CRC Code Research Articles

TS-WHT: A Two-Step Walsh–Hadamard Transform Approach for Blind Error Correcting Code Classification

Blind classification of error-correcting codes is essential for intelligent wireless communication, cognitive radio, and non-cooperative communication. The object of this paper is to investigate the differences in the Walsh spectrum between linear block code, convolutional code, and Turbo code. Based on these differences, a two-step Walsh Hadamard transform (TS-WHT) strategy is proposed for error-correcting code blind classification. Further, according to the detection theory, the peak detection threshold is detailed and optimized with the cyclic feature of CRC codes. The results indicate that when employing the optimized threshold, probabilities of correct classification are superior for linear block codes and convolutional codes. Moreover, with the help of the fault-tolerant feature of the WHT technique, the classification accuracy of the proposed method is substantially better than existing approaches under low signal-to-noise ratio(SNR) conditions.

Integer codes correcting single errors and detecting burst errors within two bytes

AbstractThe need for fast communication has led to the introduction of 100 Gbps optics in local area networks (LANs). To avoid retransmission of corrupted packets, in some LANs, the CRC codes were replaced by Reed‐Solomon (RS) codes. Although this solution has met its primary goal, it has two drawbacks: first, the payload size was significantly reduced, and second, each network node had to be equipped with much more complex hardware, which increased the cost of the network. Starting from the fact that the vast majority of corrupted packets contain only single errors, in this paper, a simple strategy for error control in 100 Gbps optical LANs is presented. More precisely, instead of protecting the data with the CRC or RS codes, using integer codes is suggested that can correct single errors and detect two types of burst errors: l‐bit burst errors within one b‐bit byte and h‐bit burst errors within two b‐bit bytes (1 ≤ h < l < b). Unlike the CRC or RS codes, the proposed ones use integer and lookup table operations, which are supported by all processors. In addition, the proposed codes can be interleaved without delay and without using any additional hardware. This feature allows the decoder not only to process larger data sets in parallel, but also to detect all double burst errors.

Forward Backward Syndrome Computation: A Reduced Complexity CRC Code Decoder

This paper proposes a simplified method for correcting a single-bit error event in a message protected by a CRC code thanks to a new decoding method called Forward-Backward Syndrome Computation. The two key ideas of the paper are to show that: 1) syndrome computation can be performed in any direction (Forward and Backward), 2) consequently, it is possible to create a partition of the received message in a set of disjoint syndromes, thus allowing a direct identification of the error position. Evaluation in terms of number of 2-input binary logic functions shows a complexity reduction by a factor of up to 3 compared to conventional methods.

Back-compatible Color QR Codes for colorimetric applications

Color correction techniques in digital photography often rely on the use of color correction charts, which require including this relatively large object in the field of view. We propose here to use QR Codes to pack these color charts in a compact form factor, in a fully compatible manner with conventional black and white QR Codes; this is, without losing any of their easy location, sampling and digital data storage features. First, we present an algorithm to build these new colored QR Codes that preserves the original QR Code functionality - much more than other coloring proposals based on the random substitution of black and white pixels by colors - that relies on the ability of the native CRC code to correct and counteract these alterations. Second, we demonstrate that, as a result, these QR Codes can allocate far many more colors than the conventional color correction charts, enabling much more accurate color correction schemes in a more convenient and usable format.

Polar Coding for Channels With Deletions

Deletion errors are notoriously difficult to correct. The capacity of the binary deletion channel is not yet fully known, and there is no completely satisfactory solution even to the specific problem of correcting only two deletion errors. In this paper, we show that polarization theory, along with the powerful methods of polar coding and successive-cancellation decoding, can be extended to channels with deletion errors. Our results further generalize to channels corrupted by insertion errors and/or combinations of insertions and deletions. It was recently proposed to correct deletion errors using polar codes designed for the BEC, precoded with a sufficiently powerful CRC code. In this approach, in order to correct ${d}$ deletions in a codeword of length ${n}$ , successive-cancellation decoding is run separately for each of the $\binom {n}{d}$ possible deletion patterns, while treating the deleted symbols as erasures. The resulting complexity of decoding is ${O}\bigl ({n}^{d+1}\log {n}\bigr)$ . In contrast, we develop herein a natural generalization of successive-cancellation decoding to channels with deletion errors. Our algorithm is based on the recursive structure of polar codes, processing the channel output in ${m} = \log _{2} {n}$ decoding layers, just like the conventional successive-cancellation decoder. Every node in each decoding layer propagates its uncertainty about the deletion pattern to the nodes in the next layer and, eventually, the correct deletion pattern becomes visible at the decoder output, with high probability. This makes it possible to consider at most $\binom {d+2}{2}$ scenarios at each node, and the overall decoding complexity is only ${O}\bigl ({d}^{3} {n} \log {n}\bigr)$ , which scales polynomially rather than exponentially with the number of deletions. Based on the proposed algorithm, we also investigate the channel polarization phenomenon. We are able to prove weak polarization where the number of deletions ${d}$ is small, i.e. ${d}={o}({n})$ . We also conjecture the strong polarization for when ${d}={O}(1)$ and provide some evidence to support this conjecture. The strong polarization, if proven, implies that our coding scheme achieves the capacity of any binary-input symmetric channel which furthermore introduces a constant number of deletions.

Deep Ensemble of Weighted Viterbi Decoders for Tail-Biting Convolutional Codes.

Tail-biting convolutional codes extend the classical zero-termination convolutional codes: Both encoding schemes force the equality of start and end states, but under the tail-biting each state is a valid termination. This paper proposes a machine learning approach to improve the state-of-the-art decoding of tail-biting codes, focusing on the widely employed short length regime as in the LTE standard. This standard also includes a CRC code. First, we parameterize the circular Viterbi algorithm, a baseline decoder that exploits the circular nature of the underlying trellis. An ensemble combines multiple such weighted decoders, and each decoder specializes in decoding words from a specific region of the channel words’ distribution. A region corresponds to a subset of termination states; the ensemble covers the entire states space. A non-learnable gating satisfies two goals: it filters easily decoded words and mitigates the overhead of executing multiple weighted decoders. The CRC criterion is employed to choose only a subset of experts for decoding purpose. Our method achieves FER improvement of up to 0.75 dB over the CVA in the waterfall region for multiple code lengths, adding negligible computational complexity compared to the circular Viterbi algorithm in high signal-to-noise ratios (SNRs).

Fast Block Sequential Decoding of Polar Codes

A reduced complexity sequential decoding algorithm for polar (sub) codes is described. The proposed approach relies on a decomposition of the polar (sub) code being decoded into a number of outer codes, and on-demand construction of codewords of these codes in the descending order of their probability. Construction of such codewords is implemented by fast decoding algorithms, which are available for many codes arising in the decomposition of polar codes. Further complexity reduction is achieved by taking hard decisions of the intermediate LLRs, and avoiding decoding of some outer codes. Data structures for sequential decoding of polar codes are described. The proposed algorithm can be also used for decoding of polar codes with CRC and short extended BCH codes. It has lower average decoding complexity compared with the existing decoding algorithms for the corresponding codes.

Performance Analysis of CRC-Polar Concatenated Codes

Polar code has been proven to obtain Shannon capacity for Binary Input Discrete Memoryless Channel (BIDMC) and its use has been proposed as the channel coding in 5G technology. However, its performance is limited in finite block length, compared to Turbo or LDPC codes. This research proposes the use of various CRC codes to complement Polar codes with finite block length and analyses the performance based on Block Error Rate (BLER) to Es/N0 (dB). The CRC codes used are of degrees 11 and 24, with 3 different polynomial generators for each degree. The number of bits in the information sequence is 32. The list sizes used are 1, 2, 4, and 8. Simulation results show that the concatenation of CRC and Polar codes will yield good BLER vs Es/N0 performance for short blocks of codeword, with rates 32/864 and 54/864. Concatenating CRC codes with Polar codes will yield a BLER performance of 10-2 with Es/N0 values of -9.1 to -7.5 dB when CRC codes of degree 11 is used, depending on the SC list used. The use of CRC codes of degree 24 enables a BLER performance of 10-2 with Es/N0 values of -7 to -6 dB when the SC list used is 1 or 2. The use of CRC codes of degree 24 combined with SC list with sizes 4 or 8 will improve the BLER performance to 10-2 with Es/N0 values of -8 to -7.5 dB

Spatial and temporal variations in Spain in the standardised ratio of in-hospital mortality due to colorectal cancer, 2008\u20132014

BackgroundColorectal cancer (CRC) is the second cause of tumour mortality in Spain and Europe. To date, no studies have been conducted in Spain to evaluate the spatial and temporal distribution of the excess risk of death during hospitalisation for CRC.MethodsA cohort was constructed of all episodes of hospitalisation in Spain due to CRC (codes 153 and 154 of the International Classification of Diseases, 9th edition, Clinical Modification) during the period 2008–2014, based on the minimum basic data set published by the Ministry of Health. Mortality ratios were calculated per region for each of the years analyzed (spatial or cross-sectional analysis) and during the overall study period, for each region independently (temporal or longitudinal analysis). In the first of these analyses, particular note was taken of the regions and years in which the limits of two and three standard deviations were exceeded.ResultsTwo hundred and fifty eight thousand, nine hundred and twenty seven episodes of CRC were analysed. The patients were predominantly male (60.6%), with an average hospital stay of 13.16 days. Half underwent surgery during admission and on average presented more than six diagnoses at discharge. The spatial analysis revealed mortality ratios that deviated by at least three standard deviations in the following regions: Islas Canarias, Asturias, Valencia, Extremadura, País Vasco and Andalucía. The longitudinal analysis showed that most regions presented one or more years when CRC mortality was at least 15% higher than expected during the period; outstanding in this respect were Asturias, Navarra and La Rioja, where this excess risk was detected in at least 2 years.ConclusionsGeographic and temporal patterns of the distribution of the excess risk of mortality from CRC in Spain are described using SMRs. We conclude that during the study period, the geographic pattern of mortality in Spain did not coincide with the excess risk of mortality calculated using the SMR method described by Jarman and Foster. This method of risk estimation can be a useful tool for the study of mortality risk and its spatial variations.

Исследование реализаций алгоритмов контрольной суммы CRC32

Objective: To give comparative assessment of the basic ways of generating a checksum (CRC code) based on direct, table and matrix algorithms. Methods: Algorithms were compared by means of mathematical methods. In order to achieve the result Java Development Kit software version 1.8 and NetBeans IDE8.2 development environment were used. Results: The methods of generating checksums by means of algorithms were described in detail. For each method under consideration, the time characteristics of their work were given. The comparison of the analyzed methods was conducted. Practical importance: Based on the results of the experiment, it was concluded which method was optimal for the generation of checksums.

A new lightweight RFID mutual authentication protocol based on CRC code

A new lightweight RFID mutual authentication protocol which the security keys updated dynamically is presented by using cyclic redundancy check code (CRC code) operations and some simple logic operations. GNY logic proof and security analysis show that the protocol not only achieve the mutual authentication requirements effectively between the readers and the tags, but also prevent many security privacy problems such as eavesdropping attack, replay attack, replication attack, the problem of tag tracking on the basis of without increasing computation and communication traffic of the RFID system. Especially, the new protocol meets the EPC Class1 Gen2 standard, compared with the same safety degree of other existing protocols, the new protocol is low complexity in the aspect of hardware implementation which consumes only 18.75% of the reserved space in terms of label storage and it is suitable for low-cost RFID system. Hence, the new protocol can realise the combination of high security and low tag cost.

A new lightweight RFID mutual authentication protocol based on CRC code

A new lightweight RFID mutual authentication protocol based on CRC code

Improving Person Re-Identification by Efficient Pairwise-Specific CRC Coding in the XQDA Subspace

Improving Person Re-Identification by Efficient Pairwise-Specific CRC Coding in the XQDA Subspace

Generic Soft-Error Detection and Correction for Concurrent Data Structures

Recent studies indicate that transient memory errors ( soft errors ) have become a relevant source of system failures. This paper presents a generic software-based fault-tolerance mechanism that transparently recovers from memory errors in object-oriented program data structures. The main benefits are the flexibility to choose from an extensible toolbox of easily pluggable error detection and correction schemes, such as Hamming and CRC codes. This is achieved by a combination of aspect-oriented and generative programming techniques. Furthermore, we present a wait-free synchronization algorithm for error detection in data structures that are used concurrently by multiple threads of control. We give a formal correctness proof and show the excellent scalability of our approach in a multiprocessor environment. In a case study, we present our experiences with selectively hardening the eCos operating system and its benchmark suite. We explore the trade-off between resiliency and performance by choosing only the most vulnerable data structures for error recovery. Thereby, the total number of system failures, manifesting as silent data corruptions and crashes, is reduced by 69.14 percent at a negligible runtime overhead of 0.36 percent.

The theory of parallel crc codes based on automaton models

A theoretical representation of the CRC codes with the aid of the finite automata in the Galois fields is proposed –linear finite-state machine (LFSM). We examined three types of LSC, which are distinguished by its characteristic matrices: type 1, type 2, and type 3. Three methods of the parallel CRC computation are proposed. The theorems are proven about the estimation of duration of the CRC computation by tuple-parallel method for three types of LFSM. We presented algorithms for the CRC tuple-parallel computation by the tables lookup for different types of LFSM and demonstrated their hardware implementation. For the symbolic-parallel and symbolic-tuple-parallel methods of CRC computation, symbolic LFSM over the non-binary Galois fields are applied. In contrast to the known approaches, which consider the problem of high speed only, present article stresses the need for taking into account a capability to detect errors with the aid of generator polynomials of the CRC codes. We proposed the Hamming codes and the Abramson codes in the binary and non-binary Galois fields for this purpose. Within the framework of single mathematical apparatus – theory of LFSM – it is possible to solve two important tasks simultaneously: to carry out mathematical substantiation of performance efficiency of the methods for parallel CRC computation and to estimate the detection and correction capability of the CRC codes based on LFSM graph models. Such comprehensive approach is relevant in the practice of various data transmission, storage and compression systems.

MODELLING OF SAFETY-RELATED COMMUNICATIONS FOR RAILWAY APPLICATIONS

Purpose. The authors according to requirements of the standard valid for safety-related communication between interlocking systems (EN 50159) analyze the situation of message undetected which is transmitted across BSC channel. The main part is orientated to description of model with CRC code used for messages assurance across noise communication channel. Methodology. For mathematical description of encoding, decoding, error detection and error correction is used algebra of polynomials. Findings. The determination of the intensity of dangerous failure caused by the electromagnetic interference was calculated. Originality. To obtain information on the probability of undetected error in transmission code and safety code and on the intensity of dangerous failure from the motel it was created program with graphical interface. To calculate the probability of undetected error for any block code (n, k) was created a supporting program that displays the probabilityof undetected error for selected interval of error bit rate. Practical value.From the measured and calculated values obtained by the simulation one can see that with increasing error bit rate is increasing also intensity of dangerous failures. Transmission code did not detect all corrupted messages therefore it is necessary to use safety code independent on transmission code in safety-related applications. CRC is not able to detect errors if all bits are logical 0.

Fountain Coded Wireless Transmission Model

Background/Objectives: Rateless codes have recently attracted much engrossment in community communication research as the Luby Transform codes and Raptor codes are most renowned. The Fountain codes do not have the requirement of the receiver to inform the transmitter of the identity of the packets that are corrupted resulting in expelling the need of automatic repeat request transmission. Analysis: The fountain codes have high performance near to shanon limit and they have very low bit error rates as compared to channel codes used, they further have an advantage over these codes in criterion like overhead and space required for storage. Erasure codes based Forward Error Correction schemes and transfer protocols that support such codes approach to distribution of data. Wireless communication system requires error control coding at the physical layer in order to communicate over noisy channels, but they can also benefit from erasure correcting coding at the higher layer. The aggregation of good channel code with CRC code can transform a channel with noise characteristics into erasure channel by discarding the packet that fail. Results/Findings: This paper revisits the fountain codes and proposes a research work which uses fountain coding in encoding processes for wireless channel transmissions for various future studies and analysis. Conclusion/Applications: The use of fountain codes seems to be the best solution for wireless transmission over the channels with vast packet failures and disturbances. These findings may be used for deployment in MIMO models and other wireless transmission techniques.

Circuit Module Design of High-Voltage Side for Optical Current Transformer

The Field Programmable Gate Array (FPGA) technology has been used for designing the signal processing circuit of A/D device, CRE code and Manchester module of high -voltage side for optical current transformer in this paper. The A/D controlling module, CRC code module, Manchester encoding module and the storage in FPGA has been designed and simulated. The signal processing circuit design program of high-voltage side for optical current transformer in this paper has been proved from the simulating results and can meet the requirements of optical current transformer on communication of data, which is rapid and reliability.

Theory and practice of crc codes: new results based on automaton models

Рассмотрены теоретические основы кодов CRC с помощью математического аппарата линейных последовательностных схем (ЛПС). Проанализирована интерпретация CRC как контрольной суммы (Cyclic Redundancy Check) и как укороченных циклических кодов (Cyclic Redundancy Code). Даны рекомендации по выбору порождающих полиномов для CRC . Предложен метод параллельного вычисления CRC с сокращением количества итераций в () раз для произвольного полинома степени .

고속 SDRAM에서 실시간 Matrix형 CRC

고속동작용 반도체 메모리 제품에 추가된 CRC는 DDR4와 같은 제품에서 데이타의 신뢰도를 증가시킨다. 기존의 CRC 방식은 부가회로 면적이 커고 많은 지연시간이 발생되어, CRC 계산을 위한 내부 타이밍 마진의 부족을 유발한다. 따라서 메모리 제품 설계에서 데이터 입출력 설계에 심각한 문제를 유발한다. 본 논문에서는 오류검출 회로설계를 위한 CRC 코드 방식을 제시하고, 실시간 matrix형 CRC 방법을 제안하였다. 데이터 비트오류 발생시 오류여부를 실시간으로 시스템에 피드백(feedback) 가능하도록 하였다. 제안한 방식은 기존방식(XOR 6단, ATM-8 HEC코드)대비 부가회로 면적을 60% 개선할 수 있으며, XOR 단 지연시간을 33%개선 할 수 있다. 또한 실시간 에러 검출 방식은 전체 데이터 비트(UI0~UI9)에 대해 평균 50% 이상 오류 검출 속도를 향상시켰다. CRC feature in a high-speed semiconductor memory devices such as DDR4/GDDR5 increases the data reliability. Conventional CRC method have a massive area overhead and long delay time. It leads to insufficient internal timing margins for CRC calculation. This paper, presents a CRC code method that provides error detection and a real-time matrix type CRC. If there are errors in the data, proposed method can alert to the system in a real-time manner. Compare to the conventional method(XOR 6 stage ATM-8 HEC code), the proposing method can improve the error detection circuits up to 60% and XOR stage delay by 33%. Also the real-time error detection scheme can improve the error detection speed to agerage 50% for the entire data bits(UI0~UI9).