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In this paper, we construct new families of classical convolutional codes (CCC’s) and new families of quantum convolutional codes (QCC’s). The CCC’s are derived from (block) algebraic geometry (AG) codes. Furthermore, new families of CCC’s are constructed by applying the techniques of puncturing, extending, expanding, and by the direct product code construction applied to AG codes. In addition, utilizing the new CCC’s constructed here, we obtain new families of QCC’s. The parameters of these new codes are good. More precisely, in the classical case, a family of almost near maximum distance separable (MDS) codes is presented; in the quantum case, we construct a family of MDS (optimal) quantum convolutional codes. [ABSTRACT FROM AUTHOR]

Supporting parallel multi-priority data streams is vital for maintaining the quality of service (QoS) in vehicle-to-everything (V2X) communication. Hence, the European telecommunications standard institute (ETSI) has defined four packet types, with varying priority levels, to be used as broadcast packets in such communication. This paper studies the medium access control (MAC) layer performance of IEEE 802.11p and cellular-V2X (C-V2X) Mode 4 using discrete-time Markov chain (DTMC) based models, while considering parallel multi-priority data streams. The overall model consists of four queue models with their respective traffic generators, which are appropriately linked with the DTMCs modeling the MAC layer operations of IEEE 802.11p and C-V2X Mode 4. Closed-form solutions for the steady-state probabilities of the models are obtained, which are then utilized to derive expressions for key performance indicators at the MAC layer. Numerical results are provided to draw insights on the MAC layer performance of the two technologies. IEEE 802.11p is comparatively superior in average delay, and at maintaining fairness among multi-priority data streams, whereas C-V2X Mode 4 exhibits better collision resolution, which leads to its higher throughput. The paper also includes design insights on possible performance enhancements for future releases. [ABSTRACT FROM AUTHOR]

This paper presents a new construction of error correcting codes which achieves optimal recovery of a streaming source over a packet erasure channel. The channel model considered is the sliding-window erasure model, with burst and arbitrary losses, introduced by Badr et al. We present a simple construction, when the rate of the code is at least 1/2, which achieves optimal error correction in this setup. Our proposed construction is explicit and systematic. It uses off-the-shelf maximum distance separable (MDS) codes and maximum rank distance (MRD) Gabidulin block codes as constituent codes and combines them in a simple manner. This is in contrast to other recent works, where the construction involves a careful design of the generator or parity check matrix from first principles. The field size requirement which depends on the constituent MDS and MRD codes is also analyzed. [ABSTRACT FROM AUTHOR]

Vehicle-to-vehicle (V2V) communication plays a pivotal role in intelligent transport systems (ITS) with cellular-vehicle to everything (C-V2X) and IEEE 802.11p being the two competing enabling technologies. This paper presents multi-dimensional discrete-time Markov chain (DTMC) based models to study the medium access control (MAC) layer performance of the IEEE 802.11p standard and C-V2X Mode 4, considering periodic cooperative awareness messages (CAMs) and event-driven decentralized environmental notification messages (DENMs). Closed-form solutions for the steady-state probabilities of the models are obtained, which are then utilized to derive expressions for several key performance metrics. Numerical results are provided to draw insights on the performance. In particular, a performance comparison between IEEE 802.11p and C-V2X Mode 4 in terms of the average delay, the collision probability, and the channel utilization is presented. The results show that IEEE 802.11p is superior in terms of average delay, whereas C-V2X Mode 4 excels in collision resolution. The paper also includes design insights on possible future MAC layer performance enhancements of both standards. [ABSTRACT FROM AUTHOR]

In this paper, we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check (CRC) codes, separated by an interleaver, in the short and moderate block length regimes. The analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding. The analysis is carried out by introducing the concatenated polar code (CPC) ensembles, whose distance properties can be analyzed (for sufficiently short block lengths) by means of the uniform interleaver approach. At moderate block lengths, we resort to the Monte Carlo simulations. Results show that if the inner polar code possesses a low minimum distance and the outer CRC code has a sufficiently large amount of redundancy, then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting CPC. [ABSTRACT FROM AUTHOR]

$Z$ -complementary code set (ZCCS), an extension of perfect CCs, refers to a set of 2-D matrices having zero correlation zone properties. ZCCS can be used in various multi-channel systems to support, for example, quasi-synchronous interference-free multicarrier code-division multiple access communication and optimal channel estimation in multiple-input multiple-output systems. Traditional constructions of ZCCS heavily rely on a series of sequence operations which may not be feasible for rapid hardware generation particularly for long ZCCSs. In this paper, we propose a direct construction of ZCCS using the second-order Reed–Muller codes with efficient graphical representation. Our proposed construction, valid for any number of isolated vertices present in the graph, is capable of generating optimal ZCCS meeting the set size upper bound. [ABSTRACT FROM AUTHOR]

Memory-based LT encoders (MBLTEs) have been shown to have better performance than the regular LT encoder in terms of bit error rate (BER) and decoding convergence speed. In this paper, we explore the entire family of MBLTEs for finite block-length codes over the binary erasure channel (BEC). We propose an algorithm to extend the first and second order MBLTE approach to an arbitrary $i$ -th order MBLTE. We analyze the performance of such encoders mathematically by characterizing the expected accumulated number of recovered variable nodes at each decoding round. We define the threshold of the memory-based encoding method (MBEM) and show that the performance of MBLTEs increases as the memory order increases up to the point where this threshold is achieved. Beyond this point, we show that the performance of MBLTEs saturates if the channel erasure probability is zero and degrades otherwise. We formulate an optimization problem to solve for the optimal memory order based on whether or not the MBEM threshold is achieved. We present an extensive set of numerical results. These show agreement between our analysis and computer simulations. They also show that our optimization problem is efficient in determining the optimal memory order of MBLTEs in terms of decoding convergence speed, BER/frame-error-rate, and error floor. [ABSTRACT FROM AUTHOR]

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

Remote entanglement distribution plays a crucial role in large-scale quantum networks, and the key enabler for entanglement distribution is quantum routers (or repeaters) that can extend the entanglement transmission distance. However, the performance of quantum routers is far from perfect yet. Amongst the causes, the limited quantum memories in quantum routers largely affect the rate and efficiency of entanglement distribution. To overcome this challenge, this paper presents a new modeling for the maximization of entanglement distribution rate (EDR) on a memory-limited path, which is then transformed into entanglement generation and swapping sub-problems. We propose a greedy algorithm for short-distance entanglement generation so that the quantum memories can be efficiently used. As for the entanglement swapping sub-problem, we model it using an Entanglement Graph (EG), whose solution is yet found to be at least NP-complete. In light of it, we propose a heuristic algorithm by dividing the original EG into several sub-problems, each of which can be solved using dynamic programming (DP) in polynomial time. By conducting simulations, the results show that our proposed scheme can achieve a high EDR, and the developed algorithm has a polynomial-time upper bound and reasonable average runtime complexity. [ABSTRACT FROM AUTHOR]

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

Unique-word (UW) OFDM is a new OFDM variant that offers several advantages over cyclic prefix OFDM, such as better bit-error-rate (BER) performance in frequency selective channels and much lower out-of-band radiation. However, all results with respect to the BER performance are based on Monte-Carlo simulations only, and offer little insight in the behaviour of the system performance as function of its parameters. In this paper, we theoretically analyse the performance of the UW-OFDM system for both a deterministic channel and a fading channel, and offer a systematic way to generate the UW-OFDM signal to optimize the performance of the system. When the fading channel is known at the transmitter, we show that we can obtain a diversity order that is of the same order of magnitude as the theoretically maximum diversity order, but when the channel is not known at the transmitter, it will be very hard to achieve maximum diversity order, although the obtained diversity order is larger than one. [ABSTRACT FROM AUTHOR]

Erasure codes, such as LT and Raptor codes, are designed for the purpose of erasure-resilient distribution of data over computer networks. To achieve a small reception overhead, however, LT and Raptor codes must be used with a large design length $k$, making these codes unsuitable for real-time applications. In this paper, we propose a new class of erasure codes based on Reed–Solomon codes that unlike other Reed–Solomon-based erasure codes are rateless and also, unlike other rateless codes, guarantee zero overhead even for small $k$. Moreover, they have a reasonable computational complexity of coding when $k$ is not too large. In fact, a practical implementation of subfield subcodes of Reed-Solomon codes with arbitrarily large block lengths is presented. [ABSTRACT FROM AUTHOR]

The Fletcher checksum (FC) is a means of error detection that is analogous to cyclic redundancy checking but easier to implement in software. Now FC actually has two versions, one is Fletcher’s original paper version termed as FC and the other is RFC1146 version termed as FC-RFC. This paper shows that the FC is a cyclic code. This is done by introducing a cyclic code known as the cyclic Fletcher code (CFC) and then showing that the CFC is equivalent to the FC. The clear algebraic structure of cyclic codes is used to analyze the code block length and code distance, and then the single error correction, double error detection, and burst error detection of the CFC are analyzed. Also given is the mathematical relationship between the CFC and a modified FC. Finally, the issue of implementation is considered, and a new scheduling for the CFC is proposed to prevent data hazards. A simulation shows that the proposed scheduling offers improvements of at least 46% in throughput for the CFC and FC-RFC. [ABSTRACT FROM AUTHOR]

In this paper, we devise a fast encoding algorithm for triply extended Reed–Solomon codes. The proposed approach requires approximately two XORs per bit, which improves the prior result of three XORs per bit established by certain maximum distance separable (MDS) array codes. We also prove that, for MDS codes with two and three parities, the scheduling algorithms require at least two XORs per bit. To the best of our knowledge, this is the first provable scheduling algorithm for the triple-parity MDS codes to approach the theoretical lower bounds. The implementation with SIMD instructions is provided. The simulations show that the proposed approach is competitive, as compared with other cutting edge implementations. [ABSTRACT FROM AUTHOR]

In this paper, linear index codes with multiple senders are studied, where every receiver receives encoded messages from all senders. A new fitting matrix for the multiple senders is proposed and it is proved that the minimum rank of the proposed fitting matrices is the optimal codelength of linear index codes for the multiple senders. In addition, a new type of a side information graph related with the optimal codelength is proposed and whether given side information is critical or not is studied. Furthermore, linear index codes for the cellular network scenario are studied, where each receiver can receive a subset of sub-codewords. Since some receivers cannot receive the entire codeword in the cellular network scenario, the encoding method based on the fitting matrix has to be modified. In the cellular network scenario, we propose another fitting matrix and prove that an optimal generator matrix can be found based on these fitting matrices. In addition, some properties on the optimal codelength of linear index codes for the cellular network case are studied. [ABSTRACT FROM AUTHOR]

Low density lattice codes (LDLCs) are high-dimensional lattices with a sparse inverse generator matrix that can be decoded efficiently using iterative decoding. In the iterative LDLC decoder, the messages are Gaussian mixtures, and for any implementation, the Gaussian mixtures must be approximated. This paper describes a parametric LDLC decoding algorithm, where internally at the variable node, infinite Gaussian mixtures are approximated with three or two Gaussians, while the messages between nodes are single Gaussians. Strengths of the algorithm include its simplicity and suitability for analysis. Analysis is performed by evaluating the Kullback–Leibler divergence between the true messages and the three/two Gaussian approximation. The approximation using three or two Gaussians is more accurate than previously proposed approximations. Also, noise thresholds for the proposed LDLC decoder are presented, and the proposed decoder reduces the noise thresholds 0.05 dB compared with previous parametric decoders. The numerical results show that for $n=100$ and $n=1000$ , the two-Gaussian approximation is the same as the full-complexity decoder. But when the dimension is $n=10\,000$ , a three-Gaussian approximation is needed. [ABSTRACT FROM PUBLISHER]

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

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

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

A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size. [ABSTRACT FROM AUTHOR]

Locally repairable codes are desirable for distributed storage systems to improve the repair efficiency. In this paper, a connection between locally repairable codes with multiple disjoint repair sets and packings is derived under the condition that each repair set contains exactly one check symbol. Particularly, conditions under which an optimal locally repairable code corresponds to a packing are also characterized. As an application of this connection, some optimal locally repairable codes can be obtained by packings. Specifically, two constructions of locally repairable codes are proposed which not only generalize some known explicit constructions but also give optimal locally repairable codes with flexible new parameters. [ABSTRACT FROM AUTHOR]

Reed–Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) and may use any polar or RM decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation (SC)-, SC-list (SCL)- and belief propagation (BP)-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-ML performance for the RM(3,7)-code (e.g., 0.04 dB away from the ML bound at BLER of 10−3) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of RM codes and SC decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes. [ABSTRACT FROM AUTHOR]

In this paper, we propose primitive rateless (PR) codes. A PR code is characterized by the message length and a primitive polynomial over $\mathbf {GF}(2)$ , which can generate a potentially limitless number of coded symbols. We show that codewords of a PR code truncated at any arbitrary length can be represented as subsequences of a maximum-length sequence ($m$ -sequence). We characterize the Hamming weight distribution of PR codes and their duals and show that for a properly chosen primitive polynomial, the Hamming weight distribution of the PR code can be well approximated by the truncated binomial distribution. We further find a lower bound on the minimum Hamming weight of PR codes and show that there always exists a PR code that can meet this bound for any desired codeword length. We provide a list of primitive polynomials for message lengths up to 40 and show that the respective PR codes closely meet the Gilbert-Varshamov bound at various rates. Simulation results show that PR codes can achieve similar block error rates as their BCH counterparts at various signal-to-noise ratios (SNRs) and code rates. PR codes are rate-compatible and can generate as many coded symbols as required; thus, demonstrating a truly rateless performance. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a rateless code design for a two-time slot two-way relay channel (TWRC), where network coding operation is employed at the relay. Rateless codes are used at both users and the relay nodes as they can cope with various channel conditions. We consider a general TWRC framework, where the two users may use different rateless code generator matrices, thus, the relay cannot directly recover the physical-layer network coded (PNC) message of the two users. We propose a hybrid estimate and forward (EF)/decode and forward (DF) protocol with rateless codes for the TWRC. In the proposed scheme, the relay uses a joint belief propagation decoding process to decode the messages of two users. Depending on whether the relay recovers both users’ messages, only one user’s message or no users’ messages, the relay will generate three different types of signals, namely, NC, soft NC, and soft PNC, respectively. We design the degree distributions of the rateless codes for the three nodes of the TWRC, to make them suitable not only for decoding at two users at high signal-to-noise ratio (SNR), but also for decoding at the relay at low SNR. We show that the proposed hybrid EF/DF protocol with the designed rateless codes outperforms other schemes significantly in terms of bit error rate performance. [ABSTRACT FROM AUTHOR]

In this paper, we study systematic Luby transform (SLT) codes over binary input additive white Gaussian noise (BIAWGN) channels. To improve the bit-error-rate (BER) performance of SLT codes on the BIAWGN channel, this paper presents a novel optimization design of degree distributions. First, we apply the Gaussian approximation (GA) to analyze the asymptotic BER performance of SLT codes and calculate overhead thresholds for successful decoding. Second, we derive an approximate closed-form expression for lower bound on BER by applying the GA and further simplify the expression in low and high signal-to-noise ratio regimes, respectively. Third, we adopt the conventional linear programming (CLP) constrained by the GA to optimize the degree distribution of SLT codes. The objective of CLP is to minimize the average degree of the degree distribution that may result in bad error floor. To improve the error floor caused by CLP, we put forth a novel optimization model constrained by the lower bound on BER, and using the model we can design the degree distribution more flexibly with any desired BER. Simulation results show that the proposed degree distribution can provide better BER performance of SLT codes over the BIAWGN channel. [ABSTRACT FROM AUTHOR]

The problem of identifying the channel code from a received sequence of noise-affected codewords is known as the blind reconstruction of channel codes. Blind reconstruction of channel codes is an important problem in military surveillance applications to identify the channel code used by an adversary. In this paper, we consider the problem of the blind reconstruction of the binary cyclic codes of unknown length from an unsynchronized bitstream (i.e., when the location of codeword boundaries is not known). For the blind reconstruction of cyclic codes, it is sufficient to identify the correct synchronization, the length, and the factors of the generator polynomial of the code. Toward this, we study the distribution of the syndromes (remainders) of the received polynomials with respect to a candidate factor of the generator polynomial. We prove that the probability of zero syndrome is maximum when all the parameters are correct. Using this result, the problem of the blind reconstruction of cyclic codes is formulated and solved as a hypothesis testing problem. [ABSTRACT FROM AUTHOR]

Cyclic redundancy check (CRC) codes check if a codeword is correctly received. This paper presents an algorithm to design CRC codes that are optimized for the code-specific error behavior of a specified feedforward convolutional code. The algorithm utilizes two distinct approaches to computing undetected error probability of a CRC code used with a specific convolutional code. The first approach enumerates the error patterns of the convolutional code and tests if each of them is detectable. The second approach reduces complexity significantly by exploiting the equivalence of the undetected error probability to the frame error rate of an equivalent catastrophic convolutional code. The error events of the equivalent convolutional code are exactly the undetectable errors for the original concatenation of CRC and convolutional codes. This simplifies the computation because error patterns do not need to be individually checked for detectability. As an example, we optimize CRC codes for a commonly used 64-state convolutional code for information length k=1024, demonstrating significant reduction in undetected error probability compared to the existing CRC codes with the same degrees. For a fixed target undetected error probability, the optimized CRC codes typically require 2 fewer bits. [ABSTRACT FROM PUBLISHER]

Locally recoverable codes play a crucial role in distributed storage systems. Many studies have only focused on the constructions of optimal locally recoverable codes with regard to the Singleton bound. The aim of this paper is to construct optimal binary locally recoverable codes meeting the alphabet-dependent bound. Using a general framework for linear codes associated to a set, we provide a new approach to constructing binary locally recoverable codes with locality 2. We turn the problem of designing optimal binary locally recoverable codes into constructing a suitable set. Several constructions of optimal binary locally recoverable codes are proposed by this new method. Finally, we propose constructions of optimal binary locally recoverable codes with locality 2 and locality parameters $(\text {r},\delta)$ by Griesmer codes. [ABSTRACT FROM AUTHOR]

Unique word- (UW-) orthogonal frequency division multiplexing (OFDM) is a new multicarrier technique that is shown to be superior to the cyclic prefix- (CP-) OFDM system in terms of bit-error-rate performance in fading channels. Furthermore, in the literature, it is shown by simulations that UW-OFDM has a considerably lower out-of-band (OOB) radiation compared with CP-OFDM. In this paper, we derive an analytical expression to investigate the spectral characteristics of UW-OFDM to be able to determine the effect of the different system parameters on the OOB radiation. In addition, we include in our analysis the effects of the interpolation filter and the unique word. We show that the OOB radiation in UW-OFDM is mainly determined by the number of non-modulated subcarriers reserved as the guard band, and the ratio of the UW sequence length to the number of subcarriers. By increasing either of these two parameters, we show that we can suppress the OOB radiation to any amount. This is an important advantage of UW-OFDM over classic CP-OFDM systems, and shows that UW-OFDM is an excellent candidate for cognitive radio systems. [ABSTRACT FROM AUTHOR]

By introducing several levels of recoverability for locally recoverable codes (LRCs), LRCs with hierarchical locality (H-LRCs) are defined for correcting different numbers of erasures. There are two major ingredients in this paper. The first is to investigate some properties and existence conditions of optimal H-LRCs with respect to a generalized Singleton-like bound for H-LRCs. The second ingredient is to propose several constructions of optimal H-LRCs by employing cyclic codes. Notably, the parameters of these optimal H-LRCs are flexible. [ABSTRACT FROM AUTHOR]

This paper analyzes and explicitly constructs quasi-cyclic (QC) codes for correcting multiple bursts via matrix transformations. Our analysis demonstrates that the multiple-burst-correction capability of QC codes is determined by sub-matrices in the diagonal of their transformed parity-check matrices. By well designing these sub-matrices, the proposed QC codes are able to achieve optimal or asymptotically optimal multiple-burst-correction capability. Moreover, it proves that these codes can be QC low-density parity-check (QC-LDPC) codes, if the diagonal sub-matrices of their transformed parity-check matrices are Hadamard powers of base matrices. Analysis and simulation results show that our QC-LDPC codes perform well over not only random symbol error/erasure channels, but also burst channels. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new linear vector physical-layer network coding (NC) scheme for spatial multiplexing multiple-input multiple-output (MIMO) two-way relay channel (TWRC) where the channel state information (CSI) is not available at the transmitters. In this scheme, each user transmits $M$ independent quadrature amplitude modulation signal streams respectively from its $M$ antennas to the relay. Based on the receiver-side CSI, the relay determines a NC generator matrix for linear vector network coding, and reconstructs the associated $M$ linear combinations of all messages. We present an explicit solution for the generator matrix that minimizes the error probability at a high SNR, as well as an efficient algorithm to find the optimized solution. We propose a novel typical error event analysis that exploits a new characterization of the deep fade events for the TWRC. We derive a new closed-form expression for the average error probability of the proposed scheme over a Rayleigh fading MIMO TWRC. Our analysis shows that the proposed scheme achieves the optimal error rate performance at a high SNR. Numerical results show that the proposed scheme significantly outperforms existing schemes, and match well with our analytical results. [ABSTRACT FROM AUTHOR]

RS-LDPC codes, constructed based on the codewords of Reed-Solomon (RS) codes with two information symbols, are an important class of LDPC codes. In this paper, we present two representations, namely, quasi-cyclic (QC) representation and vector representation, for RS-LDPC codes. Under the first representation, most part of the parity-check matrix of a full-length RS-LDPC code consists of circulant permutation matrices and zero matrices. As a result, the class of codes can enjoy the advantages in hardware implementation as QC-LDPC codes. In addition, the base matrix under the QC representation of an RS-LDPC code can be explicitly given such that the rank of its parity-check matrix can be analyzed combinatorially. Under the second representation, each permutation matrix in the parity-check matrix of an RS-LDPC code is defined by a nonbinary vector, whose entries are a permutation of entries in the field from which the RS code is constructed. Then, the “affine invariance” property is proved for full-length RS-LDPC codes, which can facilitate the structural analysis of the codes. [ABSTRACT FROM PUBLISHER]

Motivated by applications involving hot data, the notion of (r, t) -availability was introduced, where a symbol is said to have (r, t) -availability if it can be reconstructed, respectively, from t disjoint repair alternatives of other symbols, each of size at most r . The key to constructing a family of locally repairable codes with (r, t) -availability is the design of a membership matrix for dividing global parities into local ones such that each repair group contains only one parity symbol. Although explicit designs of membership matrices are available, it remains unclear whether membership matrices with significantly better parameters can be constructed, which was a question left open. To tackle the open problem, this paper first provides a connection between the design of membership matrices and the design of a combinatorial object in the well-known combinatorial design theory, called resolvable configurations. This paper then proposes several direct designs of membership matrices based on Euclidean geometry, finite fields, circulant permutation matrices, affine permutation matrices, and Reed–Solomon codes. By leveraging the notion of resolvable configurations and the Kronecker product, this paper further proposes a two-step design of membership matrices. The proposed direct and two-step designs of membership matrices are shown to have significantly better parameters than those recently available. [ABSTRACT FROM AUTHOR]

It has been shown that a class of spatially coupled low-density generator-matrix (SC-LDGM) code ensembles displays distortion saturation for the lossy binary symmetric source coding problem with the belief propagation guided decimation (BPGD) algorithm, i.e., the BPGD distortion approaches the optimal expected distortion of the underlying ensemble asymptotically in code length. We investigate the distortion performance of a practical class of protograph-based SC-LDGM code ensembles and demonstrate distortion saturation numerically. Moreover, taking advantage of the convolutional structure of the SC-LDGM codes, we propose an efficient windowed encoding (WE) algorithm with two decimation techniques for lowering the WE complexity that maintain distortion performance close to the rate-distortion bound. [ABSTRACT FROM AUTHOR]

We study application of convolutional codes to the randomized encoding scheme introduced by Wyner as a way of confusing the eavesdropper over a wiretap channel. We describe optimal and practical sub-optimal decoders for the main and the eavesdropper’s channels, and estimate the security gap, which is used as the main metric. The sub-optimal decoder works based on the trellis of the code generated by a convolutional code and its dual, where one encodes the data bits and the other encodes the random bits. By developing a code design metric, we describe how these two generators should be selected for optimal performance over a Gaussian wiretap channel. We also propose application of serially concatenated convolutional codes to this setup so as to reduce the resulting security gaps. Furthermore, we provide an analytical characterization of the system performance by extending existing lower and upper bounds for coded systems to the current randomized convolutional coding scenario. We illustrate our findings via extensive simulations and numerical examples, which show that the newly proposed coding scheme can outperform the other existing methods in the literature in terms of security gap. [ABSTRACT FROM PUBLISHER]

Low density parity check (LDPC) lattices were the first family of lattices equipped with iterative decoding algorithms. We introduce quasi-cyclic LDPC (QC LDPC) lattices as a special case of LDPC lattices with one binary QC-LDPC code as their underlying code. These lattices are obtained from the Construction A of lattices providing us to encode them efficiently using shift registers. To benefit from an encoder with linear complexity in the lattice dimension, we obtain the generator matrix of these lattices in quasi-cyclic form. We generalize the proposed quasi-cyclic form of the generator matrix for other Construction A lattices, namely the LDA lattices, with a non-binary QC-LDPC code as their underlying code. We provide a low-complexity decoding algorithm of QC LDPC-lattices based on the sum product algorithm. To design lattice codes, QC LDPC-lattices are combined with the nested lattice shaping that uses the Voronoi region of a sublattice for shaping. The shaping gain and the shaping loss of our lattice codes with dimensions 40, 50, and 60 using an optimal quantizer, are presented. The guidelines for applying efficient shaping methods, like hypercube shaping, for QC LDPC-lattices are also given. Consequently, we establish a family of lattice codes that perform practically close to the sphere bound. [ABSTRACT FROM PUBLISHER]

The A^\ast algorithm for decoding linear block codes is an efficient tree-search method for decoding short linear block codes. The information contained in the parity bits are potentially useful for reducing the edges needed in the tree search. We propose to extract the information embedded in the parity bits by considering sums of parity bits which can be decided by part of the message bits. Analysis for the number of required message bits for obtaining sums of parity bits is provided. The information contained in the parity bits can help to reduce the number of search tree edges. [ABSTRACT FROM PUBLISHER]

Random interleavers are widely employed in digital communication systems to combat channel fading and burst errors. In applications such as grant-free access by Internet of Things (IoT) devices, accurately identifying a specific irregular interleaving pattern within an interleaver period is vital to both terminal recognition and data recovery. In this work, we investigate effective approaches for random interleaving pattern identification in Reed-Solomon (RS) coded data streams. We first propose an algorithm of low computational complexity to detect positions of code symbols belonging to the same RS codeword group (RSCG) under modest bit error rate. We further develop another low-complexity algorithm to successfully identify random interleaving patterns for RS code symbols within each RS codeword under moderate to high error rate applications. Our theoretical analysis and simulation results corroborate to demonstrate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

The nearest lattice point problem in $\mathbb {R}^{n}$ is formulated in a distributed network with $n$ nodes. The objective is to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node communication. Algorithms with a single as well as an unbounded number of rounds of communication are considered for the case $n=2$. For the algorithm with a single round, expressions are derived for the error probability as a function of the total number of communicated bits. We observe that the error exponent depends on the lattice structure and that zero error requires an infinite number of communicated bits. In contrast, with an infinite number of allowed communication rounds, the nearest lattice point can be determined without error with a finite average number of communicated bits and a finite average number of rounds of communication. In two dimensions, the hexagonal lattice, which is most efficient for communication and compression, is found to be the most expensive in terms of communication cost. [ABSTRACT FROM AUTHOR]

A novel method to recursively construct a set of precoded polar codes of various rates and short-to-moderate lengths is presented. The proposed code design method minimizes the successive cancellation (SC) decoding error probability estimate under three constraints. The first constraint is the minimum distance requirement to improve the maximum-likelihood (ML) performance of the resulting code and therefore the performance under the SC list (SCL) decoding. The other two constraints introduce preselected supercode and subcode, where the supercode ensures fast computation of the minimum distance and the subcode ensures reduction of the search space size. The supercode is given by the Plotkin sum of shorter codes, which are nested to simplify computation of low-weight codewords. These low-weight codewords are needed to satisfy the minimum distance constraint. The simulation results indicate that the proposed precoded polar codes of lengths 128 and 256 provide a better frame error rate (FER) than polar codes with CRC and e-BCH polar subcodes under the SCL decoding algorithm with the list size $8-128$. [ABSTRACT FROM AUTHOR]

An $m\times (k+r)$ binary maximum distance separable (MDS) array code contains $k$ information columns and $r$ parity columns with each entry being a bit, where any $k$ out of $k+r$ columns can recover the $k$ information columns. When there is a failed column, it is critical to minimize the repair bandwidth that is the total number of bits downloaded from $d$ out of $k+r-1$ surviving columns in repairing the failed column. In this article, we first propose two explicit constructions of binary MDS array codes that have asymptotically optimal repair bandwidth for any information column, where $r\geq 2$ and $d=k+r-1$ for the first construction, and $r\geq 4$ is an even number and $d=k+\frac {r}{2}-1$ for the second construction. By applying a generic transformation for the proposed two classes of binary MDS array codes, we then obtain two classes of new binary MDS array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column. [ABSTRACT FROM AUTHOR]

We construct polar codes for the specific purpose of incremental redundancy hybrid automatic repeat request (IR-HARQ) schemes. The rate compatibility of our scheme is ensured by both puncturing and extending of the code. A new puncturing algorithm for polar codes is proposed, and we develop an algorithm for finding good extending sequences for polar codes from any arbitrary punctured rate, with the goal of improving the throughput as much as possible. Simulation results for different types of puncturing and extending algorithms are presented. We show how the proposed extending algorithm, when properly operated with a good puncturing algorithm and a well-chosen puncturing rate, yields IR-HARQ coding schemes which can operate within 1 dB of Shannon capacity over a very wide range of signal-to-noise ratios. [ABSTRACT FROM PUBLISHER]

Regenerating codes provide an efficient way to recover data at failed nodes in distributed storage systems. It has been shown that regenerating codes can be designed to minimize the per-node storage (called MSR) or minimize the communication overhead for regeneration (called MBR). In this work, we propose new encoding schemes for error-correcting MSR and MBR codes that generalize our earlier results on error-correcting regenerating codes. General encoding schemes for product-matrix MSR and MBR codes are derived such that the encoder based on Reed-Solomon (RS) codes is no longer limited to the Vandermonde matrix proposed earlier. Furthermore, MSR codes and MBR codes with the least update complexity can be found. A decoding scheme is proposed that utilizes RS codes to perform data reconstruction for MSR codes. The proposed decoding scheme has better error correction capability and incurs least number of node accesses when errors are present. A new decoding scheme is also proposed for MBR codes that is more capable and can correct more error-patterns. Simulation results are presented that exhibit the superior performance of the proposed schemes. [ABSTRACT FROM PUBLISHER]

We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM codes have a smaller error probability than polar codes under MAP decoding. This motivates us to introduce a family of codes that “interpolates” between RM and polar codes, call this family \cal Cinter=\C\alpha:\alpha\in[0,1]\, where C\alpha\vert\alpha=1 is the original polar code, and C\alpha\vert\alpha=0 is an RM code. Based on numerical observations, we remark that the error probability under MAP decoding is an increasing function of $\alpha$. MAP decoding has in general exponential complexity, but empirically the performance of polar codes at finite block lengths is boosted by moving along the family {\cal C}_{\rm inter} even under low-complexity decoding schemes such as, for instance, belief propagation or successive cancellation list decoder. We demonstrate the performance gain via numerical simulations for transmission over the erasure channel as well as the Gaussian channel. [ABSTRACT FROM AUTHOR]