proposed codes with finite blocklength is exemplified by comparing their error performance with that of existing SC-TCs via computer simulations. [ABSTRACT FROM AUTHOR]

We propose a new type of ultimate-Shannon-limit-approaching codes called spatially coupled protograph-based low-density parity-check Hadamard convolutional codes (SC-PLDPCH-CCs), which are constructed by spatially coupling PLDPC-Hadamard block codes. We develop an efficient decoding algorithm that combines pipeline decoding and layered scheduling for the decoding of SC-PLDPCH-CCs, and analyze the latency and complexity of the decoder. To estimate the decoding thresholds of SC-PLDPCH-CCs, we first propose a layered protograph extrinsic information transfer (PEXIT) algorithm to evaluate the thresholds of spatially coupled PLDPC-Hadamard terminated codes (SC-PLDPCH-TDCs) with a moderate coupling length. With the use of the proposed layered PEXIT method, we develop a genetic algorithm to find good SC-PLDPCH-TDCs in a systematic way. Then we extend the coupling length of these SC-PLDPCH-TDCs to form good SC-PLDPCH-CCs. Results show that our constructed SC-PLDPCH-CCs can achieve comparable thresholds to the block code counterparts. Simulations illustrate the superiority of the SC-PLDPCH-CCs over the block code counterparts and other state-of-the-art low-rate codes in terms of error performance. For the rate-0.00295 SC-PLDPCH-CC, a bit error rate of 10−5 is achieved at $E_{b}/N_{0} = -1.465$ dB, which is only 0.125 dB from the ultimate Shannon limit. [ABSTRACT FROM AUTHOR]