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In this paper, we propose an image tampering localization algorithm using semi-fragile watermarking and Error-Locating codes in the DWT domain. By introducing different families of codes, we show the benefit in terms of image tampering localization and complexity of using control code error localization as an authentication function. Indeed, we first experimentally show that error localization block codes is as precise as using classical error correcting codes (Reed-Solomon and BCH codes) to locate image tampering. However, their corresponding decoding algorithms complexity is at least quadratic which make them impractical for some real time applications. To solve this problem, we introduce error-control codes called Error-Locating codes where error localization is reduced to a single syndrome computation performed with low number of binary operations (detailed later in the paper). We provide comparisons of image quality and tampering localization performances using error-detection, error-localization and error-correction approaches with different error control codes.

BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity. ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper.

Error Correction Codes (ECCs) are commonly used to protect memories against soft errors with an impact on memory area and delay. For large memories, the area overhead is mostly due to the additional cells needed to store the parity check bits. In terms of delay, the overhead is mostly needed to detect and correct errors when the data is read from the memory. Most ECCs that can correct more than one error have a complex decoding process and so are limited in high speed memory applications. One exception is One Step Majority Logic Decodable (OS-MLD) codes for which decoding can be done in parallel at high speed. Unfortunately, there are only a few OS-MLD codes that provide a limited choice in terms of block sizes, error correction capabilities and code rate. Therefore, there is considerable interest in a novel construction of OS-MLD codes to provide additional choices for protecting memories. In this paper, a new method to construct Double Error Correction (DEC) OS-MLD codes is presented. This method is based on the use of parity check matrices in which two bits have at most two parity check equations in common; the proposed method provides codes that require a smaller number of parity check bits than existing codes like Orthogonal Latin Square (OLS) codes. The drawback of the proposed Two Bit Overlap (TBO) codes is that they require slightly more complex decoding than OLS codes. Therefore, they provide an intermediate solution between OLS and non OS-MLD codes in terms of decoding delay and number of parity check bits. The proposed TBO codes have been implemented for some block sizes and compared to both OLS and BCH codes to illustrate the trade off in delay and memory overhead. Finally, this paper discusses the generalization of the proposed scheme to codes with larger error correction capabilities. Publicado

It is known that the bit errors of polar codes with successive cancellation (SC) decoding are coupled. However, existing concatenation schemes of polar codes with other error correction codes rarely take this coupling effect into consideration. To achieve a better error performance of concatenated systems with polar codes as inner codes, one can divide all bits in an outer block into different polar blocks to completely de-correlate the possible coupled errors in the transmitter side. We call this interleaving a blind interleaving (BI) which serves as a benchmark. Two BI schemes, termed BI-DP and BI-CDP, are proposed in the paper. To better balance performance, memory size, and the decoding delay from the de-interleaving, a novel interleaving scheme, named the correlation-breaking interleaving (CBI), is proposed. The CBI breaks the correlated information bits based on the error correlation pattern proposed and proven in this paper. The proposed CBI scheme is general in the sense that any error correction code can serve as the outer code. In this paper, Low-Density Parity-Check (LDPC) codes and BCH codes are used as two examples of the outer codes of the interleaving scheme. The CBI scheme 1) can keep the simple SC polar decoding while achieving a better error performance than the state-of-the-art (SOA) direct concatenation of polar codes with LDPC codes and BCH codes; 2) achieves a comparable error performance as the BI-DP scheme with a smaller memory size and a shorter decoding delay. Numerical results are provided to verify the performance of the BI schemes and the CBI scheme.

Embedded RAM blocks (BRAMs) in field programmable gate arrays (FPGAs) are susceptible to single event effects (SEEs) induced by environmental factors such as cosmic rays, heavy ions, alpha particles and so on. As technology scales, the issue will be more serious. In order to tackle this issue, two different error correcting codes (ECCs), the shortened Hamming codes and shortened BCH codes, are investigated in this paper. The concrete design methods of the codes are presented. Also, the codes are both implemented in flash-based FPGAs. Finally, the synthesis report and simulation results are presented in the paper. Moreover, the heavy-ion experiments are performed, the experimental results indicate that the error cross-section using the shortened Hamming codes can be reduced by two orders of magnitude compared with the device without mitigation, and no errors are discovered in the experiments for the device using the shortened BCH codes.

Semiconductor superlattice secure key distribution (SSL-SKD) has been experimentally demonstrated to be a novel scheme to generate and agree on the identical key in unconditional security just by public channel. The error correction in the information reconciliation procedure is introduced to eliminate the inevitable differences of analog systems in SSL-SKD. Nevertheless, the error correction has been proved to be the performance bottleneck of information reconciliation for high computational complexity. Hence, it determines the final secure key throughput of SSL-SKD. In this paper, different frequently-used error correction codes, including BCH codes, LDPC codes, and Polar codes, are optimized separately to raise the performance, making them usable in practice. Firstly, we perform multi-threading to support multi-codeword decoding for BCH codes and Polar codes and updated value calculation for LDPC codes. Additionally, we construct lookup tables to reduce redundant calculations, such as logarithmic table and antilogarithmic table for finite field computation. Our experimental results reveal that our proposed optimization methods can significantly promote the efficiency of SSL-SKD, and three error correction codes can reach the throughput of Mbps and provide a minimum secure key rate of 99%.

BCH codes are being widely used in commercial NAND flash controllers, and the decoding algorithm based on the Berlekamp-Massey (BM) algorithm is a classic solution for solving the key equation used for error correction. The latency of BM decoding is the bottleneck of the Bose-Chaudhuri Hocquenghem (BCH) decoder when correcting a high number of bit errors. However, the flash memory has an error distribution that degrades with usage: few errors occur in the new memory and a low number of errors occur within a code block. With usage, the system performance degrades and BM decoding needs t iterations in order to correct a larger number t of errors. In an attempt to improve the system performance for high speed applications, early termination of the BM decoding is necessary to overcome this degradation. In this paper, a practical solution for early termination checking for BM algorithm is provided. The analysis of proposed method is presented by means of considering the weight distribution of BCH code and deriving the probability of malfunction as the event of undetectable error. The proposed method is presented to be effective by the numerical results and the probability of malfunction for the proposed method is lower than 10−26. As a result, the FPGA testing on a USB device validate the reliability of the proposed method for applying to a commercial product.

In the heritage domain, capturing facts and knowledge for preventive conservation of Built Cultural Heritage (BCH) requires access to a large variety of data. It is a multidisciplinary activity and uses heterogeneous terminologies. In this regard, the BCH-ontology has been developed to facilitate integration and exchange of heterogeneous built cultural heritage information. The BCH-ontology reuses three already developed ontologies: Geneva City Geographic Markup Language (Geneva CityGML), Monument Damage ontology (Mondis), and CIDOC Conceptual Reference Model (CIDOC-CRM). Additionally, it provides a complete semantic framework by defining some classes and properties for improving BCH management. This paper presents the validation of the BCH-ontology ontological model to determine whether the ontology is able to represent BCH data under a preventive conservation approach. The San Luis seminary is a historical building built in the late XIX century in Cuenca-Ecuador and it is employed as use case. This validation allowed the identification of further use cases where the ontology offers a potential additional value in the BCH-domain.

Let p be an odd prime, and Fp(m) is the finite field of pm elements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2ps over Fp(m) are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2ps over finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2ps over Fp(m). To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.

Facing the challenge of enormous data sets variety, several machine learning-based algorithms for prediction (e.g, Support vector machine, multi layer perceptron and logistic regression) have been highly proposed and used over the last years in many fields. Error correcting codes (ECCs) are extensively used in practice to protect data against damaged data storage systems and against random errors due to noise effects. In this paper, we will use machine learning methods, especially multi-class logistic regression combined with the famous syndrome decoding algorithm. The main idea behind our decoding method which we call logistic regression decoder (LRDec) is to use the efficient multi-class logistic regression models to find errors from syndromes in linear codes such as bose, ray-chaudhuri and hocquenghem (BCH), and the quadratic residue (QR). Obtained results of the proposed decoder have a significant benefit in terms of bit error rate (BER) for random binary codes. The comparison of our decoder with many competitors proves its power. The proposed decoder has reached a success percentage of 100% for correctable errors in the studied codes.

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., Accepted for publication in IEEE Transaction on Communications, July 2021

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

Forward error correction coding is the most common way of channel coding and the key point of error correction coding. Therefore, the recognition of which coding type is an important issue in non-cooperative communication. At present, the recognition of FEC codes is mainly concentrated in the field of semi-blind identification with known types of codes. However, the receiver cannot know the types of channel coding previously in non-cooperative systems such as cognitive radio and remote sensing of communication. Therefore, it is important to recognize the error-correcting encoding type with no prior information. In the paper, we come up with a neoteric method to identify the types of FEC codes based on Recurrent Neural Network (RNN) under the condition of non-cooperative communication. The algorithm classifies the input data into Bose-Chaudhuri-Hocquenghem (BCH) codes, Low-density Parity-check (LDPC) codes, Turbo codes and convolutional codes. So as to train the RNN model with better performance, the weight initialization method is optimized and the network performance is improved. The experimental result indicates that the average recognition rate of this model is 99% when the signal-to-noise ratio (SNR) ranges from 0 dB to 10 dB, which is in line with the requirements of engineering practice under the condition of non-cooperative communication. Moreover, the comparison of different parameters and models show the effectiveness and practicability of the algorithm proposed.

Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman sums., 16 pages. To appear in Advances in Mathematics of Communications

The increase in capacity demand for vehicular communication is generating interest among researchers. The standard spectra allocated to VANET tend to be saturated and are no longer enough for real-time applications. Millimeter-wave is a potential candidate for VANET applications. However, millimeter-wave is susceptible to pathloss and fading, which degrade system performance. Beamforming, multi-input multi-output (MIMO) and diversity techniques are being employed to minimize throughput, reliability and data rate issues. This paper presents a tractable channel model for VANET in which system performance degradation due to error is addressed by concatenated Alamouti space-time block coding (ASTBC) and Bose–Chaudhuri–Hocquenghem (BCH) coding. Two closed-form approximations of bit error rate (BER), one for BCH in Rayleigh fading and the second for BCH with ASTBC, are derived. These expressions comprise SNR and code rate and can be utilized in designing VANET architectures. The results show that the BER using concatenated ASTBC and BCH outmatches the conventional BER ASTBC expression. The analytical results are compared with numerical results, thereby showing the accuracy of our closed-form expressions. The performance of the proposed expressions is evaluated using different code rates.

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

Polar codes have drawn much research attention in the last ten years for their capacity-achieving property. However, their conventional successive cancellation decoding method performs not well at a short or moderate length. In order to improve the performance, concatenation with other error-correction codes has been proved an effective approach, whereas current concatenation schemes using rate-optimized method are too complex to implement with long decoding latency. In this paper, we propose a critical set protected BCH-Polar code with its corresponding decoding architecture. In the proposed concatenation scheme, we only provide extra protection to partial information bits in the critical set, which is constructed based on the channel reliability. For its corresponding decoding architecture, we redesign some components and adopt the Look-up table decoding method for BCH codes, resulting in much degradation of decoding latency. Compared with existing decoders, the hardware implementation shows low decoding latency and high throughput-area efficiency.

This paper deals with a comparative analysis of the performances of two different communications standards for wireless body area networks (WBAN): ETSI SmartBAN and Bluetooth low energy (BLE). The performance comparison is carried out in additive white Gaussian noise (AWGN) channel as well as in multipath fading channel. Aggregate interference is also taken into account to show which technology tolerates interference better. Both SmartBAN and BLE are possible wireless solutions to implement services like delivering individual’s health-related data. The results show that the SmartBAN can outperform BLE in both AWGN and fading channel, in particular, when interference is present, since SmartBAN can take advantage of BCH and repetition features.

Ensemble models are widely used to solve complex tasks by their decomposition into multiple simpler tasks, each one solved locally by a single member of the ensemble. Decoding of error-correction codes is a hard problem due to the curse of dimensionality, leading one to consider ensembles-of-decoders as a possible solution. Nonetheless, one must take complexity into account, especially in decoding. We suggest a low-complexity scheme where a single member participates in the decoding of each word. First, the distribution of feasible words is partitioned into non-overlapping regions. Thereafter, specialized experts are formed by independently training each member on a single region. A classical hard-decision decoder (HDD) is employed to map every word to a single expert in an injective manner. FER gains of up to 0.4dB at the waterfall region, and of 1.25dB at the error floor region are achieved for two BCH(63,36) and (63,45) codes with cycle-reduced parity-check matrices, compared to the previous best result of the paper "Active Deep Decoding of Linear Codes".

Self-orthogonal codes are an interesting type of linear codes due to their wide applications in communication and cryptography. It is known that self-orthogonal codes are often used to construct quantum error-correcting codes, which can protect quantum information in quantum computations and quantum communications. Let $\mathcal C$ be an $[n, k]$ cyclic code over $\mathbb {F}_{q}$ , where $\mathbb {F}_{q}$ is the finite field of order $q$ . The hull of $\mathcal C$ is defined to be the intersection of the code and its dual. In this paper, we will employ the defining sets of cyclic codes to present two general characterizations of the hulls that have dimension $k-1$ or $k^\perp -1$ , where $k^\perp $ is the dimension of the dual code $\mathcal C^\perp $ . Several sufficient and necessary conditions for primitive and projective BCH codes to have $(k-1)$ -dimensional (or $(k^\perp -1)$ -dimensional) hulls are also developed by presenting lower and upper bounds on their designed distances. Furthermore, several classes of self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. The dimensions and minimum distances of some self-orthogonal codes are determined explicitly. In addition, several optimal codes are obtained.