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In this paper, we discuss the simple current orbifold of a rational Narain CFT (Narain RCFT). This is a method of constructing other rational CFTs from a given rational CFT, by 'orbifolding' the global symmetry formed by a particular primary fields (called the simple current). Our main result is that a Narain RCFT satisfying certain conditions can be described in the form of a simple current orbifold of another Narain RCFT, and we have shown how the discrete torsion in taking that orbifold is obtained. Additionally, the partition function can be considered a simple current orbifold with discrete torsion, which is determined by the lattice and the B-field. We establish that the partition function can be expressed as a polynomial, with the variables substituted by certain q -series. In a specific scenario, this polynomial corresponds to the weight enumerator polynomial of an error-correcting code. Using this correspondence to the code theory, we can relate the B-field, the discrete torsion, and the B-form to each other. [ABSTRACT FROM AUTHOR]

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e ≥ 3 , and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q > 3 . In the 1970s, this was proved for q a prime power, for q = 2 r 3 s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q. [ABSTRACT FROM AUTHOR]

In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the combinatorial setting where the maximum number of errors inflicted by an adversary is proportional to the number of transmissions, which goes to infinity. Without feedback, it is known that the rate of optimal asymmetric-error-correcting codes for the error fraction $\tau \ge 1/4$ vanishes as the blocklength grows. In this paper, we give an efficient feedback encoding scheme with $n$ transmissions that achieves a positive rate for any fraction of errors $\tau < 1$ and $n\to \infty $. Additionally, we state an upper bound on the rate of asymptotically long feedback asymmetric error-correcting codes. [ABSTRACT FROM AUTHOR]

We study the application of Hamming codes to wet paper steganography. To that end, we propose the use of decoding algorithms that do not verify the minimum distance property and present one of these algorithms. We study its properties and show results of some numerical experiments. [ABSTRACT FROM AUTHOR]

The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$. [ABSTRACT FROM AUTHOR]

In this paper, we use multivariate polynomial rings to construct quantum error-correcting codes (QECCs) via Hermitian construction. We establish a relation between linear codes and ideals of multivariate polynomial rings. We give a necessary and suffcient condition for a multivariate polynomial to generate a Hermitian dual-containing code. By comparing with the literatures in recent years, we construct 31 new QECCs over F q , where q = 3 , 4 , 5 , 7 . Some of them reach quantum singleton bound and some of them exceed quantum GV bound. [ABSTRACT FROM AUTHOR]

High temperatures have dramatic negative effects on interconnect performance. In a bus, whenever the state transitions from "0" to "1", or "0" to "1", joule heating causes the temperature to rise. A low-power error-correcting cooling (LPECC) code, introduced in Chee et al. (IEEE Trans Inf Theory 64:3062–3085, 2018), is a coding scheme which can be used to control the peak temperature, the average power consumption of on-chip buses and error-correction for the transmitted information, simultaneously. In this paper, we show upper bounds and some lower bounds of LPECC codes by some combinatorial configurations, and also present several families of optimal LPECC codes. [ABSTRACT FROM AUTHOR]

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG Γ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG Γ (v , k , t , λ , μ) with μ = 1 , including some mixed Moore graphs and corresponding codes arising from them, in this paper. [ABSTRACT FROM AUTHOR]

Sidon sets have several applications in mathematics and in real-world problems, including the generation of secret keys in cryptography, error-correcting codes, and the physical problem of compression of signals in telecommunications. In particular, in cryptography, the design of cryptographic functions with optimal properties like nonlinearity and differential uniformity plays a fundamental role in the development of secure cryptographic systems. Based on the construction of Bose-type Sidon sets, in this paper we present the construction of a new cryptographic function with good properties of nonlinearity and differential uniformity. [ABSTRACT FROM AUTHOR]

Secret-sharing is a vital topic of cryptography and has widely used in information protection. One technique to the development of secret-sharing schemes is primarily based on error-correcting codes. In this paper, we determine the minimal codewords of relative four-weight codes and have the utility in the secret sharing schemes. [ABSTRACT FROM AUTHOR]

BCH codes are an interesting class of cyclic codes with good error-correcting capability and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Let F q be the finite field of size q and n = q m - 1 , where m is a positive integer. Let C (q , m , δ) be the primitive narrow-sense BCH codes of length n over F q with designed distance δ . Denote s = m - t , r = m mod s and λ = ⌊ t / s ⌋ . In this paper, we mainly investigate the dimensions and Bose distances of the codes C (q , m , δ) with designed distance of the following two types: δ = q t + h , ⌈ m 2 ⌉ ≤ t < m , 0 ≤ h < q s + ∑ i = 1 λ - 1 q r + i s ; δ = q t - h , ⌈ m 2 ⌉ < t < m , 0 ≤ h < (q - 1) ∑ i = 1 s q i . This extensively extends the results on Bose distance in Ding et al (IEEE Trans Inf Theory 61(5):2351–2356, 2015). Moreover, the parameters of the hulls of the BCH code C (q , m , q t) are studied in some cases. [ABSTRACT FROM AUTHOR]

There are various metrics for researching error-correcting codes. Especially, high-density data storage system gives the existence of inconsistency for the reading and writing process. The symbol-pair metric is motivated for outputs that have overlapping pairs of symbols in a certain channel. The Rosenbloom–Tsfasman (RT) metric is introduced since there exists a problem that is related to transmission over several parallel communication channels with some channels not available for the transmission. In this paper, we determine the minimum symbol-pair weight and RT weight of repeated-root cyclic codes over R = F p m [ u ] / ⟨ u 4 ⟩ of length n = p k . For the determination, we explicitly present third torsional degree for all different types of cyclic codes over R of length n. [ABSTRACT FROM AUTHOR]

Low-density parity check (LDPC) codes, are a family of error-correcting codes, their performances close to the Shannon limit make them very attractive solutions for digital communication systems. There are several algorithms for decoding LDPC codes that show great diversity in terms of performance related to error correction. Also, very recently, many research papers involved the genetic algorithm (GA) in coding theory, in particular, in the decoding linear block codes case, which has heavily contributed to reducing the bit error rate (BER). In this paper, an efficient method based on the GA is proposed and it is used to improve the power of correction in terms of BER and the frame error rate (FER) of LDPC codes. Subsequently, the proposed algorithm can independently decide the most suitable moment to stop the decoding process, moreover, it does not require channel information (CSI) making it adaptable for all types of channels with different noise or intensity. The simulations show that the proposed algorithm is more efficient in terms of BER compared to other LDPC code decoders. [ABSTRACT FROM AUTHOR]

Error-correcting codes are known to define chiral 2d lattice CFTs where all the U(1) symmetries are enhanced to SU(2). In this paper, we extend this construction to a broader class of length-n codes which define full (non-chiral) CFTs with SU(2)n symmetry, where n = c + c ¯ . We show that codes give a natural discrete ensemble of 2d theories in which one can compute averaged observables. The partition functions obtained from averaging over all codes weighted equally is found to be given by the sum over modular images of the vacuum character of the full extended symmetry group, and in this case the number of modular images is finite. This averaged partition function has a large gap, scaling linearly with n, in primaries of the full SU(2)n symmetry group. Using the sum over modular images, we conjecture the form of the genus-2 partition function. This exhibits the connected contributions to disconnected boundaries characteristic of wormhole solutions in a bulk dual. [ABSTRACT FROM AUTHOR]

While several papers have explored the application of deep learning for COVID-19 detection in chest X-ray images, the consideration of images taken under emergency conditions remains limited. In this paper, we define emergency conditions as situations where inevitable errors occur during X-ray imaging in severe respiratory conditions of COVID-19 patients. These errors include positioning errors, protocol parameter errors, and motion artifacts. Our research aims to propose an intelligent tool based on deep neural networks and ensemble learning to detect COVID-19 cases from chest X-ray images in both standard and emergency conditions. As comprehensive datasets incorporating emergency images are scarce, we first generated a local dataset by investigating COVID-19 cases in our local hospital. We specifically selected cases with the aforementioned errors caused by emergency conditions. This dataset comprises patients in acute situations, as well as technical radiographic errors resulting from a significant number of people requiring radiography images. We evaluated the performance of recent popular pre-trained deep neural networks on our local dataset, which highlighted the importance of considering such emergency images. Subsequently, we transfer-learned 11 popular deep neural networks using our local dataset. Furthermore, we identified the transferred neural networks with the highest accuracies (exceeding 95% accuracy) and implemented ensemble learning methods such as Majority Voting and Error-Correcting Output Codes (ECOC). Our proposed methods excel in identifying COVID-19 in both emergency and non-emergency chest X-rays with high accuracy. [ABSTRACT FROM AUTHOR]

In order to improve the multi-class assessment performance of transient stability in power systems, a multi-class assessment model that combines the CLV-GAN algorithm with an improved error-correcting output coding technique is proposed in the paper. To address the issue of the small number of unstable samples in power systems, a sample generation model is constructed by combining a dual-encoder VAE with a GAN network. The model generates effective artificial samples to balance the sample ratio between categories by learning the latent distribution of aperiodic and oscillatory unstable samples from the distribution. The decomposition method based on an improved error-correcting output coding algorithm is applied to convert the multi-class problem into a decision fusion issue for binary models. This method improves the overall performance of the multi-class model, particularly significantly increasing the recognition accuracy of discrimination against oscillatory unstable samples and reducing the safety hazards in the operation of power systems. The simulation validation was conducted on the IEEE 39-bus and IEEE 140-bus systems to confirm the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of quantum error-correcting (QEC) codes, which can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and by using preshared entangled states between the sender and the receiver. In this paper, we investigate EAQEC codes of length n = 2 (q 2 + 1) a , where q is an odd prime power, a = m 2 + 1 and m is an odd integer. The resulting EAQEC codes are entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes when the minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]

This paper proposes Value-Aware solid-state drive (SSD) with fast access speed and low power consumption by eliminating error-correcting code (ECC). Value-Aware SSD utilizes the error tolerance of image recognition application using a deep neural network (DNN) to enhance reliability. In a previous paper, which proposes Value-Aware SSD, fast ECC decoder is implemented and SSD is evaluated with the 32-bit floating-point data format. On the other hand, in this paper, the proposed Value-Aware SSD is analyzed with 32-bit and 8-bit fixed-point data format and achieves the higher reliability even without ECC by newly proposed two techniques, Critical Bit Error Reduction (CBER) and Middle & Lower Page Error Reduction (M&L-PER). CBER and M&L-PER are proposed for 32-bit and 8-bit data format of application, respectively. These techniques modulate the threshold voltage ($V_{\mathrm {TH}}$) distribution of memory cells by recognizing the importance of each stored bit. By proposed CBER, as much as 15% bit error rate (BER) of NAND flash is allowed while the application provides high image recognition accuracy. Even if bit precision is truncated to 8 bit, 3.9% BER is accepted by M&L-PER. The fast read access and low power consumption are realized because ECC is not required. Finally, this paper analyzes the Value-Aware techniques with 3-D multi-level cell (MLC) NAND flash to compare the effects for 3D-MLC and triple-level cell (TLC) NAND flash. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to construct pliable fractional repetition (FR) codes that are locally recoverable for distributed storage systems (DSSs). FR codes are integral in constructing a class of distributed storage codes with exact repair by transfer. Pliable FR codes are a new type of FR codes in which both the per-node storage and repetition degree can easily be adjusted simultaneously; thus, pliable FR codes are vital for DSSs in which parameters can dynamically change over time. However, the constructions of pliable FR codes with repair locality remain unknown. In addition, the tradeoffs between the code minimum distance of an FR code and its repair locality are not fully understood. To address these problems, this paper first presents general results regarding FR codes. Subsequently, this paper presents an improved Singleton-like bound for locally recoverable FR codes under an additional requirement that each node must be part of a local structure that, upon failure, allows it to be exactly recovered by a simple download process. Moreover, this paper proposes a construction of locally recoverable FR codes that can achieve the proposed Singleton-like bound; this construction is based on bipartite graphs with a given girth. In particular, this paper also proposes a general bipartite-graph-based approach to constructing optimal pliable FR codes with and without repair localities; in this approach, a new family of bipartite graphs, called matching-feasible graphs, is introduced. Finally, this paper proposes the explicit constructions of optimal pliable FR codes by using a family of matching-feasible graphs with arbitrary large girth. Notably, in addition to attaining a Singleton-like bound for FR codes, the explicit pliable FR codes are optimal locally recoverable FR codes from two perspectives of repair locality. The explicit pliable FR codes can also be used as FR batch codes to provide load balancing in DSSs. [ABSTRACT FROM AUTHOR]

Material identification is playing an increasingly important role in various sectors such as industry, petrochemical, mining, and in our daily lives. In recent years, material identification has been utilized for security checks, waste sorting, etc. However, current methods for identifying materials require direct contact with the target and specialized equipment that can be costly, bulky, and not easily portable. Past proposals for addressing this limitation relied on non-contact material identification methods, such as Wi-Fi-based and radar-based material identification methods, which can identify materials with high accuracy without physical contact; however, they are not easily integrated into portable devices. This paper introduces a novel non-contact material identification based on acoustic signals. Different from previous work, our design leverages the built-in microphone and speaker of smartphones as the transceiver to identify target materials. The fundamental idea of our design is that acoustic signals, when propagated through different materials, reach the receiver via multiple paths, producing distinct multipath profiles. These profiles can serve as fingerprints for material identification. We captured and extracted them using acoustic signals, calculated channel impulse response (CIR) measurements, and then extracted image features from the time–frequency domain feature graphs, including histogram of oriented gradient (HOG) and gray-level co-occurrence matrix (GLCM) image features. Furthermore, we adopted the error-correcting output code (ECOC) learning method combined with the majority voting method to identify target materials. We built a prototype for this paper using three mobile phones based on the Android platform. The results from three different solid and liquid materials in varied multipath environments reveal that our design can achieve average identification accuracies of 90% and 97%. [ABSTRACT FROM AUTHOR]

In this paper, we present a construction of quantum error-correcting codes (QECCs) codes and entanglement-assisted quantum error-correcting (EAQECCs) using Euclidean hulls and sums of cyclic codes of length n over a family of ring R s = F q + v 1 F q + v 2 F q + ⋯ + v s F q , where q is an odd prime power and v i 2 = v i , v i v j = v j v i = 0 , for i , j = 1 , 2 , 3 , ⋯ , s and i ≠ j . The study delves into various aspects of this construction. We explore the generator polynomials, the dimension of both Euclidean hulls and the sums of cyclic codes over the ring R s . Further, we determine several new QECCs and EAQECCs. This paper claims that our obtained codes have improved parameters (e.g. higher minimum distance or greater dimension) than the existing quantum codes. Moreover, we present some detailed examples that effectively illustrate our findings. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes such as generalized Reed-Solomon codes. In this paper, we define a new extended generalized Reed-Solomon code, i.e., lengthened generalized Reed-Solomon code, which has good algebraic structure and many excellent properties, thus we extend the error-correcting pair to the case for lengthened generalized Reed-Solomon codes. Firstly, we give some sufficient conditions for which an LGRS code is non-GRS, and a necessary and sufficient condition for an LGRS code to be MDS or AMDS, respectively. And then, we constructively determine the existence of the error-correcting pair for lengthened generalized Reed-Solomon codes and give several examples to support our main results. [ABSTRACT FROM AUTHOR]

Iterative stochastic decoding is an alternative to standard fixed-point decoding of low-density parity check (LDPC) codes that can be used to minimize inter-node routing. A flexible field-programmable gate array (FPGA)-based stochastic decoding (SD) hardware architecture is presented in this paper. The architecture is designed to decode different code rates of LDPC codes that comply with the fifth-generation (5G) New Radio (NR) standard. This decoder's runtime flexibility is desirable as it switches to a better-performing code rate automatically based on the channel conditions without the extra time needed to reprogram the FPGA. An offline design method is implemented to generate the hardware description language (HDL) code description of the decoder for the required code rate set, which is further synthesized and integrated into a Xilinx Kintex-7-series FPGA board to determine the hardware resource utilisation and processing throughput. Synopsys design tools were employed during both the simulation and synthesis stages in combination with TSMC 65 nm CMOS standard cell technology to facilitate comparative analysis. Compared with state-of-the-art designs, the proposed architecture reduces hardware utilization by up to 26% and increases energy efficiency by 52%. [ABSTRACT FROM AUTHOR]

In 2006, Brun, Devetak and Hsieh developed the theory of entanglement-assisted quantum error-correcting codes (EAQECCs), which generalizes that of quantum stabilizer codes. The study of EAQECCs is currently a hot topic in coding theory and quantum information theory. Let q = p e be a prime power and let ℓ be an integer with 0 ≤ ℓ ≤ e - 1 . The ℓ -Galois hulls of MDS codes are an effective tool for constructing MDS EAQECCs. In this paper, we construct twenty-two families of q-ary MDS codes with ℓ -Galois hulls satisfying 2 (e - ℓ) ∣ e , which are not covered by those available in Cao (Galois hulls of MDS codes and their quantum error correction, 2020. https://arxiv.org/abs/2002.12892v2, IEEE Trans Inf Theory 67(12):7964–7984, 2021) when ℓ ≠ e 2 . From these families of MDS codes with ℓ -Galois hulls, we obtain twenty-two families of q-ary MDS EAQECCs. Compared with the MDS EAQECCs from Cao (Galois hulls of MDS codes and their quantum error correction, 2020. https://arxiv.org/abs/2002.12892v2, IEEE Trans Inf Theory 67(12):7964–7984, 2021), some of the MDS EAQECCs constructed in this paper have much larger dimensions. As an application, we give new families of p-ary entanglement-assisted concatenated quantum codes (EACQCs) by combining an arbitrary p-ary [ [ n ′ , e , d ′ ; c ′ ] ] p EAQECC with the q-ary MDS EAQECCs constructed in this paper. It turns out that the parameters of these EACQCs are very flexible, e.g., the possibility of the lengths of these EACQCs is far greater than the one of the lengths of these MDS EAQECCs. [ABSTRACT FROM AUTHOR]

Ensemble learning, online learning and deep learning are very effective and versatile in a wide spectrum of problem domains, such as feature extraction, multi-class classification and retrieval. In this paper, combining the ideas of ensemble learning, online learning and deep learning, we propose a novel deep learning method called deep error-correcting output codes (DeepECOCs). DeepECOCs are composed of multiple layers of the ECOC module, which combines several incremental support vector machines (incremental SVMs) as base classifiers. In this novel deep architecture, each ECOC module can be considered as two successive layers of the network, while the incremental SVMs can be viewed as weighted links between two successive layers. In the pre-training procedure, supervisory information, i.e., class labels, can be used during the network initialization. The incremental SVMs lead this procedure to be very efficient, especially for large-scale applications. We have conducted extensive experiments to compare DeepECOCs with traditional ECOC, feature learning and deep learning algorithms. The results demonstrate that DeepECOCs perform, not only better than existing ECOC and feature learning algorithms, but also related to deep learning ones in most cases. [ABSTRACT FROM AUTHOR]

This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions that are able to mask u partially stuck cells while correcting at the same time t random errors. The process of "masking" determines a word whose entries coincide with writable levels at the (partially) stuck cells. For u > 1 and alphabet size q > 2 , our new constructions improve upon the required redundancy of known constructions for t = 0 , and require less redundancy for masking partially stuck cells than former works required for masking fully stuck cells (which cannot store any information). Second, we show that treating some of the partially stuck cells as erroneous cells can decrease the required redundancy for some parameters. Lastly, we derive Singleton-like, sphere-packing-like, and Gilbert–Varshamov-like bounds. Numerical comparisons state that our constructions match the Gilbert–Varshamov-like bounds for several code parameters, e.g., BCH codes that contain all-one word by our first construction. [ABSTRACT FROM AUTHOR]

Monte Carlo simulation is performed with uniformly distributed U(0,1) pseudo-random numbers. Because the numbers are generated from a mathematical formula, they will contain some serial correlation, even if very small. This serial correlation becomes embedded in the correlation structure of the response variable. The response variable becomes an asynchronous time series. This leads to hidden errors in the response variable. The purpose of this paper is to illustrate how this happens and how it can be corrected. The method is demonstrated for the case of a simple queue for which the time in the system is known exactly from theory. The paper derives the correlation between an exponential random variable and its antithetic counterpart obtained by power transform with an infinitesimal negative exponent. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of integer codes capable of correcting l-bit burst errors within one b-bit byte and h-bit burst errors within two b-bit bytes (1 ≤ h < l < b). The presented codes are constructed with the help of a computer and have the potential to be used in various optical networks. To evaluate the performance of the proposed codes, we analyze the probability of incorrect decoding for different error rates. In addition, the paper shows that the presented codes are suitable for software implementation and that can be easily transformed into codes capable of correcting double burst errors. [ABSTRACT FROM AUTHOR]

Free-space optical (FSO) communication is of supreme importance for designing next-generation networks. Over the past decades, the radio frequency (RF) spectrum has been the main topic of interest for wireless technology. The RF spectrum is becoming denser and more employed, making its availability tough for additional channels. Optical communication, exploited for messages or indications in historical times, is now becoming famous and useful in combination with error-correcting codes (ECC) to mitigate the effects of fading caused by atmospheric turbulence. A free-space communication system (FSCS) in which the hybrid technology is based on FSO and RF. FSCS is a capable solution to overcome the downsides of current schemes and enhance the overall link reliability and availability. The proposed FSCS with regular low-density parity-check (LDPC) for coding techniques is deliberated and evaluated in terms of signal-to-noise ratio (SNR) in this paper. The extrinsic information transfer (EXIT) methodology is an incredible technique employed to investigate the sum-product decoding algorithm of LDPC codes and optimize the EXIT chart by applying curve fitting. In this research work, we also analyze the behavior of the EXIT chart of regular/irregular LDPC for the FSCS. We also investigate the error performance of LDPC code for the proposed FSCS. [ABSTRACT FROM AUTHOR]

The construction of a quasi-cyclic low density parity-check (QC-LDPC) matrix is usually carried out in two steps. In the first step, a prototype matrix is defined according to certain criteria (size, girth, check and variable node degrees, and so on). The second step involves the expansion of the prototype matrix. During this last phase, an integer value is assigned to each non-null position in the prototype matrix corresponding to the right-rotation of the identity matrix. The problem of determining these integer values is complex. The state-of-the-art solutions use either some mathematical constructions to guarantee a given girth of the final QC-LDPC code, or a random search of values until the target girth is satisfied. In this paper, we propose an alternative/complementary method that reduces the search space by defining large equivalence classes of topologically identical matrices through row and column permutations using additive, structural, and multiplicative transformations. Selecting only a single element per equivalence class can reduce the search space by a few orders of magnitude. Then, we use the formalism of constraint programming to list the exhaustive sets of solutions for a given girth and a given expansion factor. An example is presented in all sections of the paper to illustrate the methodology. [ABSTRACT FROM AUTHOR]

In this paper, we study two families of codes with availability, namely, private information retrieval (PIR) codes and batch codes. While the former requires that every information symbol has $k$ mutually disjoint recovering sets, the latter imposes this property for each multiset request of $k$ information symbols. The main problem under this paradigm is to minimize the number of redundancy symbols. We denote this value by $r_{P}(n,k)$ and $r_{B}(n,k)$ , for PIR codes and batch codes, respectively, where $n$ is the number of information symbols. Previous results showed that for any constant $k$ , $r_{P}(n,k) = \Theta (\sqrt {n})$ and $r_{B}(n,k)= {\mathcal{ O}}(\sqrt {n}\log (n))$. In this paper, we study the asymptotic behavior of these codes for non-constant $k$ and specifically for $k=\Theta (n^\epsilon)$. We also study the largest value of $k$ such that the rate of the codes approaches 1 and show that for all $\epsilon < 1$ , $r_{P}(n,n^\epsilon) = o(n)$ and $r_{B}(n,n^\epsilon) = o(n)$. Furthermore, several more results are proved for the case of fixed $k$. [ABSTRACT FROM AUTHOR]

Since their rediscovery in the early 1990s, low-density parity-check (LDPC) codes have become the most popular error-correcting codes owing to their excellent performance. An LDPC code is a linear block code that has a sparse parity-check matrix. Cycles in this matrix, particularly short cycles, degrade the performance of such a code. Hence, several methods for counting short cycles in LDPC codes have been proposed, such as Fan's method to detect 4-cycles, 6-cycles, 8-cycles, and 10-cycles. Unfortunately, this method fails to count all 6-cycles, i.e., ignores numerous 6-cycles, in some given parity-check matrices. In this paper, an improvement of this algorithm is presented that detects all 6-cycles in LDPC codes, as well as in general bipartite graphs. Simulations confirm that the improved method offers the exact number of 6-cycles, and it succeeds in detecting those ignored by Fan's method. [ABSTRACT FROM AUTHOR]

The paper presents a novel efficient encoding and decoding method for self‐correction and anti‐interference used in big data product tracing system based on the cyclic code theory. The principle of constructing code set is first illustrated. Then the specific encoding and decoding methods suitable for the algebra rules are invented in sequence for efficiency. At last, to test the new coding system, the paper accelerates the decoding time by OpenMP more than 36 times. The error‐correcting code presents its perfect recovery property with a recovery rate of more than 95% under 60% damage rate. In analyzing its efficiency, the encoding process only costs 0.0045 second in average while the time of decoding method will increase with the damage rate's ascending, but it can be controlled within 16 seconds. In comparison to other coding patterns in the references, the newly error‐correcting and anti‐interference coding method achieves a balance in complexity, encoding rate, and decoding time with its priority in recovering rate under the damage rate of 60%. [ABSTRACT FROM AUTHOR]

This review paper aims to examine the use of polar codes in public key cryptosystems, providing a comprehensive overview of the state-of-the-art in this field at present. This study thoroughly searches databases such as IEEE Xplore, Scopus, and the ACM digital library to locate relevant research articles. In this study, we demonstrate the use of polar codes in public key cryptography and provide valuable insights for researchers interested in this field of research. The paper identifies several areas for further research and development, such as improved security and reliability of polar code-based cryptosystems. An analysis of the review highlights the major challenges and open research questions in this area, including the need for efficient key generation algorithms and the trade-offs between security and performance. An analysis of various existing encryption techniques as well as their security proofs is provided in the tables. [ABSTRACT FROM AUTHOR]

This research highlights the importance of error-correcting codes in ensuring secure and efficient data transmission over noisy channels. This paper aims to address the issue of limited information regarding the factors that contribute to the effectiveness of the implementation of Convolutional Codes in GPS systems. The research problem revolves around the insufficiency of scholarly sources elucidating the rationale behind the utilization of convolutional codes, specifically in GPS systems, rather than others. Through an in-depth analysis of these factors, this study strives to achieve a comprehensive understanding of the application of Convolutional Codes in GPS. To tackle this research problem, a novel methodology involving comparative analysis is employed. The coding techniques commonly used in satellite communication systems (such as BCH, LDPC, and turbo codes) are carefully examined and compared to the advantages and suitability of Convolutional codes and the Viterbi algorithm for GPS systems. Each coding technique is evaluated based on factors such as error detection and correction capabilities, bandwidth efficiency, computational complexity, and resilience to noise. The key findings of this study shed light on the unique advantages offered by Convolutional codes and the Viterbi algorithm for GPS systems. The analysis reveals that these coding techniques exhibit superior error detection and correction capabilities, efficient bandwidth utilization, and the ability to withstand noise in the GPS communication channel. The results also highlight the computational complexity associated with these techniques, providing valuable insights for the implementation of convolutional codes in GPS systems. Overall, this article contributes to the existing knowledge by providing a comprehensive understanding of the reasons behind the suitability of convolutional codes for GPS systems. The findings of this study serve as a resource for researchers, engineers, and practitioners in the field of satellite communication, aiding in the comprehensive understanding, advancement, and optimization of GPS system designs. [ABSTRACT FROM AUTHOR]

The user datagram protocol is the most widely used protocol for streaming multimedia data over the Internet. Like other protocols at the network and transport layers, it uses the Internet Checksum (IC) to detect channel errors. The consequence of this is that each corrupted packet is dropped, which can degrade the quality of service perceived by a user. In this paper, the authors present a solution to this problem based on integer codes capable of correcting single and double adjacent errors. The presented codes are a generalization of the IC, which makes them easy to implement in software. [ABSTRACT FROM AUTHOR]

Quantum key distribution (QKD) can provide information-theoretically secure keys for two parties of legitimate communication, and information reconciliation, as an indispensable component of QKD systems, can correct errors present in raw keys based on error-correcting codes. In this paper, we first describe the basic knowledge of information reconciliation and its impact on continuous variable QKD. Then we introduce the information schemes and the corresponding error correction codes employed. Next, we introduce the rate-compatible codes, the hardware acceleration of the reconciliation algorithm, the research progress of information reconciliation, and its application in continuous variable QKD. Finally, we discuss the future challenges and conclude. [ABSTRACT FROM AUTHOR]

Motivated by the application of high-density data storage technologies, Cassuto and Blaum introduced codes for symbol-pair read channels in 2011, and Yaakobi et al. generalized the coding framework to that for b-symbol read channels where b ≥ 2 in 2016. In this paper, we establish a b-sphere-packing bound and present a recurrence relationship for the b-weight enumerator. We determine all parameters of linear perfect b-symbol e-error-correcting codes over F q for e < 2 b and show that for 2 b ≤ e < 3 b , there exist at most finite such codes for a given b ,   e , and q. We construct a family of linear perfect b-symbol b-error-correcting codes over F q using constacyclic codes. [ABSTRACT FROM AUTHOR]

Topological quantum error-correcting codes are an important tool for realizing fault-tolerant quantum computers. Heavy hexagonal coding is a new class of quantum error-correcting coding that assigns physical and auxiliary qubits to the vertices and edges of a low-degree graph. The layout of heavy hexagonal codes is particularly suitable for superconducting qubit architectures to reduce frequency conflicts and crosstalk. Although various topological code decoders have been proposed, constructing the optimal decoder remains challenging. Machine learning is an effective decoding scheme for topological codes, and in this paper, we propose a machine learning heavy hexagonal decoder based on a convolutional neural network (CNN) to obtain the decoding threshold. We test our method on heavy hexagonal codes with code distance of three, five, and seven, and increase it to five, seven, and nine by optimizing the RestNet network architecture. Our results show that the decoder thresholding accuracies are about 0.57% and 0.65%, respectively, which are about 25% higher than the conventional decoding scheme under the depolarizing noise model. The proposed decoding architecture is also applicable to other topological code families. [ABSTRACT FROM AUTHOR]

Due to the emergence of extremely high density memory along with the growing number of embedded memories, memory yield is an important issue. Memory self-repair using redundancies to increase the yield of memories is widely used. Because high density memories are vulnerable to soft errors, memory error correction code (ECC) plays an important role in memory design. In this paper, methods to exploit spare columns including replaced defective columns are proposed to improve memory ECC. To utilize replaced defective columns, the defect information needs to be stored. Two approaches to store defect information are proposed—one is to use a spare column and the other is to use a content-addressable-memory. Experimental results show that the proposed method can significantly enhance the ECC performance. [ABSTRACT FROM AUTHOR]

In this paper, we present generalized edge-pairings for the family of hyperbolic tessellations {4λ, 4} with the purpose of obtaining the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations {4λ, 4λ}, implying that the associated codes achieve the least error probability or, equivalently, that these codes are optimum codes. We also consider the topological quantum error-correcting codes on surfaces with genus g ≥ 2, as proposed in other papers. From that, we focus on the class of topological quantum maximum distance separable codes, showing that the only such codes are precisely the ones with minimum distance d ≤ 2. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel class of real number codes termed as ℝ-spectral codes, derived from real symmetric operators. Our focus is on addressing the challenges associated with error correction in communication systems utilizing real number codes. Our study initiates with a meticulous definition and analysis of the fundamental properties of ℝ-spectral codes. These properties serve as the foundation for comprehending the distinctive characteristics that ℝ-spectral codes offer, ensuring the integrity and reliability of transmitted data. Our exploration extends to the construction of maximum distance separable (MDS) ℝ-spectral codes, which boast the maximum error correction capability. As a culmination of our research, we present an efficient decoding technique tailored for ℝ-spectral codes. Leveraging an MDS ℝ-spectral code and our decoding technique, we achieve optimal error correction and ensure good numerical stability. This innovative approach opens new avenues for enhancing the performance of real-number codes in practical applications. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code 풞 with minimum distance 2ℓ + 2, if it has an ℓ-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases. [ABSTRACT FROM AUTHOR]

Quantum error-correcting codes and entanglement-assisted quantum error-correcting codes have important applications in quantum computing and quantum communication. In this paper, we construct several classes of quantum MDS codes and entanglement-assisted quantum MDS codes by using generalized Reed-Solomon codes. The parameters of most of the codes constructed are new. [ABSTRACT FROM AUTHOR]

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C 1 , C 2) such that C 1 contains C 2 , C 2 is even, and the shortening of the dual of C 1 with respect to the support of each codeword of C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C 1 , C 2) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. [ABSTRACT FROM AUTHOR]

Given the growing urge for plastic management and regulation in the world, recent studies have investigated the problem of plastic material identification for correct classification and disposal. Recent works have shown the potential of machine learning techniques for successful microplastics classification using Raman signals. Classification techniques from the machine learning area allow the identification of the type of microplastic from optical signals based on Raman spectroscopy. In this paper, we investigate the impact of high-frequency noise on the performance of related classification tasks. It is well-known that classification based on Raman is highly dependent on peak visibility, but it is also known that signal smoothing is a common step in the pre-processing of the measured signals. This raises a potential trade-off between high-frequency noise and peak preservation that depends on user-defined parameters. The results obtained in this work suggest that a linear discriminant analysis model cannot generalize properly in the presence of noisy signals, whereas an error-correcting output codes model is better suited to account for inherent noise. Moreover, principal components analysis (PCA) can become a must-do step for robust classification models, given its simplicity and natural smoothing capabilities. Our study on the high-frequency noise, the possible trade-off between pre-processing the high-frequency noise and the peak visibility, and the use of PCA as a noise reduction technique in addition to its dimensionality reduction functionality are the fundamental aspects of this work. [ABSTRACT FROM AUTHOR]

The homogeneous weight (metric) is useful in the construction of codes over a ring of integers Z p l (p prime and l ≥ 1 an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be Z 4 . This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of n-tuples over Z p l . Necessary and sufficient conditions for existence of an (n, k) linear code over Z p l correcting the error patterns with respect to the homogeneous weight are derived. [ABSTRACT FROM AUTHOR]

The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]