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We consider in this letter the problem of blind frame synchronization of Low Density Parity-Check (LDPC) codes. We present and compare new methods of blind frame synchronization based on the calculation of Likelihood functions of the syndrome. Simulation results show that the four proposed synchronization techniques are effective once applied to LDPC codes. The method that involves the maximization of the Likelihood Difference (LD) function of the parity-checks, presents the best results in terms of probability of false synchronization.

In non-cooperative communications, blind identification of the widely used LDPC codes is an emerging research hot spot. This is different from and more difficult than blind identification of channel coding in cooperative communications. Common disadvantages of the existing algorithms are poor fault tolerance, high number of iterations, and only being applicable to short codes. Our proposed algorithm is based on the idea presented by Sicot et al. (2009) for binary phase-shift keying (BPSK) signals over the additive white Gaussian noise (AWGN) channel. Firstly, to sort the codewords according to a proper threshold, we present a threshold function and derive its zero point, which is verified to be the optimal threshold through experiments. This threshold improves the fault tolerance of our algorithm. Secondly, an operation called bidirectional Gaussian column elimination (BGCE) is proposed to replace Gaussian column elimination (GCE). This operation speeds up the process of deriving parity-checks, and considerably reduces the number of iterations. Thirdly, we use an existing technique for finding low-weight codewords as proposed by Canteaut and Chabaud (1998) to make the linearly independent parity-checks sparse, thereby realizing reconstruction of the sparse parity-check matrix of the LDPC code. The results of comparative experiments demonstrate that our algorithm outperforms existing algorithms.

In cognitive radio context, the coding parameters are unknown at the receiver. The design of an intelligent receiver is essentially to identify these parameters from the received data blindly. In this paper, we are interested in the blind identification of binary linear block codes from received noisy data. In order to recognize the code length, the concept of the normalized column weight vector is defined and cosine similarity is used to measure the difference between linear block codes and random codes. Then, the correct code length could be obtained by finding the local minimum of cosine similarity. The proposed code length recognition method needs no prior knowledge about the codes, which results in completely blind identification. To reconstruct the parity check matrix, the concept of association rules mining is introduced to the problem of blind identification of channel codes for the first time. Furthermore, five criteria are proposed to reduce the redundant rules mined by the association rules mining algorithm and to recognize the parity check vectors effectively. Simulations show that the proposed two methods have excellent performance even in a high error rate transport environment. The performance comparisons with existing methods validate the advantages of our two proposed methods.

Recently, erasure coding techniques are considered as essential schemes for the reliability of the modern distributed storage systems (DSSs) with the frequent node-level failure. Especially, locally repairable codes (LRCs) are widely adopted by the practical advantage of reducing the latency for repair process. However, recent researches show that many cases for system failure are also originated from the silent disk errors. For the conventional LRCs with low error correction capability, repair process from erasure coding can propagate the silent errors and thus, the DSSs become more vulnerable compared to the cases only with node failure. Therefore, we propose a mean time to data loss (MTTDL) from the modified Markov chain model in order to evaluate effects by silent disk errors. Also, new design of binary error-resilient locally repairable codes (ER-LRCs) with high error and erasure correction capabilities are proposed, which have larger values of bit-wise minimum Hamming distance than the existing LRCs. Here, ER-LRCs can be constructed by modifying the parity check matrix from well-known optimal binary and nonbinary LRCs. From the numerical analysis using the proposed Markov model with empirical parameters, it is shown that the proposed ER-LRCs have better MTTDL values when compared to the existing LRCs.

Syndrome-Trellis Code (STC) is a near-optimal convolutional method for adaptive steganography. Hitherto, the existing adaptive steganography commonly depends on the carefully designed distortion cost function, which controls the embedding position of the message in the cover signal. From another point of view, we implement adaptive steganography by improving the STC coding process (named Adaptive-STC). The parity-check matrix is the key to the encoding and extraction process of STC. In this work, we prove that the average embedding change probability of corresponding elements can be changed by adjusting its submatrix. Following that, we specially design an adaptive parity-check matrix to replace the designed distortion cost to restrict the embedding position. The generation of the adaptive parity-check matrix can be formulated as a multi-constrained integer programming problem in which the width of the submatrix is allocated at a fixed height. To solve this particular problem, we propose a targeted intelligent optimization algorithm (named GOAS) that can adaptively generate the parity-check matrix according to different audio cover. The experimental results show that the proposed method outperforms the state-of-the-art adaptive steganography with reduced embedding changes and improved audio quality while ensuring the ability against steganalysis.

Belief propagation (BP) is a soft-decision (SD) decoder that is commonly used to obtain near-optimal decoding performance for linear codes defined over sparse parity-check matrix. Nevertheless, the BP yields poor performance when used in the decoding of algebraic block codes, usually described by a dense parity-check matrix. Hence, enhanced BP decoders transform the parity check matrix of such codes at each iteration for efficient decoding. In this article, the performance of the transformed parity-check matrix of the iterative SD decoder is analysed for the class of binary cyclic codes using a perfect knowledge model (PKM). The PKM computes a list of candidate matrices and selects a baseline parity-check matrix according to a distance metric. The selected matrix is optimal since it minimizes the probability of error over various choices in the list. Results show that, for a given channel condition, the conventional transformed matrix obtained by Gaussian elimination is sub-optimal and does not necessarily contain unitary weighted columns at corresponding columns of the unreliable bits. Moreover, PKM can be used to verify the performances of newly developed iterative SD decoders for binary cyclic codes based on parity-check equations instead of maximum-likelihood decoding.

This paper proposes an algorithm to design rate-adaptive irregular LDPC codes with improved Bit error rate (BER) performance. It focuses on achieving better BER results for higher order coding rates which are in high demand for 5G mobile communications. The algorithm generates a parity check matrix based on the message block size. The encoding complexity of the irregular LDPC code is comparatively reduced due to the proposed parity check matrix. Simulation results compare the BER performances of irregular LDPC codes for various code rates and modulation schemes with the Parity check matrix (PCM) generated by the proposed algorithm. The BER performance of improved irregular LDPC code is analyzed in a real-time scenario using Software Defined Radio platform as well.

The manuscript presents the construction of a diagonally structured Parity check matrix using lower and upper (LU) decomposition for regular Quasi Cyclic-Low Density Parity Check Codes (QC-LDPC) with girth 12. This diagonally structured parity check matrix reduces the sparsity characteristics of QC-LDPC codes. The regular Quasi Cyclic-LDPC code has been constructed using circulant shifting and randomization method in which uniform distribution of non-zero elements, XOR operation, and pairing of row and column has been done. A Simplified Log Domain Sum-Product Algorithm (SLDSPA) is used for the decoding of the constructed code. To reduce the CPU run time of the constructed code, a column-wise loop optimization technique is applied. The frame error rate and CPU run time is calculated for different code lengths such as code1 (816, 408), code2 (1008, 504), code3 (1032, 516), code4 (1944, 972), code5 (672, 336) and code6 (504, 252). The construction of regular QC-LDPC codes on half code rate is obtained for the weight of each row is 6 and each column is given by 3. The results showing that the constructed code is better performing in terms of Low FER (10−1 to 10−7), Low SNR (0 to 3.5 dB) and Less CPU run time in milliseconds.

This paper proposes a novel adaptive image steganography method combining matrix coding, which can achieve better visual quality according to the given embedding rate compared with other existing schemes. The embedding processes are divided into two phases: preparation and implementation. In the preparation phase, all 128 combinations of a 7-bit binary number are classified into eight groups by a parity check matrix based on matrix coding. Thus, each group corresponds to an octal number, and contains 16 candidates. Correspondingly, all pixels of a cover image are classified into three categories by a predefined embedding rate level: low, middle, and high. In the implementation phase, each pixel in the high level can be embedded with 3 bits by substituting its 7 LSBs (Least Significant Bit) using a close candidate from the group determined by the to-be-embedded 3 bits. For the middle level, pixel pairs are used to hide 3-bit data by substituting its (3, 4) LSBs combination using a close candidate form the group determined by the to-be-embedded 3 bits. For pixels in the low level, a triple pixel combination is used to embed 3-bit data by substituting its (2, 2, 3) LSBs combination using a close candidate form the group determined by the to-be-embedded 3 bits. The experimental results indicate that the proposed scheme is optimal compared to other existing matrix-coding data hiding schemes in terms of visual quality, while providing an identical embedding capacity.

Single bit errors occur more frequently in memory elements. So for the avoidance of these single bit errors the Hamming code technique was recognized as the best among all other coding techniques. The most effective coding technique was Hamming [16, 11, 5] 2 which results in single error detection and correction as well as double adjacent error detection and correction. In this paper, a method to design SEC-DAEC codes with optimized decoding is presented and evaluated. Extended Hamming code is developed by adding an extra parity bit. The implementation is evaluated and compared by its application in an audio signal authentication in matlab with introduced errors The evaluations show there is a slight increase in area, power and delay of the decoder.

Neural Normalized MinSum (N-NMS) decoding delivers better frame error rate (FER) performance on linear block codes than conventional normalized MinSum (NMS) by assigning dynamic multiplicative weights to each check-to-variable message in each iteration. Previous N-NMS efforts have primarily investigated short-length block codes (N < 1000), because the number of N-NMS parameters to be trained is proportional to the number of edges in the parity check matrix and the number of iterations, which imposes am impractical memory requirement when Pytorch or Tensorflow is used for training. This paper provides efficient approaches to training parameters of N-NMS that support N-NMS for longer block lengths. Specifically, this paper introduces a family of neural 2-dimensional normalized (N-2D-NMS) decoders with with various reduced parameter sets and shows how performance varies with the parameter set selected. The N-2D-NMS decoders share weights with respect to check node and/or variable node degree. Simulation results justify this approach, showing that the trained weights of N-NMS have a strong correlation to the check node degree, variable node degree, and iteration number. Further simulation results on the (3096,1032) protograph-based raptor-like (PBRL) code show that N-2D-NMS decoder can achieve the same FER as N-NMS with significantly fewer parameters required. The N-2D-NMS decoder for a (16200,7200) DVBS-2 standard LDPC code shows a lower error floor than belief propagation. Finally, a hybrid decoding structure combining a feedforward structure with a recurrent structure is proposed in this paper. The hybrid structure shows similar decoding performance to full feedforward structure, but requires significantly fewer parameters., This paper has been accepted by ISTC2021

In another work [S. Huang and K. R. Brown, Phys. Rev. Lett. 127, 090505 (2021)] we introduced syndrome extraction methods for Calderbank-Shor-Steane quantum error-correction codes that interpolate between the well-known Shor and Steane syndrome extraction methods. Here we provide detailed proofs of the main theorems and show that up to gate ordering there is a one-to-one correspondence between extraction gadgets and partitions of the parity check matrix. Operationally, all the circuits in our framework can be obtained by merging ancilla qubits of Shor error correction with certain rules, which enables us to design fault-tolerant syndrome extraction circuits for given specific codes. We then apply our construction to the toric code and provide a detailed analysis of the time overhead of fault tolerance. In particular, we construct a syndrome extraction family whose time overhead smoothly varies from Shor to Steane error correction. To understand the potential advantage, we consider two simplified error models: no errors on the ancilla block and uncorrelated errors on the ancilla block. We study the threshold behavior of the family for both error models and show its potential advantage for quantum architectures with long coherence times.

Previous studies revealed that low-density parity-check convolutional codes (LDPC-CC) with rational parity-check matrices may outperform LDPC-CC with ordinary polynomial parity-check matrices. However, such a performance improvement relies on a dynamic scheduling-aided decoding scheme with high processing latency which may prevent LDPC-CC with rational parity-check matrices from practical applications. In this paper, we find that conventional belief-propagation algorithms suitable for high-speed parallel implementation can still provide the satisfactory performance gain for LDPC-CC with rational parity-check matrices as long as some virtual channel outputs which can accelerate the convergence of iterative decoding are properly supplied. Criteria for generating virtual channel outputs are investigated from the viewpoint of Tanner graph. Simulation results are also provided for performance verification.

Motivated by applications in unsourced random access, this paper develops a novel scheme for the problem of compressed sensing of binary signals. In this problem, the goal is to design a sensing matrix $A$ and a recovery algorithm, such that the sparse binary vector $\mathbf{x}$ can be recovered reliably from the measurements $\mathbf{y}=A\mathbf{x}+\sigma\mathbf{z}$, where $\mathbf{z}$ is additive white Gaussian noise. We propose to design $A$ as a parity check matrix of a low-density parity-check code (LDPC), and to recover $\mathbf{x}$ from the measurements $\mathbf{y}$ using a Markov chain Monte Carlo algorithm, which runs relatively fast due to the sparse structure of $A$. The performance of our scheme is comparable to state-of-the-art schemes, which use dense sensing matrices, while enjoying the advantages of using a sparse sensing matrix., Comment: Accepted to Entropy Special Issue on "Information-Theoretic Aspects of Non-Orthogonal and Massive Access for Future Wireless Networks"

Aiming at the problem that quasi-cyclic low density parity check (QC-LDPC) codes may have the error floor in the high signal to noise ratio (SNR) region, a new construction method of the QC-LDPC codes with the low error floor is proposed. The basic matrix of the method is based on the progressive edge growth (PEG) algorithm and the improved eliminate elementary trapping sets (EETS) algorithm so as to eliminate the elementary trapping sets in the basic matrix, then the Zig-Zag method is used to construct the cyclic shift matrix which is used to extend the basic matrix in order to construct the parity check matrix. The method not only can improve the error floor in the high SNR region, but also can flexibly design the code length and code rate. The simulation results show that at the bit error rate of 10−6, the PEG-trapping-Zig-Zag (PTZZ)-QC-LDPC(3024,1512) codes with the code rate of 0.5, compared with the PEG-Zig-Zag (PZZ)-QC-LDPC(3024,1512) codes and the PEG-QC-LDPC(3024,1512) codes, can respectively improve the net coding gain of 0.1 dB and 0.16 dB. The difference among the bit error rate performance curves will become better with the increase of the SNR. In addition, the PTZZ-QC-LDPC(3024,1512) codes have no error floor above the SNR of 2.2 dB.

In recent years, there has been a growing interest in codes that can correct adjacent bit errors in memories. This is due to the increasing percentage of radiation induced errors that affect multiple cells due to technology scaling. The cells affected by the errors are physically close and in many cases adjacent. This means that in the absence of interleaving, adjacent bits will be likely to be affected by an error. A number of Single Error Correction and Double Adjacent Error Correction (SEC-DAEC) codes have been proposed to deal with those errors. These codes have a low overhead in terms of number of additional parity check bits compared to Double Error Correction (DEC) codes. A problem is that they can introduce a significant penalty in terms of delay compared to a Single Error Correction (SEC) or a Single Error Correction and Double Error Detection (SEC-DED) code. This is due to the more complex decoding that needs to check approximately twice the number of syndrome patterns. In this paper, low delay SEC-DAEC codes are presented and the parity check matrices for 16, 32 and 64 bit data words are provided. To reduce the delay, a number of techniques are used in the construction of the code and in the implementation of the decoder. The evaluation shows that this results in significant savings compared to existing SEC-DAEC codes. Therefore, the proposed codes can be useful for high speed designs on which adjacent error correction is required.

This paper considers the problem of source coding with side information at the decoder, also called Slepian-Wolf source coding scheme. In practical applications of this coding scheme, the statistical relation between the source and the side information can vary from one data transmission to another, and there is a need to adapt the coding rate depending on the current statistical relation. In this paper, we propose a novel rate-adaptive code construction based on LDPC codes for the Slepian-Wolf source coding scheme. The proposed code design method allows to optimize the code degree distributions at all the considered rates, while minimizing the amount of short cycles in the parity check matrices at all rates. Simulation results show that the proposed method greatly reduces the source coding rate compared to the standard LDPCA solution., Comment: 25 pages, 7 figures, submitted to IEEE Transactions on Communications

The increasing importance of Multiple Cell Upsets (MCUs) in modern memories has spurred research on error correction codes that can correct adjacent bit errors. For example, Double Adjacent Error Correction (DAEC), 3-bit burst and 4-bit burst codes have been proposed in the last years. However, as the error correction capabilities are increased so is the complexity of the encoder and decoder circuits. This directly impacts the encoding and decoding delay and thus limits the use of the codes in high speed memories. To reduce the complexity, some techniques have been proposed recently. Those include more advanced optimization programs to find codes with a lower number of ones in the parity check matrix or the interleaving of data and parity check bits to simplify the decoding. In this paper, those techniques are combined to design more efficient 3-bit burst error correction codes. The encoders and decoders for the proposed codes have been implemented and compared with the state of the art 3-bit burst error correction codes. The results show that the new codes reduce the encoder and decoder delay significantly. Therefore, the proposed codes provide memory designers with a more efficient option to implement protection against 3-bit burst errors for high speed memories.

This work proposes a new code which can be considered as the modification of linear block code with code rate 1/2 and less than 1/2. The theme of the work is derived from the controllability of discrete time system and accordingly the generator matrix is chosen. A procedure is proposed in this paper which can transform the lower dimensional space to higher dimensional space. A Code is subspace of a space, so, with the help of addition of an equivalent identity matrix, between I (Identity matrix) and P (Parity Check) matrix of the established generator matrix, the lower dimensional space is transformed to higher dimensional space. Multiple numbers of errors could be corrected with the help of this designed code. It is also shown that decoding of this code can be established using the methods like syndrome detection and nearest neighbor decoding. The proposed code satisfies all the bounds. The minimum distance d, which could be achieved here, is at least equal to the total no. of information bits wh...

When the proton released due to the decay of radioactive atoms in the chip, strikes the memory, the contents in cell can be altered resulting in soft errors. This phenomenon can be even caused by cosmic rays. As the spacing between the cells is decreasing in the CMOS technology, multiple nearby cells get affected due to radiations. In this work, detection and correction mechanism using H-Matrix for multiple possibilities of double adjacent errors is presented. When 32-data bits are considered, correction of double adjacent errors up to 8 bits are correctable. Double adjacent means two burst errors each with maximum 4 error bits. Using three different parity check matrices, three different H-Matrix based codes with different numbers of parity bits are proposed and comparatively analysed for different parameters.

In this paper we address the problem of list decoding of linear codes over an integer residue ring Zq, where qis a power of a prime p. The proposed procedure exploits a particular matrix representation of the linear code over Zpr, called the standard form, and the p-adic expansion of the to-be-decoded vector. In particular, we focus on the erasure channel in which the location of the errors is known. This problem then boils down to solving a system of linear equations with coefficients in Zpr. From the parity-check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field pr−1Zpr. This yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased. We show that such an algorithm actually computes all possible erased coordinates, that is, the provided list is minimal. This work of the second and forth authors is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. The first author partially supported by Ministerio de Ciencia e Innovación via the grant with ref. PID2019-108668GB-I00. The third author is supported by the Swiss Confederation through the Swiss Government Excellence Scholarship no: 2019.0413 and by the Swiss National Science Foundation grant n. 188430.

Day by day the internet traffic is increasing rapidly. Challenges to improve these internet traffic requirements leads to evolution of 5G NR LDPC (New Radio Low Density Parity Check) codes which provides high coding gain and energy efficiency. BER Vs Eb/No dB curve defines the energy efficiency and coding gain of the any error correcting codes. LDPC codes which are adopted by 5G Standard are called 5G NR LDPC codes.In the design of the decoder,Message Passing Approach reduces the decoding complexiyt due to the sparsity of the parity check matrix.. An efficient decoding algorithm is the key to the successful code. In this paper an iterative minsum approximation message passing decoding algorithm has been implemented for both BG1 and BG2. Results show that the coding gain can be improved up to 9dB.

A new algorithm for efficient exact maximum likelihood decoding of polar codes (which may be CRC augmented), transmitted over the binary erasure channel, is presented. The algorithm applies a matrix triangulation process on a sparse polar code parity check matrix, followed by solving a small size linear system over GF(2). To implement the matrix triangulation, we apply belief propagation decoding type operations. We also indicate how this decoder can be implemented in parallel for low latency decoding. Numerical simulations are used to evaluate the performance and computational complexity of the new algorithm., Comment: Accepted to ISIT 2021

The main aim of this paper is to study various properties of extended binary Golay [24, 12, 8] code, when we know its generator matrix. Using MATLAB, we can study syndrome decoding, weight of a codeword, error correction and error detection of extended binary Golay [24, 12, 8] code.

The main aim of this paper is to study the encoding process of binary Schubert code [43,7] using MATLAB, when one knows its generator matrix. Using MATLAB, we can study syndrome decoding, weight of a codeword, error correction and error detection of the code.

The main aim of this paper is to study various properties of extended binary Hamming [8, 4, 4] code, when we know its generator matrix. Using MATLAB, we can study syndrome decoding, weight of a codeword, error correction and error detection of binary Hamming [8, 4, 4] code.

Linear codes are presented that can detect, locate and correct all repeated burst errors of length b or less which occur in multiple sub-blocks. We obtain lower and upper bounds on the number of check digits for the existence of these codes. Three examples, one for each type of code, are provided.

A new class of convolutional codes, called skew convolutional codes, that extends the class of classical fixed convolutional codes, is proposed. Skew convolutional codes can be represented as periodic time-varying convolutional codes but have a description as compact as fixed convolutional codes. Designs of generator and parity check matrices, encoders, and code trellises for skew convolutional codes and their duals are shown. For memoryless channels, one can apply Viterbi or BCJR decoding algorithms, or a dualized BCJR algorithm, to decode skew convolutional codes.

Quantum low-density generator matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities, displaying much improved performance in comparison to other sparse-graph codes. However, the nature of CSS designs and the manner in which they must be decoded limit the performance that is attainable with codes that are based on this construction. This motivates the search for quantum code design strategies capable of avoiding the drawbacks associated with CSS codes. In this article, we introduce non-CSS quantum code constructions based on classical LDGM codes. The proposed codes are derived from CSS QLDGM designs by performing specific row operations on their quantum parity check matrices to modify the associated decoding graphs. The application of this method results in performance improvements in comparison to CSS QLDGM codes, while also allowing for greater flexibility in the design process. The proposed non-CSS QLDGM scheme outperforms the best quantum low-density parity check codes that have appeared in the literature.

В даній роботі проаналізовано матриці перевірки на парність, які можуть бути використані в методі виправлення помилок на основі low density parity check (LDPC) матриць в системах квантового розподілу ключів. Система квантового розподілу ключів має неминучі помилки в просіяному ключі, які повинні бути виправлені алгоритмом виправлення помилок для створення захищеного ключа. У цьому аналізі 1000-бітні просіяні ключі розділяються на 50 частин, по 20 біт в кожній частині. Алгоритм створює 50 синдромів, по 10 біт в синдромі, що відповідають кожній частині, за допомогою перемноження матриць перевірки на парність розміром 10 × 20. Матриці перевірки на парність створюються алгоритмом, запропонованим Девідом Маккеєм та Редфордом Нілом. Процес створення синдрому складається з матричного перемноження 20-бітної частини просіяного ключа на матрицю перевірки на парність. Алгоритм посилає сформований синдром другій стороні. Під час передачі підслуховувач може перехопити синдром, але він не може дізнатись точне повідомлення з синдрому, навіть якщо він володіє матрицею перевірки на парність теж. Друга сторона також ділить її просіяний ключ на 50 частин, створює синдром з кожної частини і порівнює з отриманим синдромом. Якщо синдроми різні, ці частини просіяних ключів відкидають. Однак, через те, що довжина повідомлень складає 20 біт, а довжина синдромів 10 біт, можуть виникати ситуації, коли різні частини просіяного ключа матимуть однакові синдроми. Для даного випадку кожні 1024 повідомлень будуть мати один спільний синдром. Тому необхідно знайти таку оптимальну матрицю, яка знімає ймовірність отримання однакових синдромів від різних частин просіяного ключа. In this paper, the parity-check matrices that can be used in low density parity check (LDPC) based error correction method for quantum key distribution are analyzed. The quantum key distribution system has inevitable errors in sifted key that must be corrected by an error correction algorithm to create a secure key. In this analysis, 1000-bit sifted keys are divided into 50 parts. The algorithm creates 50 syndromes corresponding to each part by multiplying 10 × 20 bit parity-check matrices. The algorithm sends the generated syndrome to the other side, which also divides the sifted key into 50 parts, creates a syndrome from each part, and compares with the received syndrome. If the syndromes are different, these sifted key parts are discarded. However, there may be situations where different parts may have the same syndromes. Therefore, it is necessary to find such an optimal matrix that removes the probability of getting the same syndromes at different parts of the sifted key.

Steganography is the task of embedding a secret message in a cover media (e.g. image, video and voice) such that the stego media is not intuitively separable from the original ones. It is one of the interesting fields of security especially in transferring digital media as the major components of the web sites, emails and any web-based communications. Matrix embedding is a general approach used in many steganography schemes, especially where the cover size is small e.g. Voice Over IP (VOIP) packets. In this approach, the message is mapped to a series of bits (stego) to replace the same number of low significant bits in the cover. The embedding matrix is used for extracting the message by a linear combination of the stego bits. For a given matrix and secret message, there are specific series of stego-bits from which the message can be extracted. Embedding is done by an inverse problem to minimize a cost as the difference between the stego and the original bits. Hence, each message has a specific embedding cost based on the matrix. In this paper, a method is proposed to find an embedding matrix which minimizes the expectation of embedding cost for a uniform distribution of messages. To that end; a dynamic programming algorithm is proposed to efficiently find the expected cost for a given matrix. By use of this algorithm, any search method may be used to solve the problem. In this paper, a fast Hill-climbing search strategy is designed to find a local optimum matrix in an allowable time.

This paper proposes the use of systematic LDPC codes for hard decision Message Passing (hMP), which is a predecoder to Vector Symbol Decoding (VSD) for non-binary codes. VSD is a verification-based decoder whose outputs are decoded code words. LDPC are randomly generated and by nature, they are non-systematic codes. Using systematic instead of non-systematic codes allows the decoder to obtain the decoded data directly without a troublesome mapping step. In the simulations, 15 regular parity check matrices H are randomly generated and converted to their systematic forms. Results in a Gilbert-Elliot 2-state fading channel model with 6 different channel conditions show that hMP prefers codes with lower- weight H. For column weight of 9 bits and row weight of 18 bits, systematic H have lower weight after row operation and are better than the non-systematic ones for all channel conditions.

Bit-flips caused by single-event upsets (SEUs) are a well-known problem in computer memories. A SEU is a change of state (bit-flip) caused by one single ionizing particle (e.g., ions, electrons, photons) striking a sensitive node in a micro-electronic device. The error in the device output or in its operations caused as a result of the strike is called a SEU or a soft error. The conventional way to combat this is through error detection and correction (EDAC) circuitry, but for low-budget space missions a software implementation of the EDAC can be used. Algorithms of interest to this paper are Hamming codes, previously investigated as candidates for software based EDAC (see Shirvani et al.’s 2000 paper). The paper is focused on their cyclic version, in order to exploit the advantages of cyclic codes, but, in order to avoid their principal disadvantage, i.e., their error correction complexity, the implementation of a Hamming cyclic decoder based on its parity check matrix is proposed and evaluated in terms of frame error rate (FER) and union bound on FER results.

In this study, random codes are applied to the classical syndrome coding scheme to achieve secrecy of communications. By analysing the effect of the values of the columns of the parity check matrix on the resulting security level of communications, a code design method is presented which constructs a class of random codes, termed random permutation codes, which achieve high security levels and are easily generated. A theoretical analysis method is presented which determines the security level achieved by randomly chosen, linear binary codes, and compared with simulation results obtained by Monte Carlo analysis. The results verify the theoretical approach. In particular, the theoretical method is also suitable for analysis of long codes having a large number of parity check bits which are beyond evaluation by computer simulation. The results show that the security performance of any randomly chosen permutation code is close to that of the best equivocation code having the same code parameters. This has the practical advantage in syndrome coding of being able to use an ephemeral code for each communication session, thereby providing forward secrecy, a desired feature of modern, secure communication systems.

Error correction is a significant step in postprocessing of continuous-variable quantum key distribution system, which is used to make two distant legitimate parties share identical corrected keys. We propose an experiment demonstration of high speed error correction with multi-edge type low-density parity check (MET-LDPC) codes based on graphic processing unit (GPU). GPU supports to calculate the messages of MET-LDPC codes simultaneously and decode multiple codewords in parallel. We optimize the memory structure of parity check matrix and the belief propagation decoding algorithm to reduce computational complexity. Our results show that GPU-based decoding algorithm greatly improves the error correction speed. For the three typical code rate, i.e., 0.1, 0.05 and 0.02, when the block length is $10^6$ and the iteration number are 100, 150 and 200, the average error correction speed can be respectively achieved to 30.39Mbits/s (over three times faster than previous demonstrations), 21.23Mbits/s and 16.41Mbits/s with 64 codewords decoding in parallel, which supports high-speed real-time continuous-variable quantum key distribution system., 8 pages, 2 figures

This paper presents a novel linear error detection and correction approach for single and multiple bit error codes called low complexity parity check (LCPC) code. The LCPC code detects and corrects consecutive and non-consecutive bit errors. It can be used as a forward error correction scheme in the data transmission system of green wireless networks and the green Internet of Things. The proposed code improves network performance in terms of throughput, end-to-end delay, and bit error rate (BER). LCPC codes also have less complexity and lower memory requirements than Reed Solomon (RS) and low-density parity check (LDPC) codes because they have less non-zero elements in the generator matrix and the parity check matrix. Unlike LDPC codes, LCPC codes do not require reiteration in the decoding process. Various code rates of the LCPC code are proposed to reduce the complexity of the encoding and decoding process, which in turn decreases energy consumption. Simulation results show that the proposed LCPC (9, 4) code outperforms the popular LDPC codes. Compared with the LDPC (8, 4) with the decode bit flip algorithm, LCPC (9, 4) offers a coding gain of nearly 3 dB at a BER equal to 10−5.

In this paper, we design an irregular low-density parity-check (LDPC) codes for unequal inter-symbol interference channels based on the modified extrinsic information transfer chart. During transmission, each LDPC codeword will be divided and transmitted to the different channels. The aim of this scheme is to avoid transmitting the whole codeword in a high error rate channel. The simulation results show that the proposed code optimized for both PR1 and PR2 channels can achieve a 0.2 coding gain over the codes designed specifically for each individual channel. When this simple code is applied to a perpendicular channel with these two targets, the coding gains of the proposed code is about 0.08 dB at the frame error rate of $10^{-3}$ .

The key challenges in real time voice communication in long term evolution mobile are reduction in complexity and latency. Efficient encoding and decoding algorithms can cater to these. The implementation of such polar code based efficient algorithms is proposed in this paper. The overall latency of 3.8 ms is needed to process 8 bit block length. The novel sub-matrix near to identity matrix is presented. This resulted into minimization of loops among least reliable bits due to iterated parity check matrix. Look-up table based memory mapping is used in encoder to reduce latency while Euclidian decoding technique is used in decoder. The number of iterations is reduced by 50%. The experimentation is performed with additive white Gaussian noise and QPSK modulation. The proposed modified iterative decoding algorithm requires SNR of 5.5 dB and 192 computations for targeted bit error rate of 10−4. The second proposed method needs 9 dB, 2 iterations for 384 computations. The penalty paid is quantization error of 0.63% due to restricting computations to fourth order series of hyperbolic function with same 8 bit block length.

In view of the problems that the encoding complexity of quasi-cyclic low-density parity-check (QC-LDPC) codes is high and the minimum distance is not large enough which leads to the degradation of the error-correction performance, the new irregular type-II QC-LDPC codes based on perfect cyclic difference sets (CDSs) are constructed. The parity check matrices of these type-II QC-LDPC codes consist of the zero matrices with weight of 0, the circulant permutation matrices (CPMs) with weight of 1 and the circulant matrices with weight of 2 (W2CMs). The introduction of W2CMs in parity check matrices makes it possible to achieve the larger minimum distance which can improve the error- correction performance of the codes. The Tanner graphs of these codes have no girth-4, thus they have the excellent decoding convergence characteristics. In addition, because the parity check matrices have the quasi-dual diagonal structure, the fast encoding algorithm can reduce the encoding complexity effectively. Simulation results show that the new type-II QC-LDPC codes can achieve a more excellent error-correction performance and have no error floor phenomenon over the additive white Gaussian noise (AWGN) channel with sum-product algorithm (SPA) iterative decoding.

In this paper, we propose a novel extending algorithm to design rate compatible (RC) QC-IRA-d codes through Tanner Graph extension. The structure of QC-IRA-d codes provides high efficiency and low complexity for graph extension. Thus, in the proposed method, there are three features: (1) close relation between the extended parts and original parity check matrix; (2) larger length of introduced cycles; (3) balanced check node degree. All of them guarantee good performance of the designed RC family codes. Simulation results show that the graph extension method outperforms existed algorithms from 0.05 to 0.47 dB for different rate codes.

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes’ propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1–2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10–25 iterations less than the standard BP decoder.

Matrix embedding (ME) can be exploited to improve the embedding efficiency for steganography by making fewer changes to the cover data. In ME, the sender and recipient agree on a parity check matrix (PCM) in advance. The PCM will be used by the sender to embed secret message into the cover data and later by the decoder to extract the embedded message. The embedding performance of ME is greatly influenced by the PCM thus the choice of PCM is crucial for ME. On the other hand, since larger sized PCM usually leads to higher embedding efficiency, how to keep the balance between the computational complexity and the embedding efficiency is also an important problem of ME. Based on these considerations, an efficient ternary ME method for large payloads data embedding is proposed in this paper. We utilize a specific matrix construction for PCM to improve embedding efficiency and a sub-optimal search strategy to reduce the computational complexity. The experimental results show that the proposed method achieves good embedding efficiency at low time cost and it outperforms some state-of-the-art works. For example, for a cover image with 512 × 512 pixels at an embedding rate of 1 bit per pixel, the proposed method can be implemented within 0.5 second in a personal computer with a rather high embedding efficiency as 3.89.

The LDPC codes are Shannon Limit codes that can achieve low bit error rates for SNR applications. The features of LDPC Codes are reduction in the decoding time, latency and as well as no error-floors at high SNRs. The proposed algorithms are SBF, MSA, and MLDD. The various decoding algorithms have been compared for these codes. The parameters are describing the which algorithm further helps in selecting the better decoder used for Medical and Signal Processing Applications. These codes are also used in Generating the Barcodes depends on the size of the Parity Check matrix.

For correcting burst errors in channels with memory are widely used various code constructions together with interleaving. For increasing the reliability of information transmission in such channels is advisable to apply algebraic interleaved convolutional codes. The principles of polynomial and matrix representation of algebraic interleaved convolutional codes with predetermined coding rate are given. The parameters of these codes which completely determined by the modified generator polynomial are presented. Algebraic decoding these codes has been a large computational complexity and limited correction capability. Algebraic interleaved convolutional codes can be represented as a long binary block codes which can be determined with using generator or parity check matrix. The combination approach to iterative decoding these codes whose purpose is to find the most probable codeword by assessing a log likelihood ratio of each code symbol is proposed. The share using information about the reliability of symbols, adaptive belief propagation and natural computing procedures are offered. According to this approach each iteration of decoding starts with finding the expected codeword using natural computing procedures. If the received vector is not codeword, then adaptive belief propagation procedures are used for updating the information about the reliability of symbols which then used in the next iteration of decoding. The basic principles of implementation of the main steps of the proposed method decoding algebraic interleaving convolution codes are presented.

An improved extension method based on protograph extrinsic information transfer (PEXIT) charts is proposed to design rate-compatible quasi-cyclic low-density parity-check (QC-LDPC) codes with more flexible rates and better performance. The masking matrix of the highest-rate code can first be constructed column by column to own the lowest decoding threshold. Then, an extension algorithm is proposed to obtain the masking matrices of lower-rate codes with flexible rates. Finally, the base matrices with small amount of short cycles are designed and the parity-check matrices of rate-compatible QC-LDPC codes are obtained through matrix extension. Simulations show that compared with the existing codes, the proposed codes have more flexible rates while achieving better performance.

Most of cryptosystems, identification and signature schemes rely on the discrete logarithm problem and the prime factorization problem. These problems would be broken through Shor's quantum factorization algorithm in the case that quantum computers would come to exist. Code-based cryptography is one of post-quantum cryptography. There are several cryptography to reduce the key size by using the public matrix with the circulant structure, but some attacks have been proposed with the circulant structure as a weak point. In this paper, we propose a code-based identification scheme using high-density parity-check (HDPC) matrices. The poposed scheme is an improvement of Stern's identification scheme based on the binary syndrome decoding (BSD) problem. In the proposed scheme, we apply HDPC matrices to the BSD problem in order to prevent attacks using iterative decoding algorithm. Also, to prevent attack using circulant structure, we construct HDPC matrices of multiple block-row with circulant submatrices of different size. Furthermore, we evaluate the security level, key size, computational cost and security against existing attacks.

LDPC is being enormously used these days in number of communication applications for its reliably and resourcefulness over a certain channel. LDPC decoding “Sum-Product Algorithm and Minimum Sum Algorithm” are implemented and compared in this paper. The parity check matrix used is unique in its construction; sparse method is used to generate the matrix. The decoder is simulated in MATLAB. Performance and effectiveness of algorithms is verified under the presence of noise.

Recently, Compressive sensing (CS) paradigm has been considered as an attractive sampling method comparing with Nyquist Sampling Theorem. This is due to its ability to have the same efficiency by using a smaller number of samples. Within the centralized Cognitive Radio (CR) network; it can be individually applied at each Secondary User (SU) which leads to reduce its collected samples into a small number of the measurements. In this paper, this reduction is recognized by using the semi-orthogonal Regular Parity Check (RPC) matrix as a measurement matrix of CS paradigm. By this way, the overall samples that are going to be processed at the Fusion Centre (FC) to detect the holes are also reduced. This detection is carried on by using the l 2 - minimization as a CS recovery algorithm. The integration between the semi-orthogonal RPC matrix and the l 2 -minimization recovery algorithm helps for reducing the complexity of the l 2 - minimization recovery algorithm. Moreover, the results show that the used matrix enhances the ability of the l 2 - minimization recovery algorithm to estimate the status of the sub-channels. This is done with high accuracies at different number of SUs for different Signal to Noise Ratio (SNR) and without any prior information about PUs’ number.