This paper focuses on the design and performance analysis of faster-than-Nyquist (FTN) signaling employing enhanced 5G low-density parity-check (LDPC) codes, oriented toward the requirements of future 6G systems. We propose the extrinsic information transfer (EXIT) chart analysis for the LDPC-coded FTN system based on the Ungerboeck observation model, where the input-output mutual information function of the detector is approximated using least squares fitting. With the proposed EXIT chart analysis, we explore the thresholds and decoding performance of different LDPC codes (regular codes, irregular codes and protograph codes) in both Nyquist and FTN systems, revealing two important observational findings for FTN signaling: 1) Unlike Nyquist systems, where certain 5G New Radio (NR)-like information puncturing can enhance the decoding threshold and performance, we observe that in the FTN setting considered in this paper such puncturing leads to performance degradation; 2) Unlike Nyquist systems, the paritycheck matrix of LDPC codes optimized for FTN signaling tends to be relatively sparser within comparable ensembles, due to the intentionally introduced inter-symbol interference (ISI). Based on these findings, we develop tailored LDPC codes for FTN signaling by applying the masking operation to the base matrix of the standard 5G LDPC codes, aiming to achieve a lower decoding threshold and thereby better decoding performance. Moreover, the raptor-like structure and rate compatibility are preserved in the proposed LDPC codes, and the encoder and decoder are reused with only minor modifications. Numerical results show that: 1) All simulation results align with the decoding thresholds obtained by the proposed EXIT chart analysis, confirming the effectiveness of the analysis; 2) For the FTN system, the tailored LDPC codes outperform standard 5G LDPC codes, achieving over 0.4 dB coding gain and approaching (slightly exceeding) the constrained Nyquist capacity; 3) Under the same spectral efficiency, FTN with tailored LDPC codes performs better than standard 5G LDPC codes with Nyquist signaling, demonstrating a coding gain of up to 0.6 dB; 4) The proposed LDPC codes with the FTN signaling achieve better performance compared to existing high-performance codes specifically designed for FTN signaling.

This paper proposes an iterative detection and decoding (IDD) scheme for asynchronous sparse code multiple access (aSCMA), referred to as aIDD, in satellite-terrestrial uplink communication scenario with the low earth orbit (LEO) satellite equipped with uniform planar array (UPA) antenna. In detector design, we first develop the extended factor graph for aSCMA by considering the memory induced by asynchronous transmission, and an asynchronous message passing algorithm (A-MPA) is proposed. In A-MPA, the noise whitening on the sampled symbols is performed to mitigate the correlation among the noise samples due to the matched filtering, and the updating rules are then designed to achieve superior performance. Furthermore, we propose an asynchronous expectation propagation algorithm (A-EPA) by exploiting the diversity gains induced by UPA, where the means and variances of the transmitted SCMA codewords are updated with high reliability. Simulation results show that the proposed A-EPA can achieve the same performance as that of A-MPA but with lower complexity at a high number of receive antennas. In decoder design, a soft-output ordered likelihood decoder (S-OLD) is proposed to generate the soft information with high reliability compared with the belief propagation (BP) decoder under low-density parity check (LDPC) code. By combining the proposed A-EPA/A-MPA and S-OLD, the proposed aIDD scheme iteratively exchanges the messages between the detector and the decoder until the maximum number of iterations of the outer loop is achieved or the decoding results of all the users are converged. Simulation results show that the proposed aIDD/A-EPA and aIDD/A-MPA have the same performance and are better than that of the synchronous IDD and joint detection and decoding (JDD) schemes.

In this paper, we propose a fixed-to-fixed length coding approach to near-lossless compression by leveraging block Markov superposition transmission (BMST) of generalized Reed-Solomon (GRS) codes. To compress non-uniform non-binary sources, we propose two schemes: multi-level coding with a natural mapper and single-level coding with a mapper called sparsifier. The multi-level coding can be proved to achieve the source entropy, while the single-level coding is more suitable for practical use. Both the natural mapper and the sparsifier map non-binary symbols with higher probability to sparser binary vectors of fixed length. When compared with variable-length coding, the most distinguished feature of the proposed coding is that the error propagation caused by a few erroneous bits can be controlled. Even more, the proposed scheme can be easily extended as joint source-channel coding (JSCC) with a wide range of code rates by fixing the input while lengthening the output. Numerical results show that the proposed codes can approach the Shannon limits for transmitting non-uniform sources over noisy channels, providing a universal way to trade off bandwidth and the transmission power.

This paper proposes an improved ordered statistics decoding (OSD) algorithm, termed OSD with pre-stored systematic matrices (PSM-OSD), which completely avoids online Gaussian elimination by pre-storing multiple systematic generator matrices (SGMs) and selecting one based on the received sequence for decoding. To further improve the search efficiency, the local-constraint mechanism is combined with the PSM-OSD, resulting in the LC-PSM-OSD. We then address three critical questions for design: how many SGMs should be pre-stored, how they should be constructed, and how to select one for decoding. For the first question, we introduce the notion of maximum reliable-information coverage, which characterizes how well the pre-stored SGMs match the reliable bit positions. For construction and selection, we introduce a Reed-Muller-based design to ensure high basis diversity, and the sum-reliability strategy provides superior decoding performance among the tested selection criteria. A saddlepoint-based analytical framework is developed to estimate both the frame error rate (FER) upper bound and the average number of searches, enabling pre-stored matrix design without exhaustive decoding simulations. We also prove that, with tailored early stopping and unlimited maximum number of searches, the proposed algorithm is a non-exhaustive-search ML decoding algorithm. Numerical results and analysis demonstrate that the proposed decoding algorithms are effective across various coding schemes, including random linear codes, BCH codes, and 5G polar codes. Moreover, the results show that the proposed LC-PSM-OSD can closely approach the finite-length capacity over almost all code rates for the considered codes, and outperforms conventional OSD, guessing codeword decoding (GCD), and LC-GCD. © 2026 IEEE.

In this paper, we prove that the sub-field image of generalized Reed-Solomon (RS) code ensemble can achieve the symmetric capacity ofp-ary memoryless channels. Unlike the totally random linear code ensemble, as a class of maximum distance separable (MDS) codes, the generalized RS code ensemble lacks the pair-wise independence among codewords and has non-identical distributions of nonzero codewords. To prove the coding theorem for thep-ary images of generalized RS code ensemble, we analyze the exponential upper bound on the error probability of the generalized RS code in terms of its spectrum using random coding techniques. In the finite-length region, we present an ML decoding algorithm for the generalized RS codes over the binary erasure channels (BECs). In particular, the algebraic structure of the generalized RS codes allows us to implement the parallel Lagrange interpolation to derive an ordered systematic matrix. Subsequently, we can reconstruct the ML codeword through a change of basis, accelerating the conventional Gaussian elimination (GE), as validated in the simulation results. Additionally, we apply this decoding technique to the ordered statistic decoding with local constraints (LC-OSD) algorithm over the additive white Gaussian noise (AWGN) channels with binary phase shift keying (BPSK) modulation and three-level pulse amplitude modulation (3PAM). Simulation results show that, in the high-rate region, generalized RS codes defined over fields of characteristic three with 3PAM perform better than those defined over fields of characteristic two with BPSK.

This paper proposes an asynchronization-aided iterative detection and decoding (AIDD) scheme for sparse code multiple access (SCMA) in satellite-terrestrial communication scenario, where the messages between the detector and decoder are iteratively exchanged with an additional degrees- of-freedom (DoF) in terms of delay. We first propose a parallel expectation propagation algorithm (P-EPA) for asynchronous multiuser detection, where a new initialization method is introduced by efficiently utilizing the prior information to enhance the performance of the detector. Furthermore, a universal soft-output decoder (S-OLD) is proposed based on the ordered likelihood decoder (OLD), which can generate the soft information with high reliability and serve as the input of P-EPA in the proposed AIDD. The iteration between the P-EPA and S-OLD is terminated when the maximum iteration number of the outer loop is achieved or the decoding results of all the users are converged. Simulation results show that the proposed P-EPA has better performance and lower latency compared to its counterparts, and the proposed AIDD also has better performance and fewer iterations than the synchronous IDD and joint detection and decoding (JDD) schemes.

This paper proposes an enhanced guessing codeword decoding (GCD) algorithm, termed GCD with most reliable cyclic basis (MRCB-GCD), specifically tailored for BCH codes. Unlike original GCD, the proposed method selects the k consecutive bits with the highest aggregate reliability as the re-encoding basis, leveraging the cyclic structure of BCH codes. To further enhance search efficiency, a local constraint (LC) mechanism is introduced, where the extended reliable bits of length k+δ enable the algorithm to skip numerous unnecessary test error patterns (TEPs). To stop the search process, we introduce two termination criteria and prove that the proposed approach with the trivial termination criterion forms a non-exhaustive-search maximum likelihood (ML) decoding algorithm. A saddlepoint-based numerical method is developed to approximately calculate the upper bound on the performance gap to the ML decoding and estimate the average number of searches, showing the advantages of the proposed MRCB-GCD over the original GCD in terms of both decoding performance and search efficiency. Simulation results demonstrate that: 1) MRCB- GCD outperforms original GCD in both frame error rate (FER) and number of searches, 2) the LC mechanism and termination criteria further reduce the number of searches, and 3) the proposed approach achieves finite-length capacity across various code rates but without requiring Gaussian elimination.

In this paper, we propose a new coding approach to near-lossless compression of$Q$-ary sources by utilizing a sparsifier and block Markov superposition transmission (BMST) codes. The symbols from a$Q$-ary source are mapped to fixedlength binary vectors by the sparsifier such that the symbols with higher probabilities are mapped to vectors of lower weights. The binary sparse sequences are then compressed in a BMST manner. The most distinguished feature of the proposed source coding is that the error propagation can be mitigated in the presence of noise. Numerical results show that the proposed scheme performs well for$Q$-ary sources, providing a universal but simple way to achieve near-lossless coding at rates approaching the source entropy.

This paper focuses on the analysis and design of improved 5G low-density parity-check (LDPC) codes for faster-than-Nyquist (FTN) signaling. We first propose the protograph-based extrinsic information transfer (PEXIT) chart analysis for the LDPC-coded FTN system using the sum-product algorithm (SPA) based on the Ungerboeck observation model, where the distribution of the output mutual information from the detector is approximately derived using least squares fitting. With the proposed PEXIT chart analysis, we then design the improved LDPC codes for the coded FTN signaling aiming to achieve a lower decoding threshold and thereby better error performance. The proposed codes are optimized based on the raptor-like structure of the 5G LDPC codes and also support rate compatibility. The proposed codes reveals two distinct LDPC code design criteria for FTN signaling, i.e., 1) no information bits should be punctured; 2) columns with high column weights should be removed in the base graph. The advantages of the proposed codes are explicitly verified by our numerical results, where noticeable coding gains compared to existing codes and coded Nyquist systems can be observed.

Successive cancellation list (SCL) decoding enables polar codes to deliver satisfactory performance in finite-length scenarios but it comes with high latency and complexity. To reduce latency, a partitioned SCL (PSCL) decoding algorithm, implemented over a PSCL decoding tree, can be utilized. In this work, we aim to lower down the complexity of the PSCL decoding, resulting in an efficient decoding algorithm with low latency and complexity for polar codes. To achieve this, we define two metrics at each level of the PSCL decoding tree. One is for evaluating the reliability of a path and the other is for evaluating the reliability of a list of paths. Then, we propose a double-threshold strategy in the PSCL decoding process where unreliable valid paths are pruned based on the first metric, and then the smallest reliable list of paths is selected to continue based on the second metric. Simulation results demonstrate that when polar codes are decoded using the proposed low-complexity PSCL decoder, both the sorting complexity and the computational complexity are reduced and significantly decrease as the signal-to-noise ratio (SNR) increases.

We propose in this paper to enhance the 5G low-density parity-check (LDPC) codes by transmitting the codewords in a block Markov superposition transmission (BMST) manner, resulting in a class of spatially coupled LDPC codes. We present a generalized extrinsic information transfer (EXIT) chart for performance analysis, showing that the decoding thresholds can be improved by increasing the memory size (coupled width)$m$. However, for m > 1, the receiver requires a relatively large window size for the sliding window decoding (SWD) algorithm, potentially causing unacceptable decoding latency. To address this issue, we propose an adaptive sliding window decoding (ASWD) algorithm, in which the decoding window size depends on the decoding state of the BMST system. The proposed EXIT chart analysis can also effectively guide the setting of the maximum decoding window in the ASWD algorithm, as well as the setting of the superposition fractions for the BMST-5G-LDPC system. Simulation results show that: 1) the ASWD algorithm can effectively reduce the average decoding window size in the medium to high signal-to-noise ratio (SNR) region, without the performance loss compared to the conventional SWD algorithm; 2) the proposed BMST-5G-LDPC code can achieve about 0.4, 0.5 and 1.7 dB performance gains compared to the 5G LDPC code over the AWGN channel, the fast fading channel, and the quasi-static block fading channel, respectively.

We propose a class of codes, referred to as staircase matrix codes (SMCs), which have staircase-like generator matrices or parity-check matrices. We illustrate that polar codes and Reed-Muller (RM) codes are (equivalent to) two instances of SMCs. The most distinguished feature of the SMCs is that the staircase-like matrices enable parallel implementation of the Gaussian elimination (GE). Then we propose a representative ordered statistics decoding (ROSD) algorithm for the SMCs. Different from the conventional OSD, which forms the most reliable basis (MRB) by selecting reliable bits globally, the proposed ROSD forms an extended basis by selecting relatively reliable bits locally at least one from each staircase. We demonstrate by simulation that the ROSD has a similar performance to the OSD with local constraints (LC-OSD) and that the proposed random SMCs can outperform the RM codes and the polar codes. To further reduce the decoding delay and improve the performance, we propose a heuristic construction of staircase generator matrix codes (SGMCs) and analyze the ensemble weight spectrum (related to performance) and the quality of MRB (related to average number of test error patterns (TEPs)) for the heuristic construction. The simulation results show that, the proposed heuristic construction can reduce the average number of TEPs of the ROSD and provide a potential reduction in average decoding delay in the high signal-to-noise ratio (SNR) region. Furthermore, the proposed heuristic construction can approach the RCU bounds in a wide range of code rates.

This paper is concerned with a new list decoding algorithm called list coset decoding. With the recently proposed linear coding framework, we first give a theorem for the optimality of the list coset decoding. Different from the commonly accepted list decoding, which attempts to find a list ofLmost probable candidate codewords, we form a list (with a constrained size) that may not contain all theLmost reliable candidate codewords but still contains the transmitted (uncoded) vector with high probability. The list coset decoding divides the space into subspaces so that the transmitted vector can be reached with high probability. We then apply the list coset decoding to the ordered statistics decoding with local constraints (LC-OSD) to implement the parallel decoding for linear block codes. With the dynamic approximate ideal (DAI) early stopping criterion, the newly proposed algorithm can reduce the decoding delay, compared with the serial implementation. To numerically evaluate the decoding delay, we propose to use the saddlepoint approximation, which is a high-accuracy counting method. Simulation results show that the estimation matches well with the simulation results in high signal-to-noise ratio (SNR) region. To further reduce the decoding delay, we propose several offline test error pattern (TEP) generation methods. Then, the parallel decoding can exploit the pre-stored TEPs for each coset list. Simulation results demonstrate that the parallel list coset decoding in LC-OSD, utilizing offline TEP generation based on empirical reliability, has minimal performance degradation. The simulation results also show that, on average, one or two trials in parallel are sufficient for near maximum-likelihood (ML) performance.

In this paper, we propose a two-stage soft-decision decoding (SDD) algorithm for BCH codes. At the first stage, we search for test error patterns (TEPs) according to the reliabilities of the received bits by using the flipping pattern tree (FPT) algorithm or the ordered reliability bits (ORB) technique, with a bounded number of searches. At the second stage, traditional algebraic hard-decision decoding (HDD), say, Berlekamp-Massey (BM) algorithm, is performed for these TEPs to find codewords. The proposed algorithm, referred to as FPT-BM algorithm or ORB-BM algorithm, can achieve near-optimal performance with a significantly small number of searches (dozens to hundreds). This enables efficient parallel implementation, ensuring low decoding latency and high throughput. To justify the sufficiency of the small number of searches, we provide qualitative reasoning and quantitative evaluation (using saddlepoint approximation) to show that the transmitted codeword under the proposed FPT-BM decoding strategy is reached much earlier in the search list than that under the single stage decoding (FPT-only). To analyze the performance, we calculate the upper bound on the performance gap between the proposed algorithm and maximum likelihood (ML) decoding based on the saddlepoint approximation, showing that the proposed algorithm can effectively approach the ML lower bound for high rate BCH codes. To reduce computational complexity, we introduce a filtering criterion, where the algebraic hard-decision decoding (BM decoding) is performed only for those candidates satisfying a sufficient number of parity-check equations. To determine the key parameters, we provide an intuitive criterion for the threshold in the filtering step based on the statistical analysis and a quantitative criterion for the maximum list size based on the upper bound on the performance gap to ML decoding. Numerical results demonstrate that: 1) By introducing the filtering criterion, the proposed FPT-BM algorithm effectively reduces the number of BM calls by half, with negligible performance loss, and outperforms the Chase-BM algorithm under the same filtering criterion; 2) The proposed decoding algorithm outperforms single stage decoding (FPT only) under limited search sizes; 3) For high rate BCH codes, the proposed decoding algorithm can approach the finite-length capacity, while for medium rate regions, the proposed decoding algorithm also exhibits better performance over FPT-only and BM-only decoding algorithms.

In this paper, we propose to transmit sparse data by exploiting block Markov superposition transmission (BMST) of high-rate low-density parity-check (LDPC) codes. The high-rate LDPC codes are taken as the basic codes to lower down the error floors, while the BMST serves as joint source-channel coding (JSCC). The most distinguished feature of the BMST-LDPC codes is their flexible construction, which requires no complicated optimization and applies to a wide range of code rates. More importantly, embedding LDPC codes into the BMST system allows us to carry the sparsity information by free-ride coding without any loss of code rate, which may find applications especially in the scenario when the source sparsity varies from block to block. Numerical results show that the proposed BMST-LDPC coded system performs well for sparse sources with entropy < 0.5 bits, providing an easy way to trade off between the transmission power and the system bandwidth.

Successive cancellation list (SCL) decoding enables polar codes and their generalizations to deliver satisfactory performance in finite-length scenarios but it comes with high latency and complexity. To reduce latency, a partitioned SCL (PSCL) decoding algorithm, implemented over a PSCL decoding tree, can be utilized. In this work, we aim to lower down the complexity of the PSCL decoding, resulting in an efficient decoding algorithm with low latency and complexity for polar-like codes. To achieve this, we define two metrics at each level of the PSCL decoding tree. One is for evaluating the reliability of a path and the other is for evaluating the reliability of a list of paths. Then, we propose a double-threshold strategy in the PSCL decoding process where unreliable valid paths are pruned based on the first metric, and then the smallest reliable list of paths is selected to continue based on the second metric. Simulation results demonstrate that when polar/CRC-polar/PAC codes are decoded using the proposed low-complexity PSCL decoder, both the sorting complexity and the computational complexity are reduced and significantly decrease as the signal-to-noise ratio (SNR) increases.

In this paper, we prove that the generalized Reed-Solomon (RS) codes are capacity-achieving over binary-input output-symmetric (BIOS) channels, in terms of frame error rate (FER) under maximum likelihood (ML) decoding. In the finite-length region, we present the ordered statistics decoding with local constraints (LC-OSD) algorithm for the generalized RS codes. In particular, the extended most reliable basis (MRB) is derived based on a systematic matrix calculated by the parallel Lagrange interpolation, accelerating the conventional Gaussian elimination (GE). Additionally, we propose a joint source-channel coding (JSCC) scheme that incorporates generalized RS codes and classified enumerative (CE) coding, where the partition of the source is optimized by the k-means++ clustering algorithm. At the transmitter, we implement the CE coding to encode the source information. Then the variable-length codeword of the CE coding is transformed into a fixed-length codeword by the multiple-rate generalized RS encoding and superimposed with a class label for transmission. At the receiver, parallel LC-OSD is performed to recover the source. Simulation results demonstrate that the proposed JSCC scheme outperforms the double polar JSCC scheme (exhibiting a coding gain of up to 1.4 dB) and the JSCC scheme based on polarizing matrix extension (exhibiting a coding gain of up to 0.6 dB), as predicted by Gallager’s JSCC bound.

This paper presents new variants of the Welch-Berlekamp algorithm that are favorable to hardware implementation. First, we derive the parallel Welch-Berlekamp (PWB) algorithm in a constructive manner based on the properties of solutions to the rational interpolation problem. The algorithm features the simultaneously performed discrepancy computation and polynomial update. Second, we explore the early-termination mechanism of the PWB algorithm for decoding of Reed-Solomon (RS) codes. By introducing the concept of incomplete error locator polynomial, we show that if$e \leq t$(whereeis the number of errors andtis the error correction capability), the PWB algorithm can be terminated at latest at the completion of the$(t+ e)$-th iteration. This leads to the early-terminating PWB (EPWB) algorithm. Finally, we develop frequency-domain versions of the PWB and EPWB algorithms, namely, FPWB and FEPWB. The key point toward the two algorithms is to replace the update of polynomial coefficients with the update of polynomial evaluations. It is worth noting that the FEPWB algorithm applies only to shortened RS codes. Furthermore, an efficient systolic architecture for the FPWB algorithm is designed, which is easily adapted for the FEPWB algorithm.

In this paper, we present a class of codes, referred to as random staircase generator matrix codes (SGMCs), which have staircase-like generator matrices. In the infinite-length region, we prove that the random SGMC is capacity-achieving over binary-input output-symmetric (BIOS) channels. In the finite-length region, we propose the generalized representative ordered statistics decoding with local constraints (LC-ROSD) algorithm for the SGMCs. The most distinguished feature of the SGMCs with LC-ROSD is that the staircase-like matrices enable parallel implementation of the Gaussian elimination (GE), avoiding the serial GE of conventional OSD and supporting a potential low decoding latency, as implied from simulations. To analyze the performance of random SGMCs in the finite-length region, we derive the ensemble weight spectrum and invoke the conventional union bound. We also derive a partially random coding union (RCU) bound, which is tighter than the conventional one and is used as a criterion to design the SGMCs. Staircase-like generator matrices allow us to derive a series of (tighter and tighter) lower bounds based on the second-order Bonferroni inequality with the incremental number of codewords. The numerical results show that the decoding performance can match well with the proposed partially RCU bound for different code rates and different profiles. The numerical results also show that the tailored SGMCs with the LC-ROSD algorithm can approach the finite-length performance bound, outperforming the 5G low-density parity-check (LDPC) codes, 5G polar codes, and Reed-Muller (RM) codes.

This paper is concerned with a guessing codeword decoding (GCD) of linear block codes, which is optimal and typically requires a fewer number of searches than the naive exhaustive search decoding (ESD). Compared with the guessing random additive noise decoding (GRAND), which is only efficient for high-rate codes, the GCD is efficient for not only high-rate codes but also low-rate codes. We prove that the GCD typically requires a fewer number of queries than the GRAND. Compared with the conventional ordered statistics decoding (OSD), the GCD does not require the online Gaussian elimination (GE). In addition to limiting the maximum number of searches, we suggest limiting the radius of searches in terms of soft weights or tolerated performance loss to further reduce the decoding complexity, resulting in the so-called truncated GCD. The performance gap between the truncated GCD and the optimal decoding can be upper bounded approximately by the saddlepoint approach or other numerical approaches. The derived upper bound captures the relationship between the performance and the decoding parameters, enabling us to balance the performance and the complexity by optimizing the decoding parameters of the truncated GCD. We also introduce a parallel implementation of the (truncated) GCD algorithm to reduce decoding latency without compromising performance. Another contribution of this paper is the application of the GCD to the polar codes. We propose a multiple-bit-wise decoding algorithm over a pruned tree for the polar codes, referred to as the successive-cancellation list (SCL) decoding algorithm by GCD. First, we present a strategy for pruning the conventional polar decoding tree based on the complexity analysis rather than the specific bit patterns. Then we apply the GCD algorithm in parallel aided by the early stopping criteria to the leaves of the pruned tree. Simulation results show that, without any performance loss as justified by analysis, the proposed decoding algorithm can significantly reduce the decoding latency of the polar codes.

This paper proposes a concatenated multi-belief ordered likelihood decoding (MB-OLD) algorithm for rate-compatible (RC) short low-density parity check (LDPC) codes, where the output log-likelihood ratios (LLRs) of belief propagation (BP) are sent to a well-designed bit-flipping decoder, which we called ordered likelihood decoder (OLD). In contrast to conventional ordered statistic decoder (OSD), the test error patterns (TEPs) sequence of OLD is generated from most likely to least likely, where the ordered reliability sequence associated with the most reliable basis (MRB) is approximated as multiple lines, and a stopping criterion (SC) is taken to reduce the decoding complexity. Furthermore, we analyze the LLR behavior of BP decoder in short block-length regimes, and propose an optimal iteration number. Based on these analyses, the output LLRs of BP within the optimal number of iterations are well combined and sent to OLD. Simulation results show that the proposed MB-OLD has the superior decoding performances in terms of error-rate and decoding complexity than its counterparts.

This paper is concerned with the SCL-GCD algorithm of polar codes, which performs the successive-cancellation list (SCL) decoding algorithm for a lower rate sub-code and the guessing codeword decoding (GCD) algorithm for a higher rate sub-code. We propose to implement the GCD algorithm in a parallel way and design early stopping criteria for reducing complexity and decoding latency without sacrificing performance. Numerical results show that, when compared with the original SCL decoding algorithm, the SCL-GCD algorithm armed with the proposed early stopping criteria has lower computational complexity and decoding latency in the high signal-noise ratio (SNR) region.

In this paper, we present an efficient decoding algorithm for a class of Reed-Solomon (RS) codes over Fermat field$\mathbb{F}_{2^{r}+1}$. We show that the Fermat number transform can be used to speed up the syndrome computation and the Chien search. The implementation architectures are designed for the two blocks. The key equation is then derived. When using the RS code in practice, there arises the issue that a$(2^{r}+1)$-ary symbol is less efficiently represented by a tuple of$(r+1)$bits. We present a nested coding scheme based on RS code and single parity-check (SPC) code to harness the inefficiency. A modified Wagner algorithm is proposed for decoding the inner (nonlinear) code and is proved to be an ML decoding over the BPSK-modulated AWGN channel. Simulation results show that the proposed RS-SPC nested coding scheme yields a considerable performance gain compared to the stand-alone RS coding scheme.

In this paper, we propose a class of codes, referred to as random staircase generator matrix codes (SGMCs), which have staircase-like generator matrices. In the infinite-length region, we prove that the random SGMC is capacity-achieving over binary-input output-symmetric (BIOS) channels. In the finite-length region, we present the representative ordered statistics decoding with local constraints (LC-ROSD) for the SGMCs. The most distinguished feature of the SGMCs with LC-ROSD is that the staircase-like matrices enable parallel implementation of the Gaussian elimination (GE), avoiding the serial GE of conventional OSD and supporting a potential low decoding latency, as implied from simulations. To analyze the performance of random SGMCs in the finite-length region, we derive the ensemble weight spectrum and invoke the conventional union bound. We also derive a partially random coding union (RCU) bound, which is tighter than the conventional one and can be used as a criterion to design the SGMCs. The numerical results show that the random SGMCs with the LC-ROSD exhibit a significant decoding time improvement compared to that with the conventional OSD. They also show that the tailored SGMCs with the LC-ROSD can approach the finite-length performance bound, outperforming the 5G low-density parity-check (LDPC) codes, 5G polar codes and Reed-Muller (RM) codes.

In this paper, we propose an approach to construction of polar codes for successive cancellation list (SCL) decoding with a preset list size. For a given code length, we construct a trellis through which a path corresponds to a polar code. Then, we employ a sequential search algorithm to find an expected path based on four path metrics and five path selection rules. Additionally, for polarization-adjusted convolutional (PAC) construction, we optimize the convolutional generator polynomial based on our constructed polar code. Numerical results show that our polar/PAC codes can achieve satisfactory performance.

We propose quasi ordered statistic decoding (quasi-OSD) of binary image of Reed-Solomon (RS) codes and explore the application of RS codes to joint source-channel coding (JSCC). Unlike the conventional OSD algorithms that use Gaussian elimination to obtain the systematic matrix, the proposed quasi-OSD algorithm utilizes the parallel Lagrange interpolation at the symbol level, which achieves a lower complexity and comparable performance compared with the locally constrained OSD (LC-OSD), for high-rate RS codes in the high signal-to-noise ratio (SNR) region. Additionally, we apply RS codes to the JSCC with known source statistics. At the transmitter, a new partition criterion is introduced in the classified enumerative (CE) coding to encode the source information. Then the multiple-rate RS coding transforms the variable-length codeword of the CE coding into a fixed-length transmitted codeword. At the receiver, parallel decoding is performed to recover the source. Simulation results demonstrate that the proposed JSCC scheme outperforms the double polar JSCC scheme by one dB, as expected by Gallager's JSCC bound.

This paper is concerned with a class of product codes, referred to as CA-polar-SPC codes, where the cyclic redundancy check (CRC) aided polar (CA-polar) codes are column component codes and single parity-check (SPC) codes are row component codes. With the help of SPCs, a decoding scheme is proposed to improve the performance of CA-polar codes at the expense of only a minor increase in decoding complexity and latency. Distinguished from the conventional iterative decoding of product codes, the proposed decoding outputs immediately the successfully decoded columns without waiting for the reception of the entire product block. Only those unsuccessfully decoded columns are further decoded with updated messages from the row decoder by treating the successfully decoded columns as determined messages instead of soft messages. Extensive simulation results show that the proposed CA-polar-SPC code outperforms the underlying CA-polar code over binary-input additive white Gaussian noise (Bi-AWGN) channels with only a minor increase in decoding latency.

This paper is concerned with a universal guessing codeword decoding (GCD) of linear block codes, referred to as locally constrained GCD (LC-GCD), which does not require the online Gaussian elimination (GE). Distinguished from the GCD algorithm, the proposed LC-GCD queries the partial error patterns using the serial list Viterbi algorithm (SLVA) over a trellis specified by a local parity-check matrix, typically reducing the number of queries. Moreover, we introduce a parallel implementation of the LC-GCD algorithm to reduce decoding latency without compromising performance. Numerical results show that the LC-GCD requires a fewer number of queries than the GCD without performance loss, indicating a lower complexity in general. The comparisons with other decoding algorithms are also provided to demonstrate the potential advantage in complexity of the LC-GCD.

A ring-based constant composition distribution matcher (R-CCDM) is designed for shaping rotationally symmetrical two-dimensional (RS-2D) constellations. The procedure of distribution matching can be divided into two steps. First, nonuniformly distributed rings are generated by CCDM to approach a probability distribution calculated from that of modulation symbols. Second, a uniform mapping is performed to select one amplitude signal point from each ring. Compared with CCDM, the proposed DM contains multiple amplitude compositions, and hence achieves less rate loss. Numerical results show that, at moderate spectral efficiency, probabilistic shaping with R-CCDM outperforms that with CCDM by about $0.2 \sim 0.3 \mathrm{~dB}$ for different modulation formats.

Although the use of full duplex (FD) multiple relays is promising in Internet of Things (IoT) systems, this scheme has low energy efficiency. In addition, it is not known whether multi-relay cooperation is always beneficial to freshness of status update. With this, a tradeoff among age of information (Aol), throughput, and energy is designed in a FD multi-relay assisted IoT system. We first propose an earliest partial FD relay selection scheme by jointing automatic repeat request (ARQ). Secondly, the employment of ARQ causes the randomness of the retransmission numbers of the two-hop relay links and the out of order update delivery. To address this complex problem, a synchronous cooperation delivery mechanism is also integrated, where a new round of the packet transmission begins only when both the FD relays and destination successively recover the received status update. Finally, we derive the average Aol by establishing four different cases. The numerical analysis shows that the increase of the number of the selected relays is not always beneficial to the information freshness.

In this paper, we propose a new ordered statistics decoding (OSD) algorithm called representative OSD (ROSD), specifically designed for polar codes. The generator matrices of polar codes are composed of the rows indexed by an information index set in a lower triangular matrix with ones on the diagonal. Therefore, the generator matrices of polar codes have a staircase-like structure that enables us to present the parallel implemen-tation of Gaussian elimination (GE) in ROSD avoiding the serial GE in conventional OSD. Different from the conventional OSD which forms the most reliable basis (MRB) by selecting reliable bits globally, the proposed ROSD forms an extended basis by selecting relatively reliable bits locally at least one from each staircase. Simulation results show that the ROSD has a similar performance to the LC-OSD and conventional OSD for 5G polar codes and Reed-Muller (RM) construction polar codes.

In this paper, we integrate the 5G low-density parity-check (LDPC) coded modulation systems with hybrid shaping, where the centroid-based geometric shaping is implemented to remedy the performance loss of the many-to-one probabilistic shaping. Taking into account the fact that the 5G parity-check matrices have an uneven density in different parts, we elaborately design a simple row-column interleaver for the bit-interleaved coded modulation with iterative decoding (BICM-ID) system to allocate the ambiguous bits caused by the many-to-one mapping to the sparser parity part, resulting in the hybrid shaping for BICM-ID (HS-BICM-ID) system. Numerical results have shown that the HS-BICM-ID can obtain shaping gains of about 0.5 dB and 1.4 dB compared to the constant composition distribution matching (CCDM) shaping and the geometric shaping, respectively, while it can obtain a shaping gain of about 1.7 dB compared to the scheme with uniform input. Our work has also shown that, at low and moderate spectral efficiency, the presented hybrid shaping can achieve a significant shaping gain and effectively remedy the performance loss of dyadic many-to-one probabilistic shaping.

This paper focuses on the design of dynamic quantization (DQ) and coded transmission schemes for federated learning (FL). In the conventional FL system, the updates divergences typically decrease with communication rounds as training goes on. We first study both the impact of the quantization bit-width in the error-free transmission scenario and the impact of bit error rate (BER) in the practical transmission scenario on the performance of FL. Then we propose a DQ scheme based on the fixed B-bit or 1-bit quantization scheme, where each device quantizes its local updates with dynamic bit-width according to the test accuracy of the global model and device-to-server signal-to-noise ratio (SNR). Due to the quantization bit-width is dynamic, the test accuracy (as a kind of extra data) is needed for devices to determine the bit-width, and the devices need to inform the server of the resultant bit-width (as another kind of extra data). To reliably transmit these extra data without consuming extra transmission resource, we utilize the free-ride coding, where the extra data are embedded into the low-density parity-check (LDPC) coded payload data. Numerical results show that in the practical scenario, B-bit (B > 1) quantization scheme shows fast convergence speed and high final accuracy (in high SNR region) while the L-bit quantization scheme exhibits greater robustness (in low SNR region). They also show that the proposed FL by DQ and free-ride coding not only can significantly reduce the communication overhead with a negligible performance gap to the upper bound (error-free scheme) even in low SNR region but also outperforms the fixed quantization coded transmission FL scheme in terms of accuracy and convergence speed.

This paper is concerned with the wireless video surveillance systems, which were widely deployed and now augmented by edge computing. Armed with the edge computing, on-device local intelligence algorithms like machine learning (ML) can be utilized to extract specific semantics such as events classification in terms of risk levels which are vital for downstream tasks, say video retrieval. Different from the emergent semantic communications, not only these extracted semantic features (for further use) but also the raw video (by legal requirement) need to be sent to the surveillance center. This application scenario is also different from those for the conventional edge computing. The main objective of this paper is to propose a cost-effective scheme for such extra semantic data transmission in a free-ride way that has mild impact on the existing communication link and requires neither bandwidth expansion nor extra transmission power. The basic idea is to superimpose in the binary field the extra bits on the coded payload data. Numerical results show that, with the 5G low-density parity-check (LDPC) codes, simultaneously transmitting semantic features along with the payload data delivers more reliable semantic features and has a negligible effect on the quality of the payload data.

In this paper, we propose a new joint source-channel coding (JSCC) for short-packet communications, especially for the uplink from the sensor to the base station. At the transmitter, the sensing information is first encoded by a two-stage description, referred to as classified enumerative (CE) coding, and then encoded by a random multiple-rate code. The two-stage CE coding describes a binary sequence by its type class indicator and its rank in the associated type class, which can approach the entropy for biased sources. The random multiple rate coding transforms the variable-length output of the CE coding into a fixed-length channel input, allocating lower energy to lower-rate component codes. At the receiver, the sensing information can be recovered by a trial-and-error (for type classes) decoding either serially or parallelly. The serial decoding has a low implementation complexity, while the parallel decoding has a low decoding delay. To alleviate the mis-correction probability and stop the decoding earlier, we turn to the cyclic redundancy check (CRC) coding. To predict the performance of the proposed JSCC scheme, we present the weighted random-coding union (RCU) bounds based on the conventional RCU bound. The proposed JSCC scheme is universal in the sense that it does not require knowledge of source statistics. Simulation results show that the performance matches well with the presented bounds, validating our analysis. Simulation results also show that the proposed JSCC scheme can outperform the double polar JSCC scheme (exhibiting a coding gain of up to 0.3 dB) and can approach the JSCC bounds (exhibiting a gap of less than 0.5 dB).

We propose a length-flexible coding scheme by defining a balanced tree. For an arbitrary code length, we first construct a balanced binary tree (BBT) where the root node represents a transmitted codeword, the leaf nodes represent either active bits or frozen bits, and a parent node is related to its child nodes by a length-adaptive$(U+V\mid V)$operation. Both the encoding and the successive cancellation (SC)-based decoding can be implemented over the constructed coding tree. For code construction, we propose a signal-to-noise ratio (SNR)-dependent method and two SNR-independent methods, all of which evaluate the reliabilities of leaf nodes and then select the most reliable leaf nodes as the active nodes. Numerical results demonstrate that our proposed codes can have comparable performance to the 5G polar codes. To reduce the decoding latency, we propose a partitioned successive cancellation (PSC)-based decoding algorithm, which can be implemented over a sub-tree obtained by pruning the coding tree. Numerical results show that the PSC-based decoding can achieve similar performance to the conventional SC-based decoding.

In this paper, we present fast encoding algorithms for Hermitian codes based on the Lin-Chung-Han fast Fourier transform (LCH-FFT). For non-systematic encoding, we extend the LCH basis to the bivariate polynomial space and develop a two-dimensional FFT algorithm. For systematic encoding, we propose a modified partial FFT algorithm and present a procedure for computing the unknown intermediates. For a Hermitian code of length $n$, the computational complexity of the presented non-systematic and systematic encoding algorithms are both $O(n{\text{log}}n)$, improving upon the currently best-known encoding complexity $O\left( {n{\text{lo}}{{\text{g}}^2}n{\text{loglog}}n} \right)$.

This paper presents an early-termination method for the Welch-Berlekamp algorithm. Specifically, if e ≤ t (where e is the number of errors and t is the error correction capability), the Welch–Berlekamp algorithm can be terminated at latest at the completion of the (t + e)-th iteration. Based on the early-termination mechanism, a new variant of the Welch–Berlekamp algorithm called eFDMA is presented, and a systolic architecture is designed for the eFDMA algorithm. This provides an efficient implementation for the key equation solver for a new class of Reed–Solomon codes recently proposed by Lin et al. [9].

It is well-known that a linearly coded vector over an erasure channel can be decoded uniquely if the sub-generator matrix formed by the unerased columns has full row rank. This property is generalized in this paper to a necessary and sufficient condition for an information bit to be retrieved uniquely from a received vector with erasures. We show that an information bit of a linear code can be uniquely decodable if and only if its corresponding row of the sub-generator matrix is linearly independent from the other rows. Then we prove that, for semi-random linear codes over binary-input output-symmetric (BIOS) memoryless channels, the information bits associated with the random part of the generator matrices can be decoded with arbitrarily small error probability as the codelength increases. This in turn is applied to prove the coding theorem for time-invariant convolutional codes.

Free-ride coding, as an approach that admits transmission of a few extra bits over a low-density parity-check (LDPC) coded link without bandwidth expansion, is applied in this paper to construct coupled LDPC codes. Firstly, we present a syndrome channel model and derive the lower and upper bounds on its capacity (referred to as accessible capacity), indicating the feasibility of the reliable transmission of extra bits. Secondly, we present the performance evaluation on both the word error rate (WER) and the bit error rate (BER) for the free-ride codes with simple lower and upper bounds. Then we propose three applications of free-ride coding to construct coupled LDPC codes, including implicit globally-coupled LDPC (GC-LDPC) codes, partial product-LDPC codes, and terminated spatially-coupled LDPC (SC-LDPC) codes, all of which have the figure of merits that they share the same code rates with the basic component LDPC codes. Simulation results show that: 1) the proposed GC-LDPC codes can outperform the component LDPC codes, yielding a coding gain of up to 0.8 dB; 2) the proposed product codes with$(3,6)$-regular LDPC component codes of length 1024 can lower the WER from$10^{-2}$down to$10^{-6}$at the SNR around 2 dB; 3) the proposed terminated SC-LDPC codes can perform as well as the conventional terminated SC-LDPC codes but without any rate loss.

In this paper, we propose a new transmission scheme for the scenario where two nodes attempt to exchange messages and the conventional low-density parity-check (LDPC) codes are implemented for error correction. In the proposed scheme, the ACK/NACK feedback information is transmitted along with the payload data by free-ride codes, while the re-transmitted codewords are superimposed (XORed) on the current codewords, both of which cost neither extra bandwidth nor transmission power. Firstly, we present a syndrome channel model and derive its capacity (referred to as accessible capacity) with a lower bound, implying that the reliable transmission of extra bits (feedback information) is possible. Then, the performance of the extra bits is analyzed by the dependency testing (DT) bound for the syndrome channels. Moreover, motivated by the DT bound, we present a low-complexity DT-like decoder for the free-ride codes. For the superposition retransmission, we present a practical implementation, where those unsuccessfully decoded codewords are sparsely interleaved and superimposed onto the current codewords. In addition, the presented transmission scheme is combined with the conventional hybrid automatic repeat request (HARQ) protocol, resulting in a throughput-enhanced conjunction HARQ scheme. Numerical results show that the word error rate (WER) of the LDPC codes can be significantly reduced by using the presented transmission scheme, but without any extra bandwidth or transmission power. They also show that the presented conjunction HARQ schemes can achieve a throughput improvement up to 80% over fading channels in comparison with the original 5G HARQ scheme.

Constructions of high-performance hard-decision decodable error correction codes are important for high-speed communication systems. In this paper, we present the generalized integrated interleaved (GII) zipper codes, in which multiple zipper codes are coupled together by the constraint of the GII code. The resulting codes are referred to as GII-zipper codes. Firstly, since the performances of GII-zipper codes are sensible to miscorrections, we propose an enhanced anchor decoder (AD) which uses the hard-decision results during the GII decoding to assign anchor reliability to reduce miscorrections. We then analyze the size and multiplicity of the minimum-sized stall patterns (MSSPs) of the GII-zipper codes. The analytical results show that the size of the MSSPs of the GII-zipper codes is larger that of the zipper codes. Finally, we present extensive simulation results to show the performance advantages of the GII-zipper codes. The results show that, when decoded with the original AD, the GII-zipper codes perform better than the comparable zipper codes and we can obtain further performance gain by the enhanced AD. Particularly, a GII-staircase code with a rate of 0.846 can achieve 0.74 dB from capacity at a bit error rate (BER) of 10−15.

We present a tandem scheme for Gaussian source compression, where a dead-zone quantizer is concatenated with a ternary low density generator matrix (LDGM) code. Both theoretical analysis and simulation results show that the LDGM codes can be universally optimal for near-lossless compression of ternary sources. Consequently, the distortion with the tandem scheme is mainly caused by the quantization, which can be negligible for high-rate quantizer. The most distinguished feature of the proposed scheme is its flexibility. The dead-zone quantizer can choose a suitable quantization level according to the distortion allowed, while the LDGM codes can adapt the code rate to approach the entropy of the quantized sequence. This is helpful to trade off between bandwith and distortion. In the meanwhile, the proposed scheme is robust when combined with the channel codes for transmission over noisy channels because of the fixed-length feature.

In this paper, we propose a new construction of product codes, where the whole information array is protected row-by-row by a low-density parity-check (LDPC) code while only a portion of the information array is protected column-by-column by an algebraic code. The most distinguished feature of the proposed product code is that, thanks to the free-ride coding technique, the additional column check bits are transmitted implicitly rather than explicitly. The constructed codes are referred to as implicit partial product-LDPC codes, which have the same rates as the row component LDPC codes. The decoding algorithm can be divided into four stages, including decoding of the free-ride codes, first-round decoding of the row codes, decoding of the column codes, and second-round decoding of the row codes by exploiting the messages associated with those successfully decoded columns. To predict the extremely low error rate of the doubly-protected (by both the row code and the column code) information bits, we derive an approximate upper bound. The simulation results show that, with a (3,6)-regular LDPC code of length 1024 as the component code, the proposed product code can lower the word error rate (WER) from 10−2down to 10−6at the SNR around 2 dB. The numerical results also show that the doubly-protected information bits are more reliable, which can have a bit error rate (BER) down to 10−15at SNR around 2.6 dB as implied by the presented approximate upper bound.

A new framework is presented in this paper for proving coding theorems for linear codes, where the systematic bits and the corresponding parity-check bits play different roles. Precisely, the noisy systematic bits are used to limit the list size of typical codewords, while the noisy parity-check bits are used to select from the list the maximum likelihood codeword. This new framework for linear codes allows that the systematic bits and the parity-check bits are transmitted in different ways and over different channels. In particular, this new framework unifies the source coding theorems and the channel coding theorems. With this framework, we prove that the Bernoulli generator matrix codes (BGMCs) are capacity-achieving over binary-input output symmetric (BIOS) channels and also entropy-achieving for Bernoulli sources.

We present a ternary source coding scheme in this paper, which is a special class of low density generator matrix (LDGM) codes. We prove that a ternary linear block LDGM code, whose generator matrix is randomly generated with each element independent and identically distributed, is universal for source coding in terms of the symbol-error rate (SER). To circumvent the high-complex maximum likelihood decoding, we introduce a special class of convolutional LDGM codes, called block Markov superposition transmission of repetition (BMST-R) codes, which are iteratively decodable by a sliding window algorithm. Then the presented BMST-R codes are applied to construct a tandem scheme for Gaussian source compression, where a dead-zone quantizer is introduced before the ternary source coding. The main advantages of this scheme are its universality and flexibility. The dead-zone quantizer can choose a proper quantization level according to the distortion requirement, while the LDGM codes can adapt the code rate to approach the entropy of the quantized sequence. Numerical results show that the proposed scheme performs well for ternary sources over a wide range of code rates and that the distortion introduced by quantization dominates provided that the code rate is slightly greater than the discrete entropy.

In this paper, we propose a new ordered statistics decoding (OSD) for linear block codes, which is referred to as local constraint-based OSD (LC-OSD). Distinguished from the conventional OSD, which chooses the most reliable basis (MRB) for re-encoding, the LC-OSD chooses an extended MRB on which local constraints are naturally imposed. A list of candidate codewords is then generated by performing a serial list Viterbi algorithm (SLVA) over the trellis specified with the local constraints. To terminate early the SLVA for complexity reduction, we present a simple criterion which monitors the ratio of the bound on the likelihood of the unexplored candidate codewords to the sum of the hard-decision vector’s likelihood and the up-to-date optimal candidate’s likelihood. Simulation results show that the LC-OSD can have a much less number of test patterns than that of the conventional OSD but cause negligible performance loss. Comparisons with other complexity-reduced OSDs are also conducted, showing the advantages of the LC-OSD in terms of complexity.

In this paper, we propose a data-aided channel estimator, which can improve the performance of the linear minimum-mean-squared-error (LMMSE) by clustering and reinforcement-learning for multiple-input multiple-output (MI-MO) systems. For clustering-based data detection, we develop a system constrained Gaussian mixture model (SCGMM), in which the a posteriori probabilities (APPs) can be calculated by the expectation-maximization (EM) algorithm. The initial centroids of the SCGMM are sensitive to the channel estimation. To obtain robust channel estimation, we design initial pilots that can reduce the estimated error of the LMMSE. To further improve the quality of channel estimation, we propose a data-aided channel estimation algorithm, which exploits the techniques of coding and reinforcement-learning to obtain soft symbol decisions. Numerical results show that the proposed method can approach the bit-error-rate (BER) performance with perfect channel state information (CSI) in the high signal-to-noise (SNR) region.

In this paper, we present a new construction of the globally-coupled LDPC (GC-LDPC) codes, referred to as implicit GC-LDPC codes, where the additional global parity-check bits are transmitted using the free-ride coding. The presented GC-LDPC codes have the same rates as the component LDPC codes, avoiding the rate reduction caused by the global parity checks for the conventional GC-LDPC codes. Moreover, the encoders of the component LDPC codes are reusable in the presented GC-LDPC codes, a distinguished feature as compared with the conventional GC-LDPC codes. The simulation results show that the proposed implicit GC-LDPC codes can improve the performance of the component LDPC codes, yielding extra coding gain of up to 0.7 dB (with (3,6)-regular LDPC codes as the component codes) and 0.5 dB (with IEEE 802.16e LDPC codes as the component codes).

This paper focuses on the real-time status update in a multiaccess vehicular network, in which multiple vehicles transmit not only the payload data (e.g., monitoring data) but also the extra data (e.g., driving intention) to a road side unit for scheduling vehicles to improve traffic efficiency and safety. A free-ride code is implemented, where the extra data is encoded by a random but fixed generator matrix and is delivered by superposition on the low-density parity-check (LDPC) coded payload data, consuming neither extra bandwidth nor extra transmit power. Considering the time slotted ALOHA random access protocol, we derive the closed-form expression for average age of information (AoI), and evaluate the AoI for both payload data and extra data. Numerical simulations demonstrate that free-ride codes can not only transmit extra data without extra transmit power, but also reduce the average AoI of extra data without affecting the average AoI of payload data.

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.

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.

In this paper, we formulate the problem to cope with the transmission of extra bits over an existing coded transmission link (referred to as coded payload link) without any cost of extra transmission energy or extra bandwidth. This is possible since a gap to the channel capacity typically exists for a practical code. A new concept, termed asaccessible capacity, is introduced to specify the maximum rate at which the superposition transmission of extra bits is reliable and has a negligible effect on the performance of the coded payload link. For a binary-input output-symmetric (BIOS) memoryless channel, the accessible capacity can be characterized as the difference between the channel capacity and the mutual information rate of the coded payload link, which can be numerically evaluated for very short payload codes. For a general payload code, we present a simple lower bound on the accessible capacity, given by the channel capacity minus the coding rate of the payload code. We then focus on the scenarios where low-density parity-check (LDPC) codes are implemented for the payload link. We propose to transmit extra bits by random superposition for encoding, and exhaustive search (with the aid of statistical learning) for decoding. We further propose, by establishing an auxiliary channel (calledsyndrome channel) induced from “zero-forcing” over the binary field, to transmit extra bits with structured codes such as repetition codes and first-order Reed-Muller (RM) codes. Numerical results show that up to 60 extra bits can be reliably transmitted along with a rate-1/2 LDPC code of length 8064.

Age-of-Information (AoI), or simply age, which measures the data freshness, is essential for real-time Internet-of-Things (IoT) applications. On the other hand, energy saving is urgently required by many energy-constrained IoT devices. This paper studies the energy-age tradeoff for status update from a sensor to a monitor over an error-prone channel. The sensor can sleep, sense and transmit a new update, or retransmit by considering both sensing energy and transmit energy. An infinite-horizon average cost problem is formulated as a Markov decision process (MDP) with the objective of minimizing the weighted sum of average AoI and average energy consumption. By solving the associated discounted cost problem and analyzing the Markov chain under the optimal policy, we prove that there exists a threshold optimal stationary policy with only two thresholds, i.e., one threshold on the AoI at the transmitter (AoIT) and the other on the AoI at the receiver (AoIR). Moreover, the two thresholds can be efficiently found by a line search. Numerical results show the performance of the optimal policies and the tradeoff curves with different parameters. Comparisons with the conventional policies show that considering sensing energy is of significant impact on the policy design, and introducing sleep mode greatly expands the tradeoff range.

In this paper, we present a new coding approach to near-lossless compression for binary sparse sources by using a special class of low density generator matrix (LDGM) codes. On the theoretical side, we proved that such a class of block LDGM codes are universal in the sense that any source with an entropy less than the coding rate can be compressed and reconstructed with an arbitrarily low bit-error rate (BER). On the practical side, we employ spatially coupled LDGM codes to reduce the complexity of reconstruction by implementing an iterative sliding window decoding algorithm. Figure of merits of the proposed scheme include its flexibility and universality. The encoder does not require the knowledge of the source statistic, while the decoder can estimate easily the source parameter as required by the iterative decoding. The implementation complexity is analyzed and the performance is simulated. Numerical results show that the proposed scheme performs well over a wide range of sources.

In this paper, we propose a new coding scheme called dual coupled polar (DC-polar) code, which is constructed by coupling two basic polar codes. We present a successive cancellation list (SCL) decoding algorithm, where a list of candidates are generated from the first basic polar code and the most likely one is identified by combining the second basic polar code. For performance analysis and code construction, we derive lower bounds and estimate the performance by the union bounds based on the low-weight codewords. We employ the adaptive SCL decoding for DC-polar codes with early termination to reduce the decoding complexity. It is shown by numerical simulations that the proposed DC-polar codes can attain a near-capacity performance in the short-length regime.

This paper focuses on the age and energy tradeoff of a generalized m-th best relay selection scheme in a dual-hop status update system, where the destination associates to the relay with the m-th smallest number of receptions. By considering the short blocklength packet and retransmission, the expressions of the average age of information (AoI) and the average energy cost (EC) are derived and analyzed. Then, the weighted sum of average AoI and average EC is introduced and is minimized to tradeoff the average AoI and average EC. Numerical results show that for the generalized m-th best relay selection scheme, the optimal block length can be found to achieve the age and energy tradeoff. In addition, the best relay is not always the most suitable one to update the status especially when the weight coefficient of EC is much larger than that of AoI.

In this work, a new coding scheme is proposed for transmitting additional bits along with low-density parity-check (LDPC) coded data at no expense of bandwidth or transmission power. At the transmitter, the LDPC coded bits are first flipped in a random-like way specified by the additional bits and then mapped into a sequence of two-dimensional signals. The modulated sequence is then partitioned into several groups, each of which is rotated by an angle that is also specified by the additional bits, resulting in a sequence of transmitted signals. At the receiver, the additional bits are detected by a serial list decoding algorithm with an early stopping criterion. Then, by removing the effect of the additional bits, the LDPC coded data can be decoded as usual. Simulation results show that, for a rate-1/2 LDPC code of length 8064, up to twenty-four additional bits can be reliably transmitted along with LDPC coded QPSK signaling.

Real-time and energy-efficient transmissions are two critical demands for status update systems, however it is difficult to meet these requirements simultaneously. This paper focuses on the tradeoff between the average age of information (AoI) and the average energy cost (EC) for two-hop decode-and-forward (DF) relaying networks with short packet transmissions. Both the partial relay selection (PRS) and max-min relay selection (MMRS) schemes are considered. The expressions for the average AoI and the average EC of two-hop relaying networks are derived, and the age-energy tradeoff is also achieved by minimizing the weighted sum of them. Numerical simulations show that both PRS and MMRS schemes have their own advantages to minimize the age and energy cost under different channel conditions, where MMRS is a fairness scheme that takes the channel conditions of two hops into account. In addition, an optimal packet length can always be found to tradeoff the average AoI and average EC in two-hop relaying networks.

We propose in this paper a new coding scheme called twisted-pair superposition transmission (TPST). The encoding is to “mix together” a pair of basic codes by superposition, while the decoding can be implemented as a successive cancellation list decoding algorithm. The most significant features of the TPST code are its predictable performance that can be estimated numerically from the basic codes and its flexible construction in the sense that it can be easily adapted to different coding rates. To construct good TPST codes in the finite length regime, we propose two design approaches – rate allocation and partial superposition. By taking tail-biting convolutional codes (TBCC) as basic codes, we show by numerical results that the constructed TPST-TBCCs have performance close to the random coding union bound in the short length regime.

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.

Age of information (AoI) and energy efficiency (EE), as two important performance metrics in status update systems, usually cannot be simultaneously optimized. This paper investigates the age and energy tradeoff for multicast networks with retransmissions, where each sensed status update is encoded as a short blocklength packet and is broadcasted to multiple destinations via independent and identically distributed error-prone channels. By considering the stopping threshold, the average AoI and EE expressions for stopping at earliest-l, stopping at preselected-l, and wait-for-all schemes are derived as a function of the packet length. On this basis, the average AoI-EE ratio is introduced and is minimized to trade off age and energy by optimizing the packet length. Numerical results indicate that among the three schemes, earliest-lscheme attains the minimum AoI when the packet length is small, and preselected-lscheme attains the maximum EE when the packet length is large. An optimal packet length can always be found for each scheme to minimize age-energy ratio. Moreover, the selection of stopping thresholdlwill also impact the age and energy performance.

In this paper, we propose a systematic low density generator matrix (LDGM) code ensemble, which is defined by the Bernoulli process. We prove that, under maximum likelihood (ML) decoding, the proposed ensemble can achieve the capacity of binary-input output symmetric (BIOS) memoryless channels in terms of bit error rate (BER). The proof technique reveals a new mechanism, different from lowering down frame error rate (FER), that the BER can be lowered down by assigning light codeword vectors to light information vectors. The finite length performance is analyzed by deriving an upper bound and a lower bound, both of which are shown to be tight in the high signal-to-noise ratio (SNR) region. To improve the waterfall performance, we construct the systematic convolutional LDGM (SysConv-LDGM) codes by a random splitting process. The SysConv-LDGM codes are easily configurable in the sense that any rational code rate can be realized without complex optimization. As a universal construction, the main advantage of the SysConv-LDGM codes is their near-capacity performance in the waterfall region and predictable performance in the error-floor region that can be lowered down to any target as required by increasing the density of the uncoupled LDGM codes. Numerical results are also provided to verify our analysis.

In this paper, we propose a novel coding scheme, which is referred to as twisted-pair superposition transmission (TPST) and can be constructed from any given basic code by "mixing together" a pair of basic codewords in a "twisted" manner. We present a successive cancellation list decoding algorithm for TPST, where a list of candidates for the first layer is generated serially and the most competitive one is identified by combining the second layer. Thresholds on empirical divergence function (EDF) are introduced for early termination to trade off performance with decoding complexity. Genie-aided bounds are derived, indicating that the performance of TPST codes can be improved by employing partial superposition. Numerical simulation results show that, by taking tail-biting convolutional codes (TBCCs) as basic codes, we can construct TPST-TBCCs with near-capacity performance in the short length regime. The construction is flexible in the sense that it can be easily adapted to a wide range of coding rates.

Age of information (AoI) and energy efficiency (EE) are two critical metrics for real-time status update systems. However, these two metrics may not be optimized simultaneously. This paper focuses on the tradeoff between the average AoI and EE of short packets based transmissions in a multicast network with automatic repeat request. The fixed redundancy coding scheme is employed, and the encoded packet is broadcasted to destinations over additive white Gaussian noise channels. The expressions of average AoI and EE are derived and analyzed. In particular, the average AoI-EE ratio is proposed, which is minimized to achieve a tradeoff between average AoI and EE. The numerical results show that there exists an optimal packet length to achieve a compromise between average AoI and EE.

In this paper, we propose a new multi-level unequal error protection (UEP) by superposition transmission (referred to as ML-UEP-by-ST) coding scheme, which provides finer error protection abilities than the unequal error protection by partial superposition transmission (referred to as UEP-by-PST) coding scheme. This new coding scheme is then applied to video transmission with the H.265 standard, where a video bitstream can be regarded as a series of one or more groups. Each group consists of either a coded video sequence (CVS) and parameter sets or a CVS only. In the ML-UEP-by-ST system, each group of an H.265 video bitstream is partitioned equally into three parts, the most important part (Part A), the less important part (Part B) and the least important part (Part C). Each of these three parts is encoded by the same low-density parity check (LDPC) code as standardized in the fifth generation mobile networks(5G). The transmission is then formed by three sections. The first transmission is coded Part A, the second transmission is the superposition of coded Part B and the interleaved version of coded Part A, and the third transmission is the superposition of coded Part C and the interleaved version of the second transmission. Simulation results show that the performance of our proposed UEP scheme is better than the traditional equal error protection (EEP) scheme and the two-level UEP-by-PST scheme over both additive white Gaussian noise (AWGN) channels and Rayleigh fading channels.

Age of Information (AoI) is a fundamentally important metric to characterize the freshness of information in real-time Internet-of-Things (IoT) monitoring systems. Another important metric is the energy cost for information sensing and transmission. In this article, we investigate the average AoI and energy cost for low-density parity-check coded status update with and without automatic repeat request (ARQ), where the fixed redundancy scheme is employed. The non-ARQ, classical ARQ, truncated ARQ, and truncated hybrid ARQ with chase combining (HARQ-CC) schemes are analyzed and compared. By using the renewal processes theory, the expressions for the average AoI as well as the average energy cost of each considered scheme are derived. Both the lower bound of age and the upper bound of energy are provided. It is shown through simulation results that the average AoI and energy cost are mainly influenced by network parameters in the low signal-to-noise ratio (SNR) region. With short code, the smaller average AoI can be obtained at the cost of more energy consumption. Compared with other schemes, the truncated HARQ-CC achieves the best average AoI and the moderate average energy cost, which is a compromise between the age and energy.

In this paper, we introduce a novel method, called doubly-recursive block Markov superposition transmission (DrBMST), to construct high-performance spatially coupled codes. An important characteristic of DrBMST codes is that the degrees of the constraint nodes in their normal graphs are at most three. As a result, DrBMST codes can be decoded with low complexity. We first prove that the error probability of an enlarged DrBMST code ensemble can be made arbitrarily small under windowed maximum-likelihood decoding (MLD) by increasing the decoding window size. This result partially explains the superior performances of DrBMST codes. Then we propose to use the extrinsic information transfer (EXIT) chart analysis to estimate the iterative windowed decoding thresholds of DrBMST codes. The EXIT chart analyses show that, with such a simple structure, DrBMST codes are comparable to BMST codes with large encoding memories in terms of decoding thresholds. Finally, we carry out comparisons to validate the advantages of DrBMST codes in terms of error performances and decoding complexities. In particular, for a decoding latency of 20,000 bits, the DrBMST code performs better than the (4, 8)-regular spatially coupled low-density parity-check (SC-LDPC) code, but with lower computational complexity. Hence, DrBMST codes can be used in communication systems with limited computational resources. In addition, we show that DrBMST can be used to construct multiple-rate codes.

This paper is concerned with block Markov superposition transmission (BMST) of tail-biting convolutional code (TBCC). We propose a new decoding algorithm for BMST-TBCC, which integrates a serial list Viterbi algorithm (SLVA) with a soft check instead of conventional cyclic redundancy check (CRC). The basic idea is that, compared with an erroneous candidate codeword, the correct candidate codeword for the first sub-frame has less influence on the output of Viterbi algorithm for the second sub-frame. The threshold is then determined by statistical learning based on the introduced empirical divergence function. The numerical results illustrate that, under the constraint of equivalent decoding delay, the BMST-TBCC has comparable performance with the polar codes. As a result, BMST-TBCCs may find applications in the scenarios of the streaming ultra-reliable and low latency communication (URLLC) data services.

In this paper, we present a new class of spatially coupled low-density parity-check (SC-LDPC) codes, which are constructed by sending codewords of LDPC block code (LDPCBC) in a block Markov superposition transmission (BMST) manner. Different from the conventional SC-LDPC codes, the proposed SC-LDPC codes can have encoder/decoder implemented with the basis of the hardware components of the corresponding LDPC-BCs. The proposed SC-LDPC codes are also a special class of BMST-LDPC codes. Distinguished from other types of BMST codes, BMST-LDPC codes have lower error floors even with an encoding memory of one and hence have lower decoding latency. Also different from the original BMST codes, partial superposition is implemented to alleviate error propagation. To analyze the bit error rate (BER) performance, we present the genie-aided (GA) bounds, which can be obtained by simulation or estimated from the performance of the basic code. Numerical results are presented to validate our analysis and demonstrate the performance advantage of the BMST-LDPC codes over the LDPC-BCs.

Age of information (AoI) is an important metric in real-time status update communication system to assess the freshness of information. Different from previous works, this paper focuses on the average AoI over additive white Gaussian noise (AWGN) channels. A fixed redundancy (FR) coding scheme is considered, which encodes each k-bits update as an n-bits packet by a low-density parity-check (LDPC) code. By using the renewal-reward theory, a closed-form expression of the average AoI under the FR scheme is derived. Simulation results show that the average AoI relies on Eb/N0, especially in the low Eb/N0region. For different Eb/N0, an optimal code length always exists to minimize the average AoI. In addition, the same average AoI can be achieved with different code lengths in the high Eb/N0region. Hence, in the high Eb/N0region, short codes are preferred especially when the transmission delay is taken into account.

In this study, the authors propose a class of exclusive OR (XOR)‐based maximum distance separable (MDS) array codes, referred to as horizontal–vertical–diagonal (HVD) codes, which can tolerate triple disk failures with optimal update complexity. The HVD code has a similar data/parity layout to the horizontal–vertical (HV) code and has many figures of merits as the HV code. Also, an efficient recovery algorithm is proposed, which can be implemented to accelerate the recovery process of double disk failures by making full use of all kinds of parities in the HVD code. The authors evaluate the encoding/decoding efficiency of the HVD code by comparing the number of XORs and the number of disk reads with other MDS array codes. Results show that the HVD code inherits most of the merits of the HV code and requires less number of disk reads when recovering single disk failure.

Internet of Vehicles (IoV) is the basis of future intelligent transportation systems. Both the control signaling and data dissemination services in IoV must be transmitted with high reliability and low latency so that safety can be guaranteed. Based on the discussion and analyses of issues in achieving high reliability low latency transmissions, we present in this paper a Polar code-based overlapping transmission scheme in which simultaneous transmissions from different nodes to the same receiving node are allowed to use the same time-frequency resource block. To effectively eliminate multiple access interference introduced by the overlapped nonorthogonal transmissions, a successive cancellation list-based improved interference elimination decoding algorithm (SCL-based IIEDA) is proposed to retrieve the Polar coded information. Numerical results show that the SCL-based IIEDA performs well on information recovery in the presented transmission scheme. In addition, it is shown that the proposed scheme not only significantly reduces the acknowledgment overhead but also greatly improves the spectral efficiency.

In this paper, we present the block Markov superposition transmission of BCH (BMST-BCH) codes, which can be constructed to obtain a very low error floor. To reduce the implementation complexity, we design a low complexity iterative sliding-window decoding algorithm, in which only binary and/or erasure messages are processed and exchanged between processing units. The error floor can be predicted by the proposed genie-aided lower bounds, while the waterfall performance can be analyzed by the density evolution method. To evaluate the error floor of the constructed BMST-BCH codes at a very low bit error rate (BER) region, we propose a fast simulation approach. Numerical results show that, at a target BER of 10-15, the proposed BMST-BCH code with hard-decision can achieve a net coding gain (NCG) of 10.55 dB with 25% overhead, while a soft-decision design can yield an NCG of 10.74 dB. The construction of BMST-BCH codes is flexible to trade off latency against performance at all overheads of interest and may find applications in optical transport networks as an attractive candidate.

Age-of-information is a novel performance metric in communication systems to indicate the freshness of the latest received data, which has wide applications in monitoring and control scenarios. Another important performance metric in these applications is energy consumption, since monitors or sensors are usually energy constrained. In this paper, we study the energy-age tradeoff in a status update system where data transmission from a source to a receiver may encounter failure due to channel error. As the status sensing process consumes energy, when a transmission failure happens, the source may either retransmit the existing data to save energy for sensing, or sense and transmit a new update to minimize age-of- information. A threshold-based retransmission policy is considered where each update is allowed to be transmitted no more than M times. Closed- form average age-of-information and energy consumption is derived and expressed as a function of channel failure probability and maximum number of retransmissions M. Numerical simulations validate our analytical results, and illustrate the tradeoff between average age-of-information and energy consumption.

This paper is concerned with the delayed bit-interleaved coded modulation with iterative decoding (DBICM-ID). We present new criteria for bit-labeling in the DBICM-ID system based on the harmonic mean of the minimum squared Euclidean distance (HMMSED) and the mean average bit-wise mutual information (MABMI) criteria. Different from the conventional HMMSED criterion, taking into account the conditions varying from no feedback to perfect feedback, we also evaluate the variance of the harmonic mean, which is then deployed to reveal the convergence rate. We take 16-QAM as an example to confirm our analysis. Numerical results show that, DBICM-ID with the designed bit-labeling scheme can obtain an extra coding gain of about 0.5 dB.

In this paper, we introduce the doubly recursive block Markov superposition transmission (DrBMST) of short code. An important characteristic of DrBMST codes is that the degrees of the constraint nodes in their normal graphical realizations are at most three. As a result, DrBMST codes can be decoded with low complexity. We propose to use an enlarged code ensemble to analyze the performance of DrBMST under windowed maximum likelihood decoding. Further, the extrinsic information transfer (EXIT) chart analysis is used to study the iterative decoding thresholds of DrBMST code ensembles. The EXIT analysis shows that the iterative decoding thresholds of DrBMST code ensembles are comparable to those of the BMST codes. We have also compared the error performance and the decoding complexity of finite-length DrBMST codes with regular spatial-coupled low-density parity-check (SC-LDPC) codes under equal decoding latency. The comparison results show that the DrBMST code performs about 0.1 dB better than a (4, 8)-regular SC-LDPC code, but with lower computational complexity.

This paper is concerned with three ensembles of systematic low density generator matrix (LDGM) codes, all of which were provably capacity-achieving in terms of bit error rate (BER). This, however, does not necessarily imply that they achieve the capacity in terms of frame error rate (FER), as seen from a counterexample constructed in this paper. We then show that the first and second ensembles are capacity-achieving under list decoding over binary-input output symmetric (BIOS) memoryless channels. We point out that, in principle, the equivocation due to list decoding can be removed with negligible rate loss by the use of the concatenated codes. Simulation results show that the considered convolutional (spatially-coupled) LDGM code is capacity-approaching with an iterative belief propagation decoding algorithm.

This paper is concerned with coded multiple-input multiple- output orthogonal frequency division multiplexing system with index modulation (MIMO-OFDM-IM) for high mobility wireless communications. To improve the bit-error rate (BER) performance, the block Markov superposition transmission (BMST) codes are constructed and combined with MIMO-OFDM-IM, where a genie-aided lower bound is also presented to predict the performance of the BMST- MIMO-OFDM-IM in the low-BER region. At the receiver, submatrix based linear minimum mean square error (LMMSE) equalization is employed in conjunction with the soft demapping algorithm. Simulation results demonstrate that the BMST-MIMO-OFDM-IM can achieve significant coding gain compared with BMST-MIMO-OFDM over doubly selective channels.

This study is concerned with coded multiple‐input multiple‐output (MIMO) visible light communication (VLC) system, where unipolar M ‐level pulse amplitude modulation is assumed along with MIMO transmission techniques, including repetition coding, spatial multiplexing and spatial modulation. In order to evaluate the performance of MIMO‐VLC system, the authors analyse the mutual information of the MIMO techniques, based on which the block Markov superposition transmission (BMST) codes are constructed to improve the bit‐error rate (BER) performance of the system. In addition, they present a genie‐aided lower bound to predict the performance of the BMST‐VLC system in the low‐BER region. Simulation results demonstrate that the proposed scheme can achieve significant coding gain compared with Reed‐Solomon‐coded MIMO‐VLC system adopted in IEEE standard 802.15.7.

Extensive studies have demonstrated the effectiveness and the flexibility of constructing capacity-approaching codes by block Markov superposition transmission (BMST). However, to achieve high performance, BMST codes typically require large encoding memories and large decoding window sizes, which result in high decoding complexity and high decoding latency. To address these issues, we introduce the recursive BMST (rBMST), in which the block-oriented feedback convolutional code is used instead of the block-oriented feedforward convolutional code of BMST. We propose to use a modified extrinsic information transfer chart analysis, which relates the mutual information to the bit error rate, to study the convergence behaviors of rBMST codes. On one hand, rBMST code shares most merits of BMST code, including near-capacity performance, low-complexity encoding, and flexible construction. On the other hand, compared with BMST code, rBMST code requires a smaller encoding memory, hence a lower decoding complexity, to approach the capacity. In particular, both analytical and simulation results show that rBMST code with encoding memory three reveals a lower error floor than the BMST code with encoding memory twelve. Furthermore, we show by analysis and simulations that rBMST with fixed encoding memory (m = 3 ) and fixed decoding delay (d = 12 ) can be used to construct capacity-approaching multiple-rate codes. Finally, the comparison between rBMST codes and spatially coupled low-density parity-check codes is carried out, which shows the advantages of rBMST codes in terms of performances and decoding complexities.

In this paper, we propose systematic block Markov superposition transmission of repetition (BMST-R) codes, which can support a wide range of code rates but maintain essentially the same encoding/decoding hardware structure. The systematic BMST-R codes resemble the classical rate-compatible punctured convolutional codes, except that they are typically non-decodable by the Viterbi algorithm due to the huge constraint length induced by the block-oriented encoding process. The information sequence is partitioned equally into blocks and transmitted directly, while their replicas are interleaved and transmitted in a block Markov superposition manner. By taking into account that the codes are systematic, we derive both upper and lower bounds on the bit-error-rate (BER) under maximum a posteriori decoding. The derived lower bound reveals connections among BER, encoding memory and code rate, which provides a way to design good systematic BMST-R codes and also allows us to make trade-offs among efficiency, performance, and complexity. Numerical results show that: 1) the proposed bounds are tight in the high signal-to-noise ratio region; 2) systematic BMST-R codes perform well in a wide range of code rates; and 3) rate 1/2 systematic BMST-R codes outperform the considered (3,6)- and (4,8)-regular spatially coupled low-density parity-check codes under an equal decoding latency constraint.

This paper is concerned with block Markov super-position transmission of BCH (BMST-BCH) codes. Compared with other BMST codes, BMST-BCH codes can achieve a lower error floor with an encoding memory of two, which is critical to reduce both delay and implementation complexity. To further reduce the implementation complexity, we propose a hard-decision sliding-window decoding algorithm, in which only binary and/or erasure messages are processed and exchanged between nodes. A fast simulation approach is proposed, with the help of the genie-aided lower bound and the density evolution analysis, to evaluate the performance of BMST-BCH codes at the BER of 10-15. BMST-BCH codes are constructed with overheads ranging from 15% to 25%, exhibiting performances comparable to staircase codes with similar latencies. The proposed construction is more flexible to trade off latency against performance, and may find applications in optical transport networks as an attractive candidate.

Extensive studies have demonstrated the effectiveness of constructing capacity-approaching codes by block Markov superposition transmission (BMST). However, to achieve high performance, BMST codes typically require large encoding memories and large decoding window sizes, which result in increased decoding complexity and decoding latency. To address this issue, we introduce the recursive BMST (rBMST), in which block-oriented feedback convolutional code is used instead of the block-oriented feedforward convolutional code. We propose to use a modified extrinsic information transfer (EXIT) chart analysis to study the convergence behavior of rBMST codes. On one hand, rBMST code shares most merits of BMST code, including near-capacity performance, low-complexity encoding, and flexible construction. On the other hand, compared with BMST code, rBMST code requires a smaller encoding memory, hence a lower decoding complexity, to approach the capacity. In particular, analytical results show that, rBMST code ensemble with encoding memory three reveals a lower error-floor than the BMST code ensemble with encoding memory twelve.

The focus of this paper is the high mobility coded wireless communication system with Doppler diversity. In order to improve the bit-error-rate (BER) performance, we combine the systematic block Markov superposition transmission (BMST) with Doppler diversity, resulting in BMST-DD, where the systematic BMST is a recently proposed rate-compatible coding scheme. In the BMST-DD system, to reduce the performance loss caused by the channel estimation error, we propose an iterative channel estimation and decoding technique for the fast time-varying fading channels. In the iteration process, the recovered symbols are fed back to the channel estimator as the new pilot symbols to improve the accuracy of the channel state information (CSI) estimation. Numerical results show that the mutual information increases with the increase of the speed due to Doppler diversity, and that the BER performance of the BMST-DD system can be improved by the iteration between the channel estimator and the decoder.

This paper proposes an interpolation based progressive algebraic Chase decoding (PACD) algorithm for Reed-Solomon (RS) codes. Based on the received information, 2η (η > 0) interpolation test-vectors are constructed. They are ordered using a reliability function, assessing their potential of yielding the intended message. The decoding is performed progressively granting priority to decode the test-vectors that are more likely to yield the intended message, and it will be terminated once the intended message is found. In the proposal, the decoding of a later test-vector utilizes the interpolation information that is generated during the decoding of the earlier ones. It results in the binary tree that represents the evolution of the interpolated polynomial sets growing in a depth-first-search manner. The PACD algorithm has the advantage of adapting its decoding computation to the channel condition, leveraging the average decoding complexity. This channel dependent feature will be validated by our simulation results which show that the PACD algorithm is less complex than various interpolation based algebraic decoding algorithms. We will also demonstrate that it can achieve a high RS decoding performance.

In this paper, we propose systematic block Markov superposition transmission of repetition (BMST-R) codes, which can support a wide range of code rates but maintain essentially the same encoding/decoding hardware structure. The systematic BMST-R codes resemble the classical rate-compatible punctured convolutional (RCPC) codes, except that they are typically non-decodable by the Viterbi algorithm due to the huge constraint length induced by the block-oriented encoding process. By taking into account that the codes are systematic, the performance of systematic BMST-R codes under maximum a posteriori (MAP) decoding can be analyzed with a simple lower bound and an upper bound with the help of partial input-redundancy weight enumerating function (IRWEF). Numerical results verify our analysis and show that systematic BMST-R codes perform well in a wide range of code rates.

One primary challenge in the high-speed train (HST) communication is that the wireless signal suffers from severe attenuation while it penetrates the sealed carriage of the train. In this letter, we show that such a nuisance can provide an opportunity of improving the uplink capacity when a full-duplex (FD) mobile relay (MR) is furnished on the train. To this end, by introducing the self-interference cancellation (SIC) level, we present an uplink channel model for the HST communication with FD MR. Then the uplink capacity is analyzed under either user equipment power constraint or total power constraint. By comparing with time division scheme, we can derive the required SIC level for the FD scheme to bring capacity improvement, which is confirmed by numerical results. Furthermore, theoretical analysis shows that, for a given SIC level, the uplink channel with severer carriage attenuation has higher capacity, which is also validated by numerical results.

This paper is concerned with the coded orthogonal frequency division multiplexing (OFDM) with index modulation (IM) for highly mobile wireless communication based on block Markov superposition transmission (BMST). In the OFDM-IM scheme, the information bits are not only mapped into the conventional two-dimensional signal points but also into the indices of the subcarriers. For the coded IM system, we choose BMST codes, which can be easily designed for any given code rate with a predictable performance lower bound. In this paper, linear minimum mean square error (LMMSE) equalization is used to mitigate the intercarrier interference (ICI) and exchange messages with the soft demapper. An iterative sliding-window decoding algorithm is also presented to the BMST decoder. To reduce the complexity, we ignore the interference between different groups of IM system. Numerical results show that the BMST-IM system performs well over time-varying frequency-selective channels and outperforms BMST-OFDM scheme at the same spectral efficiency.

This study proposes a progressive algebraic Chase decoding (PACD) algorithm for Reed–Solomon (RS) codes. On the basis of the received information, 2 η ( η > 0) interpolation test‐vectors are constructed for the interpolation‐based algebraic Chase decoding. A test‐vector reliability function is defined to assess their potential for yielding the intended message. The algebraic Chase decoding will then be performed progressively granting priority to decode the test‐vectors that are more likely to yield the message, and is then terminated once it is found. Consequently, the decoding complexity can be adapted to the quality of the received information. An enhanced‐PACD (E‐PACD) algorithm is further proposed by coupling the PACD algorithm with the adaptive belief propagation (ABP) decoding. The ABP decoding generates new test‐vectors for the PACD algorithm by enhancing the received information. It improves the Chase decoding performance without increasing the decoding complexity exponentially. It is shown that the E‐PACD algorithm's complexity can be significantly reduced by utilising the existing interpolation information of the previous Chase decodings’. Our performance evaluations show that the two proposed decoders outperform a number of existing algebraic decoding approaches. Complexity and memory analyses of the PACD algorithm are also presented, demonstrating that this is an efficient RS decoding strategy.

In this paper, we consider the asymptotic and finite-length performance of block Markov superposition transmission (BMST) of short codes, which can be viewed as a new class of spatially coupled (SC) codes where the generator matrices of short codes (referred to as basic codes) are coupled. A modified extrinsic information transfer (EXIT) chart analysis that takes into account the relation between mutual information (MI) and bit-error-rate (BER) is presented to study the convergence behavior of BMST codes. Using the modified EXIT chart analysis, we investigate the impact of various parameters on BMST code performance, thereby providing theoretical guidance for designing and implementing practical BMST codes suitable for window decoding. Then, we present a performance comparison of BMST codes and SC low-density parity-check (SC-LDPC) codes on the basis of equal decoding latency. Also presented is a comparison of computational complexity. Simulation results show that, under the equal decoding latency constraint, BMST codes using the repetition code as the basic code can outperform both (3,6)-regular SC-LDPC codes and (4,8)-regular SC-LDPC codes in the waterfall region but have a higher computational complexity.

In this paper, we propose a simple procedure to construct (decodable) good codes with any given alphabet (of moderate size) for any given (rational) code rate to achieve any given target error performance (of interest) over additive white Gaussian noise channels. We start with constructing codes over groups for any given code rates. This can be done in an extremely simple way if we ignore the error performance requirement for the time being. Actually, this can be satisfied by repetition (R) codes and uncoded (UN) transmission along with time-sharing technique. The resulting codes are simply referred to as RUN codes for convenience. The encoding/decoding algorithms for RUN codes are almost trivial. In addition, the performance can be easily analyzed. It is not difficult to imagine that the RUN code usually performs far away from the corresponding Shannon limit. Fortunately, the performance can be improved as required by spatially coupling the RUN codes via block Markov superposition transmission (BMST), resulting in the BMST-RUN codes. Simulation results show that the BMST-RUN codes perform well (within around 1 dB away from Shannon limits) for a wide range of code rates and outperform the BMST with bit-interleaved coded modulation scheme.

This paper is concerned with the spatial modulation (SM), a multiple-input multiple-output (MIMO) transmission technique, that maps information bits not only into the conventional two-dimensional signal points but also into the indices of active transmit antennas. We present a two-layer coded SM scheme, in which the spatial bits carried by the antenna indices and the signal bits carried by the conventional signals are protected separately by two error correction codes. For the ease of decoding process, the code rates are allocated according to the chain rule of the mutual information. We choose block Markov superposition transmission (BMST) codes for each layer, since they are easily designed for any given code rate with a predictable performance lower bound. An iterative sliding-window decoding algorithm is also presented by exchanging messages iteratively between the two BMST decoders and the soft-in soft-out (SISO) demapper of the SM. To reduce the computational complexity, we propose to implement the SISO demapping algorithm by employing only partial soft inputs. Numerical results show that the BMST-SM system performs well over uncorrelated Rayleigh fading channels.

Structural properties of array-based non-binary low-density parity-check (NBLDPC) codes are studied in this paper. First, we characterize graphical substructures induced by codewords of symbol weight six in array-based NBLDPC codes defined by parity-check matrices with column weight three. We also reveal necessary conditions for these graphical substructures to incur weight-6 codewords. Such conditions can be used to select nonzero elements for avoiding weight-6 codewords or reducing the multiplicity of weight-6 codewords. Second, we show that there exist weight-7 codewords in array-based NBLDPC codes defined by parity-check matrices with column weight three. As a byproduct, we find that the graphical substructure induced by a weight-7 codeword takes the graphical substructure induced by the related weight-6 codewords as a subgraph. Third, we show that there may exist codewords with symbol weight four, six, and seven in array-based NBLDPC codes defined by parity-check matrices with column weight two. These results enrich the structural analysis of array-based LDPC codes. In addition, simulation results show the performance advantage of array-based NBLDPC codes.

This paper studies the transmission of Gaussian sources through additive white Gaussian noise channels in bandwidth expansion regime, i.e., the channel bandwidth is greater than the source bandwidth. To mitigate the error propagation phenomenon of conventional digital transmission schemes, we propose in this paper a new capacity-approaching joint source channel coding (JSCC) scheme based on partially block Markov superposition transmission (BMST) of nested lattice codes. In the proposed scheme, first, the Gaussian source sequence is discretized by a lattice-based quantizer, resulting in a sequence of lattice points. Second, these lattice points are encoded by a short systematic group code. Third, the coded sequence is partitioned into blocks of equal length and then transmitted in the BMST manner. The main characteristics of the proposed JSCC scheme include: 1) entropy coding is not used explicitly and 2) only parity-check sequence is superimposed, hence, termed partially BMST. This is different from the original BMST. To show the superior performance of the proposed scheme, we present extensive simulation results which show that the proposed scheme performs within 1 dB of the Shannon limits. Hence, the proposed scheme provides an attractive candidate for transmission of Gaussian sources.

This paper studies an extension and improvement of the joint detection-decoding algorithm for nonbinary LDPC-coded modulation systems. The iterative joint detection-decoding (IJDD) algorithm in [1] combines nonbinary LDPC decoding with signal detection based on the hard-message passing strategy, resulting in significantly reduced decoding complexity. However, it applies only to majority-logic decodable nonbinary LDPC codes with high column weight. For nonbinary LDPC codes with low column weight, a noticeable performance loss will be incurred. To handle this problem, we propose a reliability-based iterative joint detection-decoding (also termed improved IJDD) algorithm, which combines the accumulated reliability of symbols based on the one-step majority-logic decoding (MLGD) algorithm and a Chase-like local list decoding algorithm. Simulation results show that the improved IJDD algorithm outperforms the IJDD algorithm by about 0.3 dB using nonbinary LDPC codes with high column weight, and by about 3 dB using nonbinary LDPC codes with low column weight (dv= 4), while maintaining the low complexity of decoding. Compared to the FFT-QSPA, the proposed algorithm has a performance degradation of 0.5 dB in the high column weight regime, and about 1 dB in the low column weight regime.

In this paper, a modified extrinsic information transfer (EXIT) chart analysis that takes into account the relation between mutual information (MI) and bit-error-rate (BER) is presented to study the convergence behavior of block Markov superposition transmission (BMST) of short codes (referred to as basic codes). We show that the threshold curve of BMST codes using an iterative sliding window decoding algorithm with a fixed decoding delay achieves a lower bound in the high signal-to-noise ratio (SNR) region, while in the low SNR region, due to error propagation, the thresholds of BMST codes become slightly worse as the encoding memory increases. We also demonstrate that the threshold results are consistent with finite-length performance simulations.

In this paper, improved Hamming sphere bounds on the maximum-likelihood decoding (MLD) performance of binary linear codes over additive white Gaussian noise (AWGN) channels are proposed. The proposed Hamming sphere bounds both on the frame and bit error probabilities are based on Gallager's first bounding technique (GFBT), where the “good” region is chosen to be an Hamming sphere, which is different from the conventional definition by the Euclidean distance. The good region is then divided into small regions to tighten the union bound on the error probability caused by the Hamming sphere. The proposed bounds require only the knowledge of the truncated weight spectrum of the code, which is helpful when the whole weight spectrum is unknown or not computable. Numerical results show that the proposed bounds are tighter than most of the upper bounds and even tighter than the tangential-sphere bound (TSB) for high code rates.

Kite codes are a class of rateless FEC codes designed for the noisy channel. In this paper, we propose a new ensemble of nonbinary rate-compatible (RC) LDPC codes, which is constructed based on Kite codes. The proposed nonbinary RC codes (nonbinary Kite codes, for simplicity) possess coding rate varying “continuously” from 0.3 to 0.9. Moreover, the degree distribution varies with the incremental redundancy. Simulation results show that the nonbinary Kite codes outperform their binary counterparts with the BPSK modulation in a wide range of coding rates over the AWGN channel. We also examine the application of nonbinary Kite codes to fountain communications. Numerical results show that the average throughput achievable with nonbinary Kite codes can be close to the capacity within a wide region of SNRs.

In this paper, asymptotic methods are used to form lower and upper bounds on the typical free distance growth rate of ensembles of periodically time-varying protograph-based spatially coupled low-density parity-check (SC-LDPC) codes over GF(q). By evaluating and comparing these bounds, we find that the typical free distance of q-ary SC-LDPC codes increases linearly with constraint length and that the bounds coincide for a sufficiently large period. In particular, we show that the free distance to constraint length ratio of (3, 6)-regular q-ary SC-LDPC code ensembles exceeds the minimum distance to block length ratio of an underlying q-ary LDPC block code (LDPC-BC) ensemble. We also show that, similar to the minimum distance growth rate of the (3, 6)-regular q-ary LDPC-BC ensemble, the free distance growth rate of (3, 6)-regular q-ary SC-LDPC code ensembles increases with the field size q up to a certain point, and then it decreases as q increases further.

In this study, the authors’ present a scheme, denoted as BMST‐MSK, which combines the block Markov superposition transmission (BMST) with the minimum shift keying (MSK) signalling. The BMST‐MSK can be implemented in two forms – the BMST with recursive MSK (BMST‐RMSK) and the BMST with non‐recursive MSK (BMST‐NRMSK). The BMST‐MSK admits a sliding‐window decoding/demodulation algorithm, where two schedules with or without iterative processing between the BMST and MSK (referred to as outer iteration) are discussed. To analyse the asymptotic performance of BMST‐MSK, the authors’ first assume a genie‐aided decoder and then derive the union bound for the equivalent genie‐aided system. Numerical results show that the performances of the BMST‐MSK match well with the derived lower bounds in the low error rate regions. From simulations, the authors’ found that the outer iterations can provide performance improvement for the BMST‐RMSK, but not for the BMST‐NRMSK. Taking a (2,1,2) convolutional code with input length of 10 000 bits as the basic code, the BMST‐NRMSK achieves a bit‐error‐rate of 10 −5 at E b / N 0 = 0.45 dB over additive white Gaussian noise channels, which is away from the Shannon limit about 0.25 dB.

In this paper, we compare the finite-length performance of protograph-based spatially coupled low-density paritycheck (SC-LDPC) codes and LDPC block codes (LDPC-BCs) over GF(q). To reduce computational complexity and latency, a sliding window decoder with a stopping rule based on a soft belief propagation (BP) estimate is used for the q-ary SC-LDPC codes. Two regimes are considered: one when the constraint length of q-ary SC-LDPC codes is equal to the block length of q-ary LDPC-BCs and the other when the two decoding latencies are equal. Simulation results confirm that, in both regimes, (3,6)-, (3,9)-, and (3,12)-regular non-binary SC-LDPC codes can significantly outperform both binary and non-binary LDPC-BCs and binary SC-LDPC codes. Finally, we present a computational complexity comparison of q-ary SC-LDPC codes and q-ary LDPC-BCs under equal decoding latency and equal decoding performance assumptions.

A construction of big convolutional codes from short codes called block Markov superposition transmission (BMST) is proposed. The BMST is very similar to superposition block Markov encoding (SBME), which has been widely used to prove multiuser coding theorems. The BMST codes can also be viewed as a class of spatially coupled codes, where the generator matrices of the involved short codes (referred to as basic codes) are coupled. The encoding process of BMST can be as fast as that of the basic code, while the decoding process can be implemented as an iterative sliding-window decoding algorithm with a tunable delay. More importantly, the performance of BMST can be simply lower bounded in terms of the transmission memory given that the performance of the short code is available. Numerical results show that: 1) the lower bounds can be matched with a moderate decoding delay in the low bit-error-rate (BER) region, implying that the iterative sliding-window decoding algorithm is near optimal; 2) BMST with repetition codes and single parity-check codes can approach the Shannon limit within 0.5 dB at the BER of 10-5for a wide range of code rates; and 3) BMST can also be applied to nonlinear codes.

This paper is concerned with the Gaussian relay channel (GRC) under the one-code constraint, where the source and the relay utilize the same code to send message. An advantage of such one-code constraint is that the error propagation resulting from re-encoding can be mitigated as the relay can forward directly the decoded “codeword” to the destination. The maximal achievable rate of the considered GRC is derived using the technique of superposition block Markov encoding based on the single code. Moreover, the forward-backward (FB) decoding strategies over the sliding window are developed both at the destination and at the relay. When LDPC codes are applied to the GRC system, a practical FB message passing decoding algorithm is presented. Simulation results show that the decoding performance can be improved as the window length increases and a small length (no greater than 4) is good enough for the FB decoding, and that re-encoding at relay may degrade the decoding performance at the destination.

In this paper, we compare the finite-length performance of non-binary spatially coupled low-density parity-check (NB SC-LDPC) codes constructed from protographs to non-binary LDPC block codes (NB LDPC-BCs). A sliding window decoding architecture with a stopping rule based on a soft bit-error-rate (BER) estimate for the NB SC-LDPC codes is considered. It is demonstrated that NB SC-LDPC codes with sliding window decoding outperform NB LDPC-BCs with no increase in decoding complexity when the decoding latency of the SC-LDPC codes equals the block length of the LDPC-BCs. We also investigate the relationship between the protograph lifting factor, the decoding window size, and the decoding performance of NB SC-LDPC codes when the decoding latency is fixed. Simulation results for several (3,6)-regular NB code examples confirm that NB SC-LDPC codes can significantly outperform both binary LDPC-BCs and binary SC-LDPC codes with the same decoding latency.

In this paper, we apply the precoding technique developed by Abbasfar et al. for ARA codes to the original rate-compatible LDPC codes introduced in [1] to obtain an improved ensemble of variable-rate LDPC codes, which perform universally well over the AWGN channel in the low to high code-rate region. The proposed codes will be named precoded Kite codes. The Extrinsic Information Transfer (EXIT) chart is used to optimize the code performance. Numerical results show that the precoded Kite codes can achieve a coding gain of up to 0.3 dB over the codes in [1]. A performance comparison is also made between the proposed codes and the Raptor codes over the AWGN channel. It is shown that the proposed codes outperform the Raptor codes universally in a wide code rate range over AWGN channels. Extensive simulation results confirm that precoded Kite codes perform close to capacity within a range of code rates from 0.1 to 0.9.

In this study, the authors consider designing low‐density parity‐check (LDPC) coded modulation systems to achieve unequal error protection (UEP). They propose a new UEP approach by partial superposition transmission (PST) called UEP‐by‐PST. In the UEP‐by‐PST system, the information sequence is distinguished as two parts, the more important data (MID) and the less important data (LID), both of which are coded with LDPC codes. The codeword that corresponds to the MID is superimposed on the codeword that corresponds to the LID. The system performance can be analysed by using discretised density evolution. Also proposed in this study is a criterion from a practical point of view to compare the efficiencies of different UEP approaches. Numerical results show that, over both additive white Gaussian noise channels and uncorrelated Rayleigh fading channels, (i) UEP‐by‐PST provides higher coding gain for the MID compared with the traditional equal error protection approach, but with negligible performance loss for the LID; and (ii) UEP‐by‐PST is more efficient with the proposed practical criterion than the UEP approach in the digital video broadcasting system.

In this study, the authors present a low‐complexity iterative joint detection–decoding algorithm for majority‐logic decodable non‐binary low‐density parity‐check (LDPC) coded modulation systems. In the proposed algorithm, a hard‐in–hard‐out decoder is combined with a hard‐decision signal detector in an iterative manner. Each iteration consists of five phases. Firstly, the detector makes hard decisions based on the iteratively updated ‘received’ signals; secondly, these hard decisions are distributed via variable nodes to check nodes; thirdly, check nodes compute hard extrinsic messages; fourthly, each variable node counts hard extrinsic messages from its adjacent check nodes and feeds back to the detection node the symbol with the most votes as well as the difference between the most votes and the second most votes; finally, these feedbacks are used to shift each ‘received’ signal point along an estimated direction to possibly reduce noise. The proposed algorithm requires only integer operations and finite field operations and consequently can be implemented with simple combinational logic circuits in practical systems. Simulation results show that the proposed algorithm performs well and hence serves as an attractive candidate for trading off performance against complexity for majority‐logic decodable non‐binary LDPC codes.

For any given short code (referred to as the basic code), block Markov superposition transmission (BMST) provides a simple way to obtain predictable extra coding gain by spatially coupling the generator matrix of the basic code. This paper presents a systematic design methodology for BMST systems to approach the channel capacity at any given target bit error rate (BER) of interest. To simplify the design, we choose the basic code as the Cartesian product of a short block code. The encoding memory is then inferred from the genie-aided lower bound according to the performance gap of the short block code to the corresponding Shannon limit at the target BER. In addition to the sliding-window decoding algorithm, we propose to perform one more phase decoding to remove residual (rare) errors. A new technique that assumes a noisy genie is proposed to upper bound the performance. Under some mild assumptions, these genie-aided bounds can be used to predict the performance of the proposed two-phase decoding algorithm in the extremely low BER region. Using the Cartesian product of a repetition code as the basic code, we construct a BMST system with an encoding memory 30 whose performance at the BER of 10-15can be predicted within 1 dB away from the Shannon limit over the binary-input additive white Gaussian noise channel.

In this paper, we consider the performance-optimized nonbinary low-density parity check code over general linear group, i.e.,$\bar{\cal C}$. A new methodology for constructing the binary representation [generalized binary representation (GBR)] of$\bar{\cal C}$is proposed, which can be optimized with regard to both degree distributions and girth. As to the decoding of the GBR, we develop a low-complexity hybrid parallel decoding process. It is shown that the decoding performance of the GBR under the proposed binary decoding process could closely approach the decoding performance of its mother code$\bar{\cal C}$under nonbinary belief propagation decoding. A simple code optimization algorithm for the GBR is also provided. Simulations show the comparative results and justify the advantages of the proposed constructions.

A new problem formulation is presented for the Gaussian interference channels with two pairs of users, which are distinguished as primary users and secondary users, respectively. The primary users employ a pair of encoder and decoder that were originally designed to satisfy a given error performance requirement under the assumption that no interference exists from other users. In the scenario when the secondary users attempt to access the same medium, we are interested in the maximum transmission rate (defined as accessible capacity) at which secondary users can communicate reliably without affecting the error performance requirement by the primary users under the constraint that the primary encoder (not the decoder) is kept unchanged. By modeling the primary encoder as a generalized trellis code (GTC), we are then able to treat the secondary link and the cross link from the secondary transmitter to the primary receiver as finite state channels. Based on this, upper and lower bounds on the accessible capacity are derived. The impact of the error performance requirement by the primary users on the accessible capacity is analyzed by using the concept of interference margin. In the case of nontrivial interference margin, the secondary message is split into common and private parts and then encoded by superposition coding, which delivers a lower bound on the accessible capacity. For some special cases, these bounds can be computed numerically by using the BCJR algorithm. Numerical results are also provided to gain insight into the impacts of the GTC and the error performance requirement on the accessible capacity.

Reed-Solomon convolutional concatenated (RSCC) code has been widely applied in wireless and space communications. However, iterative soft-decision decoding of the concatenated code is yet to be developed. This paper proposes a novel iterative soft decoding algorithm for the concatenated coding scheme. The maximum a posteriori (MAP) algorithm is used to decode the inner convolutional code. Its soft output will be deinterleaved and then passed to the soft-in-soft-out (SISO) decoding algorithm for the outer Reed-Solomon (RS) code. The outer SISO decoder integrates the adaptive belief propagation (ABP) algorithm and the Koetter-Vardy (KV) list decoding algorithm, attempting to find out the transmitted message. If it is found, the deterministic probabilities of the corresponding RS coded bits will be fed back. Otherwise, the extrinsic probabilities that are yielded by the ABP algorithm will be given as the feedback. With the proposed soft information exchange decoding mechanism, error-correction potential of the concatenated code can be better exploited. Our simulation results show that significant performance improvement can be achieved over the existing decoding algorithms.

In this paper, we present a new approach, called block Markov superposition transmission (BMST), to construct from short codes a class of convolutional codes with large constraint length. The BMST is very similar to superposition block Markov encoding (SBME), which has been widely used to prove multiuser coding theorems. We also present an iterative sliding-window decoding algorithm for the proposed transmission scheme. The extra coding gain obtained by BMST can be bounded in terms of the Markov order and with the help of the input-output weight enumerating function (IOWEF) of the BMST system, which can be computed from that of the short code by performing a trellis-based algorithm. Numerical results verify our analysis and show that an extra coding gain of 6.4 dB at bit-error rate (BER) 10-5can be obtained by BMST of the [7, 4] Hamming code.

This paper is concerned with the design and performance of nonbinary LDPC-coded differential modulation systems. A low-complexity joint detection/decoding method for noncoherent demodulation is proposed, in which the hard-message-passing strategy is used for a joint factor graph. It combines trellis-based differential detection aided with channel prediction and the reliability-based decoding of nonbinary LDPC codes introduced in [1]. The Max-Log-MAP algorithm with soft-in hard-out is used for the differential detection. Simulation results show that the proposed method can offer good performances with a greatly reduced complexity.

In this paper, new enumerating functions for linear codes are defined, including the triangle enumerating function and the tetrahedron enumerating function, both of which can be computed using a trellis-based algorithm over polynomial rings. The computational complexity is dominated by the complexity of the trellis. In addition, we show that these new enumerating functions can be used to improve existing performance bounds on the maximum likelihood decoding.

This study is concerned with the application of non‐binary low‐density parity‐check (NB‐LDPC) codes to binary input inter‐symbol interference channels. Two low‐complexity joint detection/decoding algorithms are proposed. One is referred to as max‐log‐MAP/X‐EMS algorithm, which is implemented by exchanging soft messages between the max‐log‐MAP detector and the extended min‐sum (EMS) decoder. The max‐log‐MAP/ X ‐EMS algorithm is applicable to general NB‐LDPC codes. The other one, referred to as Viterbi/GMLGD algorithm, is designed in particular for majority‐logic decodable NB‐LDPC codes. The Viterbi/GMLGD algorithm works in an iterative manner by exchanging hard‐decisions between the Viterbi detector and the generalised majority‐logic decoder (GMLGD). As a by‐product, a variant of the original EMS algorithm is proposed, which is referred to as µ ‐EMS algorithm. In the µ ‐EMS algorithm, the messages are truncated according to an adaptive threshold, resulting in a more efficient algorithm. Simulations results show that the max‐log‐MAP/ X ‐EMS algorithm performs as well as the traditional iterative detection/decoding algorithm based on the BCJR algorithm and theQ‐ary sum–product algorithm, but with lower complexity. The complexity can be further reduced for majority‐logic decodable NB‐LDPC codes by executing the Viterbi/GMLGD algorithm with a performance degradation within one dB. These algorithms provide good candidates for trade‐offs between performance and complexity.

The algebraic soft-decision decoding (ASD) algorithm is a polynomial-time soft decoding algorithm for Reed-Solomon (RS) codes. It outperforms both the algebraic hard-decision decoding (AHD) and the conventional unique decoding algorithms, but with a high computational cost. This paper proposes a progressive ASD (PASD) algorithm that enables the conventional ASD algorithm to perform decoding with an adjustable designed factorization output list size (OLS). The OLS is enlarged progressively leading to an incremental computation for the interpolation and an enhanced error-correction capability. Multiple factorizations are performed in order to find out the intended message polynomial which will be validated by a cyclic redundant check (CRC) code. The incremental interpolation constraints are introduced to characterize the progressive decoding. The validity analysis of the algorithm shows the PASD algorithm is a natural and computationally saving generalization of the ASD algorithm, delivering the same interpolation solution. The average decoding complexity of the algorithm is further theoretically characterized, revealing its dependence on the channel condition. The simulation results further validate the analysis by showing that the average decoding complexity can be converged to the minimal level in a good channel condition. Finally, performance evaluation shows the PASD algorithm preserves the error-correction capability of the ASD algorithm.

In this paper, new techniques are presented to either simplify or improve most existing upper bounds on the maximum-likelihood (ML) decoding performance of the binary linear codes over additive white Gaussian noise (AWGN) channels. Firstly, the recently proposed union bound using truncated weight spectrum by Ma et al. is re-derived in a detailed way based on Gallager's first bounding technique (GFBT), where the "good region" is specified by a sub-optimal list decoding algorithm. The error probability caused by the bad region can be upper-bounded by the tail-probability of a binomial distribution, while the error probability caused by the good region can be upper-bounded by most existing techniques. Secondly, we propose two techniques to tighten the union bound on the error probability caused by the good region. The first technique is based on pair-wise error probabilities. The second technique is based on triplet-wise error probabilities, which can be upper-bounded by the fact that any three bipolar vectors form a non-obtuse triangle. The proposed bounds improve the conventional union bounds but have a similar complexity since they involve only the Q-function. The proposed bounds can also be adapted to bit-error probabilities.

This letter is concerned with a class of nonbinary low-density parity-check (LDPC) codes, referred to as column-scaled LDPC (CS-LDPC) codes, whose parity-check matrices have a property that each column is a scaled binary vector. The CS-LDPC codes, which include algebraically constructed nonbinary LDPC codes as subclasses, admit fast encoding and decoding algorithms. Specifically, for a code over the finite field F2p, the encoder can be implemented with p parallel binary LDPC encoders followed by a series of bijective mappers, while the decoder can be implemented with an iterative decoder in which no message permutations are required during the iterations. In addition, there exist low-complexity iterative multistage decoders that can be utilized to trade off the performance against the complexity. Simulation results show that the performance degradation caused by the iterative multistage decoding algorithms is relevant to the code structure.

This paper is concerned with general interference channels characterized by a sequence of transition (conditional) probabilities. We present a general formula for the capacity region of the interference channel with two pairs of users. The formula shows that the capacity region is the union of a family of rectangles, where each rectangle is determined by a pair of spectral inf-mutual information rates. Although the presented formula is usually difficult to compute, it provides us useful insights into the interference channels. For example, the formula suggests us that the simplest inner bounds (obtained by treating the interference as noise) could be improved by taking into account the structure of the interference processes. This is verified numerically by computing the mutual information rates for Gaussian interference channels with embedded convolutional codes.

In this paper, we presented three approaches to improve the design of Kite codes (newly proposed rateless codes), resulting in an ensemble of rate-compatible LDPC codes with code rates varying “continuously” from 0.1 to 0.9 for additive white Gaussian noise (AWGN) channels. The new ensemble rate-compatible LDPC codes can be constructed conveniently with an empirical formula. Simulation results show that, when applied to incremental redundancy hybrid automatic repeat request (IR-HARQ) system, the constructed codes (with higher order modulation) perform well in a wide range of signal-to-noise-ratios (SNRs).

The extended min-sum (EMS) algorithm is redescribed as a reduced-search trellis algorithm (called M-EMS algorithm). Two variants of the M-EMS algorithm, called T-EMS algorithm and D-EMS algorithm, are presented. Simulation results show that, these three algorithms (referred to as X-EMS algorithms for convenience), combined with factor correction techniques, perform almost as well as the Q-ary sum-product algorithm (QSPA) but with a much lower complexity.

Noncontrollable finite-state channels (FSCs) are FSCs in which the channel inputs have no influence on the channel states, i.e., the channel states evolve freely. Since single-letter formulas for the channel capacities are rarely available for general noncontrollable FSCs, computable bounds are usually utilized to numerically bound the capacities. In this paper, we take the delayed channel state as part of the channel input and then define the directed information rate from the new channel input (including the source and the delayed channel state) sequence to the channel output sequence. With this technique, we derive a series of upper bounds on the capacities of noncontrollable FSCs with/without feedback. These upper bounds can be achieved by conditional Markov sources and computed by solving an average reward per stage stochastic control problem (ARSCP) with a compact state space and a compact action space. By showing that the ARSCP has a uniformly continuous reward function, we transform the original ARSCP into a finite-state and finite-action ARSCP that can be solved by a value iteration method. Under a mild assumption, the value iteration algorithm is convergent and delivers a near-optimal stationary policy and a numerical upper bound.

We correct two errors that appeared in the paper by Kaneko

This paper introduces a new class of rate less forward error correction codes named semi-random Kite (SR-Kite) codes, which can be described by a sparse semi-random parity-check matrix in systematic form. SR-Kite codes have not only rate less property, but also low error floors. We present a simulation-based greedy optimization algorithm to design the degree distribution of SR-Kite codes for independent Rayleigh fading channels. The performances of SR-Kite codes under maximum likelihood decoding are analyzed for both AWGN and independent Rayleigh fading channels via union bound. Both the analysis and simulation results show that the proposed codes perform well over AWGN and fading channels within a wide range of signal-to-noise-ratios.

A new problem formulation is presented for the Gaussian interference channels (GIFC) with two pairs of users, which are distinguished as primary users and secondary users, respectively. The primary users employ a pair of encoder and decoder that were originally designed to satisfy a given error performance requirement (EPR) under the assumption that no interference exists. In the case when the secondary users attempt to access the same medium, we are interested in the maximum transmission rate (defined as accessible capacity) at which secondary users can communicate reliably without affecting the EPR under the constraint that the primary encoder (not the decoder) is kept unchanged. The relation of the accessible capacity to the capacity region of the GIFC is revealed. By modeling the primary encoder as a generalized trellis code (GTC), we are able to treat the secondary links as finite state channels. Then upper and lower bounds on the accessible capacity are derived and computed by using the BCJR algorithm. The numerical results show us either expected or interesting facts.

In this paper, two techniques are presented to either simplify or improve most of the existing upper bounds on the maximum-likelihood decoding (MLD) performance of the binary linear codes over additive white Gaussian noise (AWGN) channels. Firstly, the recently proposed union bound using truncated weight spectra by Ma et al is re-derived in a detailed way based on Gallager's first bounding technique (GFBT). Secondly, we propose using triplet-wise error probabilities instead of pair-wise error probabilities to improve the union bound. In doing so, we prove that any three codewords form a non-obtuse triangle, which can be utilized to upper-bound the triplet-wise error probability. The proposed bounds improve the conventional union bounds but have a similar complexity since they involve only the Q-function. The proposed bounds can also be adapted to bit-error probabilities.

Kite codes, which were originally defined over the binary field, are generalized to arbitrary abelian groups in this paper. Kite codes are a special class of prefix rateless codes over groups, which can generate potentially infinite (or as many as required) random-like parity-check symbols. In this paper, we consider four kinds of Kite codes, which are binary Kite codes, Kite codes over one-dimensional lattices, Kite codes over M-PSK signal constellations and Kite codes over multi-dimensional lattices. It is shown by simulations that the proposed codes perform well over additive white Gaussian noise channels.

A modified reliability-based iterative majority-logic decoding (MRBI-MLGD) algorithm for two classes of structured LDPC codes is presented based on a recent work by Huang et al. Compared with the original one, the modified algorithm has better performance with slightly increased complexity. Then a reliability-based iterative min-sum decoding (RBI-MSD) algorithm is presented. For the presented RBI-MSD algorithm, reliability-based integer messages are processed and exchanged between variable nodes and check nodes. The main computations include only binary logical operations and integer additions. Different from the conventional min-sum algorithm, the variable nodes pass full messages rather than extrinsic messages to check nodes. This can reduce the memory loads and the computational complexity but with a little (or negligible) performance degradation. Simulation results show that, compared with the (M)RBI-MLGD algorithms, the presented RBI-MSD algorithm achieves better error performance, faster decoding convergence rate and fewer quantization bits with moderate increased computational complexity. Furthermore, the RBI-MSD algorithm is also applicable to decoding random LDPC codes, a distinct difference from the (M)RBI-MLGD algorithms. Finally, we point out that the scaling factors employed in the MRBI-MLGD algorithm and the RBI-MSD algorithm can be optimized using discretized density evolution.

In this paper, we present a low-complexity joint detection-decoding algorithm for nonbinary LDPC coded-modulation systems. The algorithm combines hard-decision decoding using the message-passing strategy with the signal detector in an iterative manner. It requires low computational complexity, offers good system performance and has a fast rate of decoding convergence. Compared to the q-ary sum-product algorithm (QSPA), it provides an attractive candidate for practical applications of q-ary LDPC codes.

This paper is concerned with the construction of a class of nonbinary irregular repeat accumulate (IRA) codes. Since they are defined on the finite field GF(q) (q>2), we will refer to the constructed codes as q-ary IRA (QIRA) codes. While preserving the excellent error correcting capability of q-ary LDPC codes, QIRA codes can be efficiently encoded like conventional binary IRA codes. By adopting the progressive edge growth (PEG) algorithm to construct the parity check matrices, we can achieve the increased girth of their factor graphs and improved decoding performance. Simulation results show that, using the sum-product algorithm on GF(q), QIRA codes outperform binary LDPC codes and turbo codes in terms of bit error ratio and frame error ratio on AWGN channels. Especially, they could achieve excellent error performance when combined with high order modulations. Feasibility study indicates, with the use of the extended min-sum (EMS) decoding algorithm, QIRA codes are competitive candidates for practical applications.

A non-controllable finite-state channel (FSC) is a finite-state channel in which the user can't control channel states. That is, the channel state of a non-controllable FSC evolves freely according to an uncontrollable probability law. Thus far, good upper bounds on capacities of general non-controllable FSCs remain unknown. Here we develop upper bounds that use delayed feedback and delayed state information, and propose dynamic programming methods to numerically evaluate the bounds.

We apply clipping to superposition coded modulation (SCM) systems to reduce the peak-to-average power ratio (PAPR) of the transmitted signal. The impact on performance is investigated by evaluating the mutual information driven by the induced peak-power-limited input signals. It is shown that the rate loss is marginal for moderate clipping thresholds if optimal encoding/decoding is used. This fact is confirmed in examples where capacity-approaching component codes are used together with the maximumaposterioriprobability (MAP) detection. In order to reduce the detection complexity of SCM with a large number of layers, we develop a suboptimal soft compensation (SC) method that is combined with soft-input soft-output (SISO) decoding algorithms in an iterative manner. A variety of simulation results for additive white Gaussian noise (AWGN) and fading channels are presented. It is shown that with the proposed method, the effect of clipping can be efficiently compensated and a good tradeoff between PAPR and bit-error rate (BER) can be achieved. Comparisons with other coded modulation schemes demonstrate that SCM offers significant advantages for high-rate transmissions over fading channels.

This paper presents a peak-power-limited superposition coding scheme based on clipping. A low-complexity soft compensation algorithm (SCA) for combating the clipping effect is investigated. It can be easily combined with soft-input soft-output (SISO) decoding algorithms in an iterative manner. Various numerical results show that the SCA can effectively mitigate the performance loss due to clipping.

In this paper, we design capacity-approaching codes for partial response channels. The codes are constructed as concatenations of inner trellis codes and outer low-density parity- check (LDPC) codes. Unlike previous constructions of trellis codes for partial response channels, we disregard any algebraic properties (e.g., the minimum distance or the run-length limit) in our design of the trellis code. Our design is purely probabilistic in that we construct the inner trellis code to mimic the transition probabilities of a Markov process that achieves a high (capacity-approaching) information rate. Hence, we name it a matched information rate (MIR) design. We provide a set of five design rules for constructions of capacity-approaching MIR inner trellis codes. We optimize the outer LDPC code using density evolution tools specially modified to fit the superchannel consisting of the inner MIR trellis code concatenated with the partial response channel. Using this strategy, we design degree sequences of irregular LDPC codes whose noise tolerance thresholds are only fractions of a decibel away from the capacity. Examples of code constructions are shown for channels both with and without spectral nulls.

In this correspondence, we investigate in a comprehensive fashion a one-layer coding/shaping scheme resembling a perfectly cooperated multiple-access system. At the transmitter, binary data are encoded by either single-level or multilevel codes. The coded bits are first randomly interleaved and then entered into a signal mapper. At each time, the signal mapper accepts as input multiple binary digits and delivers as output an amplitude signal, where the input are first independently mapped into 2-PAM signals (possibly having different amplitudes) and then superimposed to form the output. The receiver consists of an iterative decoding/demapping algorithm with an entropy-based stopping criterion. In the special cases when all the 2-PAM signals have equal amplitudes, based on an irregular trellis, we propose an optimal soft-input-soft-output (SISO) demapping algorithm with quadratic rather than exponential complexity. In the general cases, when multilevel codes are employed, we propose power-allocation strategies to facilitate the iterative decoding/dempaping algorithm. Using the unequal power-allocations and the Gaussian-approximation-based suboptimal demapping algorithm (with linear complexity), coded modulation with high bandwidth efficiency can be implemented.

We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over binary linear intersymbol interference (ISI) channels with additive white Gaussian noise (AWGN). Using the graph representations of the channel, the code, and the sum-product message-passing detector/decoder, we prove two error concentration theorems. Our proofs expand on previous work by handling complications introduced by the channel memory. We circumvent these problems by considering not just linear Gallager codes but also their cosets and by distinguishing between different types of message flow neighborhoods depending on the actual transmitted symbols. We compute the noise tolerance threshold using a suitably developed density evolution algorithm and verify, by simulation, that the thresholds represent accurate predictions of the performance of the iterative sum-product algorithm for finite (but large) block lengths. We also demonstrate that for high rates, the thresholds are very close to the theoretical limit of performance for Gallager codes over ISI channels. If C denotes the capacity of a binary ISI channel and if C/sub i.i.d./ denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables (C/sub i.i.d.//spl les/C), we prove that the maximum information rate achievable by the sum-product decoder of a Gallager (coset) code is upper-bounded by C/sub i.i.d./. The last topic investigated is the performance limit of the decoder if the trellis portion of the sum-product algorithm is executed only once; this demonstrates the potential for trading off the computational requirements and the performance of the decoder.

This is a semitutorial paper on trellis-based algorithms. We argue that most decoding/detection algorithms described on trellises can be formulated as path-partitioning algorithms, with proper definitions of mappings from subsets of paths to metrics of subsets. Thereby, the only two operations needed are path-concatenation and path-collection, which play the roles of multiplication and addition, respectively. Furthermore, we show that the trellis structure permits the path-partitioning algorithms to be formulated as forward-only algorithms (with structures resembling the Viterbi (1967) algorithm), thus eliminating the need for backward computations regardless of what task needs to be performed on the trellis. While all of the actual decoding/detection algorithms presented here are rederivations of variations of previously known methods, we believe that the exposition of the algorithms in a unified manner as forward-only path-partitioning algorithms is the most intuitive manner in which to generalize the Viterbi algorithm. We also believe that this approach may, in fact, influence the practical implementation of the algorithms as well as influence the construction of other forward-only algorithms (e.g., byte-wise forward-only detection algorithms).

We propose a numerical method to compute the capacity of a power constrained memoryless additive white Gaussian noise (AWGN) channel with finite and fixed input alphabets. The method is based on a two-part algorithm. The first part is a modified version of the Blahut-Arimoto algorithm and the second part is a simple maximization algorithm over a single parameter. The optimal input distribution we obtain can be utilized to construct probabilistic codes for this channel. These codes promise to bridge the shaping gap between the uniform-input information rate and the capacity of the channel.

Redundancy introduced using filterbank precoders at the transmitter builds a unified framework for modulation schems. Taking advantage of this diversity can offer a powerful tool for removing interblock interference and devising simple but effective precoders for suppressing the intersymbol interference (ISI) and being robust to frequency selective channels‥ In this paper, we assume that the transmitter knows the autocorrelation sequences of the channel impulse response. Under this assumption, we derive a lower and an upper bound on the free distance for a precoded channel and with this, design the precoder that maximizes the lower bound subject to a power constraint. It turns out that the optimal precoder is a unitary matrix which makes QR decomposition of its super-channel exhibit equal diagonal entries in R-factor and the lower and the upper bounds of its free distance equal. We show that for the optimal precoded channel, the detection performance using the decision feedback equalizer (DFE) based on QR decomposition is asymptotically equivalent to that of the mamximum likehood detector (MLD) when the signal to noise ratio (SNR) is large.

Another description of the two-dimensional binary (d,k) constrained array is given. Based on this, a lower bound for the capacity of the (0,k) constrained array is presented, and it is reproved that the capacity of the (d,d+1) constrained array is equal to zero.

Some properties of the minimal interpolation problem are investigated, from which a simple proof of the validity of the Welch-Berlekamp (1983) algorithm is presented. A new key equation is derived, which is closely related to the classical key equation of the syndrome-decoding algorithm and can be solved by the Welch-Berlekamp algorithm.

In this correspondence, we derive, with the method of coset decompositions, some new bounds for q-ary linear block codes, where q is the order of some finite field.