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The fundamental difference between neural networks containing and not containing feedback between elements is analyzed. It is shown that in the first of these cases, quantitative relationships describing the functioning of the neural network can be obtained based on an analogy with the theory of noise-resistant codes. In the second case, an analogy with electronic circuits that form memory cells (triggers) is valid. It is shown that feedback between elements of even the simplest neural networks can lead to the appearance of multidimensional hysteresis, when, with the same state of inputs, the system can be in several qualitatively different states, the transition between which can be abrupt. In this case, the state of the neural network outputs depends not only on the current state of its inputs, but also on the path along which this state was formed. The results obtained are used for the philosophical substantiation of a new approach to the interpretation of complex systems of various natures, which are considered analogs of neural networks. According to it, a system that can store and processing information should be considered "complex". [ABSTRACT FROM AUTHOR]

Generating quantum entanglement is plagued by decoherence. Distillation and error-correction are employed against such noise, but designing a good distillation circuit, especially on today's imperfect hardware, is challenging. We develop a simulation algorithm for distillation circuits with per-gate complexity of O (1) , drastically faster than O (N) Clifford simulators or O ( 2 N ) wavefunction simulators over N qubits. This simulator made it possible to optimize distillation circuits much larger than previously feasible. We design distillation circuits from n raw Bell pairs to k purified pairs and study the use of these circuits in the teleportation of logical qubits. The resulting purification circuits are the best-known for finite-size noisy hardware and can be fine-tuned for specific error-models. Furthermore, we design purification circuits that shape the correlations of errors in the purified pairs such that the performance of potential error-correcting codes is greatly improved. [ABSTRACT FROM AUTHOR]

The long-term preservation of large volumes of infrequently accessed cold data poses challenges to the storage community. Deoxyribonucleic acid (DNA) is considered a promising solution due to its inherent physical stability and significant storage density. The information density and decoding sequence coverage are two important metrics that influence the efficiency of DNA data storage. In this study, we propose a novel coding scheme called the DNA palette code, which is suitable for cold data, especially time-series archival datasets. These datasets are not frequently accessed, but require reliable long-term storage for retrospective research. The DNA palette code employs unordered combinations of index-free oligonucleotides to represent binary information. It can achieve high net information density encoding and lossless decoding with low sequencing coverage. When sequencing reads are corrupted, it can still effectively recover partial information, preventing the complete failure of file retrieval. The in vitro testing of clinical brain magnetic resonance imaging (MRI) data storage, as well as simulation validations using large-scale public MRI datasets (10 GB), planetary science datasets and meteorological datasets, demonstrates the advantages of our coding scheme, including high net information density, low decoding sequence coverage and wide applicability. [ABSTRACT FROM AUTHOR]

In information and security, the personal identification of individuals becomes much more important. For improving security, several biometric recognition techniques are implemented. However, in finger vein recognition, it faces the critical problem of fake finger vein images, security and less accuracy. To conquer this problem, Hybrid Feature Extraction with Linear Local Tangent Space Alignment-based dimension reduction and Support Vector Machine classifier (HFE–LLTSA–SVM) is proposed. In this hybrid, FE is considered as the combination of histogram of oriented gradients (HOG), grey-level co-occurrence matrix (GLCM), stationary wavelet transform (SWT), and local binary pattern (LBP) for extracting the hybrid feature. LLTSA perform dimension reduction in the outputs of HFE from HOG, GLCM, and LBP. Furthermore, SVM is used for classification which gives authentication based on error-correcting code. Finally, the performance parameters were calculated and the proposed method achieved better accuracy of 99.75%, when compared with existing methods. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes (EAQECCs) not only can boost the performance of stabilizer quantum error-correcting codes but also can be derived from arbitrary classical linear codes by loosing the self-orthogonal condition and using pre-shared entangled states between the sender and the receiver. It is a challenging work to construct optimal EAQECCs and determine the required number of pre-shared entangled states. Let R t = F q 2 + v F q 2 + v 2 F q 2 + ⋯ + v t F q 2 , where q is an odd prime power and v t + 1 = 1 . Based on the generalized Gray map that is provided from R t to F q 2 t + 1 , some new optimal EAQECCs are constructed from the Gray images of v-constacyclic codes over R t . Compared with the known ones, our codes have better parameters. [ABSTRACT FROM AUTHOR]

The HQC post-quantum cryptosystem enables two parties to share noisy versions of a common secret binary string, and an error-correcting code is required to deal with the mismatch between both versions. This code is required to deal with binary symmetric channels with as large a transition parameter as possible, while guaranteeing, for cryptographic reasons, a decoding error probability of provably not more than 2-128. This requirement is non-standard for digital communications, and modern coding techniques are not amenable to this setting. This paper explains how this issue is addressed in the last version of HQC: precisely, we introduce a coding scheme that consists of concatenating a Reed–Solomon code with the tensor product of a Reed–Muller code and a repetition code. We analyze its behavior in detail and show that it significantly improves upon the previous proposition for HQC, which consisted of tensoring a BCH and a repetition code. As additional results, we also provide a better approximation of the weight distribution for HQC error vectors, and we remark that the size of the exchanged secret in HQC can be reduced to match the protocol security which also significantly improves performance. [ABSTRACT FROM AUTHOR]

In this work, we investigate the evaluation of odd polynomials P defined on a finite field on the class of error-correcting codes C. We exploit the correspondence between codes and Tanner graphs. Thus, we formally define P(C), a polynomial code. Then the new notion of height of a code emerges, whose properties are studied. We extended the lower bound of Tanner on the minimum distance of a code to the case of a polynomial code, by using spectral graph theory. Computer algebra software enable us to give numerical results to illustrate the theory of polynomial codes for various classes of error-correcting codes. [ABSTRACT FROM AUTHOR]

Abstract The post-processing of quantum key distribution mainly includes error correction and privacy amplification. The error correction algorithms and privacy amplification methods used in the existing quantum key distribution are completely unrelated. Based on the principle of correspondence between error-correcting codes and hash function families, we proposed the idea of time-division multiplexing for error correction and privacy amplification for the first time. That is to say, through the common error correction algorithms and their corresponding hash function families or the common hash function families and their corresponding error-correcting codes, error correction and privacy amplification can be realized by time-division multiplexing with the same set of devices. In addition, we tested the idea from the perspective of error correction and privacy amplification, respectively. The analysis results show that the existing error correction algorithms and their corresponding hash function families or the common privacy amplification methods and their corresponding error-correcting codes cannot realize time-division multiplexing for error correction and privacy amplification temporarily. However, according to the principle of correspondence between error-correcting codes and hash function families, the idea of time-division multiplexing is possible. Moreover, the research on time-division multiplexing for error correction and privacy amplification has some practical significance. Once the idea of time-division multiplexing is realized, it will further reduce the calculation and storage cost of the post-processing process, reduce the deployment cost of quantum key distribution, and help to remote the practical engineering of quantum key distribution.

In this paper, we propose the construction of p-adic LDPC codes and develop a multi-stage decoding algorithm for these codes. The novel type of non-binary error-correcting codes operates over the p-adic integers. We evaluate the error performance of these codes through simulations and compare them with existing nonbinary LDPC and Reed-Solomon codes of the same rates. Our results show that p-adic LDPC codes achieve better error performance than the existing codes. Moreover, we demonstrate the potential use of p-adic LDPC codes in byte error correction by treating each byte as an element in the p-adic integers at a chosen precision level. The p-adic LDPC codes can be designed and optimized for specific applications that require byte error correction, with the flexibility to vary the byte under different precision levels. [ABSTRACT FROM AUTHOR]

Time crystals are a dynamical phase of periodically driven quantum many-body systems where discrete time-translation symmetry is broken spontaneously. Time-crystallinity however subtly requires also spatial order, ordinarily related to further symmetries, such as spin-flip symmetry when the spatial order is ferromagnetic. Here we define topologically ordered time crystals, a time-crystalline phase borne out of intrinsic topological order—a particularly robust form of spatial order that requires no symmetry. We show that many-body localization can stabilize this phase against generic perturbations and establish some of its key features and signatures, including a dynamical, time-crystal form of the perimeter law for topological order. We link topologically ordered and ordinary time crystals through three complementary perspectives: higher-form symmetries, quantum error-correcting codes, and a holographic correspondence. Topologically ordered time crystals may be realized in programmable quantum devices, as we illustrate for the Google Sycamore processor. The concept of a time crystal is well established, but its interplay with topological order is less explored. Wahl et al. show that time crystals may arise from topological order and that such states make the gauge theoretic perimeter law dynamic, offering a key feature to seek with quantum computers. [ABSTRACT FROM AUTHOR]

Semantic communication (SC) is a communication paradigm that has gained significant attention, as it offers a potential solution to move beyond Shannon's formulation in bandwidth-limited communication channels by delivering the semantic meaning of the message rather than its exact form. In this paper, we propose an autoencoder-based SC system for transmitting images between two machines over error-prone channels to support emerging applications such as VIoT, XR, M2M, and M2H communications. The proposed autoencoder architecture, with a semantically modeled encoder and decoder, transmits image data as a reduced-dimension vector (latent vector) through an error-prone channel. The decoder then reconstructs the image to determine its M2M implications. The autoencoder is trained for different noise levels under various channel conditions, and both image quality and classification accuracy are used to evaluate the system's efficacy. A CNN image classifier measures accuracy, as no image quality metric is available for SC yet. The simulation results show that all proposed autoencoders maintain high image quality and classification accuracy at high SNRs, while the autoencoder trained with zero noise underperforms other trained autoencoders at moderate SNRs. The results further indicate that all other proposed autoencoders trained under different noise levels are highly robust against channel impairments. We compare the proposed system against a comparable JPEG transmission system, and results reveal that the proposed system outperforms the JPEG system in compression efficiency by up to 50 % and in received image quality with an image coding gain of up to 17 dB. [ABSTRACT FROM AUTHOR]

The RSA cryptosystem has been a cornerstone of modern public key infrastructure; however, recent advancements in quantum computing and theoretical mathematics pose significant risks to its security. The advent of fully operational quantum computers could enable the execution of Shor's algorithm, which efficiently factors large integers and undermines the security of RSA and other cryptographic systems reliant on discrete logarithms. While Grover's algorithm presents a comparatively lesser threat to symmetric encryption, it still accelerates key search processes, creating potential vulnerabilities. In light of these challenges, there has been an intensified focus on developing quantum-resistant cryptography. Current research is exploring cryptographic techniques based on error-correcting codes, lattice structures, and multivariate public key systems, all of which leverage the complexity of NP-hard problems, such as solving multivariate quadratic equations, to ensure security in a post-quantum landscape. This paper reviews the latest advancements in quantum-resistant encryption methods, with particular attention to the development of robust trapdoor functions. It also provides a detailed analysis of prominent multivariate cryptosystems, including the Matsumoto–Imai, Oil and Vinegar, and Polly Cracker schemes, alongside recent progress in lattice-based systems such as Kyber and Crystals-DILITHIUM, which are currently under evaluation by NIST for potential standardization. As the capabilities of quantum computing continue to expand, the need for innovative cryptographic solutions to secure digital communications becomes increasingly critical. [ABSTRACT FROM AUTHOR]

The post-processing of quantum key distribution mainly includes error correction and privacy amplification. The error correction algorithms and privacy amplification methods used in the existing quantum key distribution are completely unrelated. Based on the principle of correspondence between error-correcting codes and hash function families, we proposed the idea of time-division multiplexing for error correction and privacy amplification for the first time. That is to say, through the common error correction algorithms and their corresponding hash function families or the common hash function families and their corresponding error-correcting codes, error correction and privacy amplification can be realized by time-division multiplexing with the same set of devices. In addition, we tested the idea from the perspective of error correction and privacy amplification, respectively. The analysis results show that the existing error correction algorithms and their corresponding hash function families or the common privacy amplification methods and their corresponding error-correcting codes cannot realize time-division multiplexing for error correction and privacy amplification temporarily. However, according to the principle of correspondence between error-correcting codes and hash function families, the idea of time-division multiplexing is possible. Moreover, the research on time-division multiplexing for error correction and privacy amplification has some practical significance. Once the idea of time-division multiplexing is realized, it will further reduce the calculation and storage cost of the post-processing process, reduce the deployment cost of quantum key distribution, and help to remote the practical engineering of quantum key distribution. [ABSTRACT FROM AUTHOR]

Inspired by classical polar codes, whose coding rate can asymptotically achieve the Shannon capacity, researchers are trying to find their analogs in the quantum information field, which are called quantum polar codes. However, no one has designed a quantum polar coding scheme that applies to quantum computing yet. There are two intuitions in previous research. The first is that directly converting classical polar coding circuits to quantum ones will produce the polarization phenomenon of a pure quantum channel, which has been proved in our previous work. The second is that based on this quantum polarization phenomenon, one can design a quantum polar coding scheme that applies to quantum computing. There are several previous work following the second intuition, none of which has been verified by experiments. In this paper, we follow the second intuition and propose a more reasonable quantum polar stabilizer code construction algorithm than any previous ones by using the theory of stabilizer codes. Unfortunately, simulation experiments show that even the stabilizer codes obtained from this more reasonable construction algorithm do not work, which implies that the second intuition leads to a dead end. Based on the analysis of why the second intuition does not work, we provide a possible future direction for designing quantum stabilizer codes with a high coding rate by borrowing the idea of classical polar codes. Following this direction, we find a class of quantum stabilizer codes with a coding rate of 0.5, which can correct two of the Pauli errors. [ABSTRACT FROM AUTHOR]

A binary constant weight code is a type of error-correcting code with a wide range of applications. The problem of finding a binary constant weight code has long been studied as a combinatorial optimization problem in coding theory. In this paper, we propose a quantum search algorithm for binary constant weight codes. Specifically, the search problem is formulated as a polynomial binary optimization problem and Grover adaptive search is used for providing the quadratic speedup. Focusing on the inherent structure of the problem, we derive an upper bound on the minimum of the objective function value and a lower bound on the exact number of solutions. By exploiting these two bounds, we successfully reduced the constant overhead of the algorithm, although the overall query complexity remains exponential due to the NP-complete nature of the problem. In our algebraic analysis, it was found that this proposed algorithm is capable of reducing the number of required qubits, thus enhancing the feasibility. Additionally, our simulations demonstrated that it reduces the average number of classical iterations by 63% as well as the average number of total Grover rotations by 31%. The proposed approach may be useful for other quantum search algorithms and optimization problems. [ABSTRACT FROM AUTHOR]

The correction notice addresses errors in the parameters and theorems of the article "Quantum error-correcting codes from the quantum construction X." Adjustments were made to Theorems 2.9, 3.2, 3.7, 3.9, Lemma 3.14, Theorems 3.15, 3.16, and 3.20 to provide accurate information on quantum error-correcting codes. Tables 2 and 3 compare and present new Quantum Error Correction (QEC) codes from the corrected theorems, offering valuable information for researchers studying quantum error correction codes. The acknowledgements thank Markus Grassl for identifying the mistakes, and the data availability section clarifies that no datasets were generated or analyzed in the study. [Extracted from the article]

In this article we will approach the soft-decision decoding for the linear block codes, is a kind of decoding algorithms used to decode data to form better original estimated received message, it is considered as a NP-hard problem. In this article we present a new decoder using memetic algorithm such metaheuristic technic operates on the dual code rather than the code itself that aims to find the error caused when sending a codeword calculated from a message of k bits of information, the resulting codeword contains n bits, including the redundancy bits, the efficiency of an error-correcting code is equivalent to the ratio k/n, the rate is belong the interval [0,1]. Hence a good code is the one that ensures a certain error correcting capability at minimum ratio. The results proved that this approach using a combination of genetic algorithm and local search algorithm provides a sufficiently good solution to an optimization problem; the new decoder is applied on linear codes where the structure is given by a parity check matrix. [ABSTRACT FROM AUTHOR]

Motivated by applications in DNA storage, we study a setting in which strings are affected by tandem-duplication errors. In particular, we look at two settings: disjoint tandem-duplication errors, and equal-length tandem-duplication errors. We construct codes, with positive asymptotic rate, for the two settings, as well as for their combination. Our constructions are duplication-free codes, comprising codewords that do not contain tandem duplications of specific lengths. Additionally, our codes generalize previous constructions, containing them as special cases. [ABSTRACT FROM AUTHOR]

Copyright of Palestine Journal of Mathematics is the property of Palestine Polytechnic University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one-dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one-dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one-dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In particular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and r, s, m be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of 2 r p s over the finite field F p m . Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we identify all maximum distance separable (MDS) cyclic codes of length 2 r p s , which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions. [ABSTRACT FROM AUTHOR]

The document is a correction notice for a publication on error-correcting codes on projective bundles over Deligne–Lusztig varieties. It provides corrections to the dimension of the obtained codes and presents mathematical formulas and corollaries related to the construction of codes on projective bundles over specific surfaces. The text emphasizes the importance of working with normalized vector bundles and provides detailed parameters for constructing codes over specific fields. The authors discuss mathematical proofs and examples related to codes in the binary case, focusing on parameters such as code length, dimension, and minimum distance. They provide specific examples for different values of q, b, n, and m, showcasing the parameters of the resulting codes and acknowledge the need for modifications to improve the information rate of certain code families. [Extracted from the article]

Weighted projective spaces are natural generalizations of projective spaces with a rich structure. Projective Reed–Muller codes are error-correcting codes that played an important role in reliably transmitting information on digital communication channels. In this case study, we explore the power of commutative and homological algebraic techniques to study weighted projective Reed–Muller (WPRM) codes on weighted projective spaces of the form ℙ(1, 1,b). We compute minimal free resolutions and thereby obtain Hilbert series for the vanishing ideal of the 픽q-rational points, and compute main parameters for these 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]

In this article, we find several novel and efficient quantum error-correcting codes ( Q ecc) by studying the structure of constacyclic ( C cc ), cyclic ( C c ), and negacyclic codes (N C c ) over the ring A k = Z p r 1 , r 2 , ⋯ , r k / ⟨ (r b (m b + 1) - r b) , r l r b = r b r l = 0 , b ≠ l ⟩ , where p = q m for m, m b ∈ N , m b | - 1 + q ∀ b , l ∈ 1 to k , q ≥ 3 is a prime, Z p is a finite field. We define distance-preserving gray map δ k . Moreover, we determine the quantum singleton defect (Q SD) of Q ecc, which indicates their overall quality. We compare our codes with existing codes in recent publications. The rings discussed by Kong et al. (EPJ Quantum Technol 10:1–16, 2023), Suprijanto et al. (Quantum codes constructed from cyclic codes over the ring F q + vF q + v 2 F q + v 3 F q + v 4 F q , pp 1–14, 2021. arXiv: 2112.13488v2 [cs.IT]), and Dinh et al. (IEEE Access 8:194082–194091, 2020) are specific cases of our work. Furthermore, we construct several novel and optimum linear complementary dual (Lcd) codes over A k. [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]

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]

Traceability via quick response (QR) codes is regarded as a clever way to learn specifics about a product's history, from its creation to its transit and preservation before reaching consumers. The QR code can, however, be easily copied and faked. Therefore, we suggest a novel strategy to prevent tampering with this code. The method is divided into two primary phases: concealing a security element in the QR code and determining how similar the QR code on the goods is to the real ones. For the first problem, error-correcting coding is used to encode and decode the secret feature in order to manage faults in noisy communication channels. A deep neural network is used to both conceal and extract the information encoded in a QR code, and the suggested network creates watermarked QR code images with good quality and noise tolerance. The network has the ability to be resilient to actual distortions brought on by the printing and photographing processes. In order to measure the similarity of QR codes, we create neural networks based on the Siamese network design. To assess whether a QR code is real or fraudulent, the hidden characteristic extracted from the acquired QR code and the outcome of QR code similarity estimation are merged. With an average accuracy of 98%, the proposed technique performs competitively and has been used in practice for QR code authentication. [ABSTRACT FROM AUTHOR]

Fault-tolerant operations based on stabilizer codes are the state of the art in suppressing error rates in quantum computations. Most such codes do not permit a straightforward implementation of non-Clifford logical operations, which are necessary to define a universal gate set. As a result, implementations of these operations must use either error-correcting codes with more complicated error correction procedures or gate teleportation and magic states, which are prepared at the logical level, increasing overhead to a degree that precludes near-term implementation. Here, we implement a small quantum algorithm, one-qubit addition, fault-tolerantly on a trapped-ion quantum computer, using the [[ 8, 3, 2]] color code. By removing unnecessary error correction circuits and using low-overhead techniques for fault-tolerant preparation and measurement, we reduce the number of error-prone two-qubit gates and measurements to 36. We observe arithmetic errors with a rate of ~1.1 x 10-3 for the fault-tolerant circuit and ~9.5 x 10-3 for the unencoded circuit. [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]

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]

Quantum comparators hold substantial significance in the scientific community as fundamental components in a wide array of algorithms. In this research, we present an innovative approach where we explore the realm of comparator circuits, specifically focussing on three distinct circuit designs present in the literature. These circuits are notable for their use of T-gates, which have gained significant attention in circuit design due to their ability to enable the utilisation of error-correcting codes. However, it is important to note that T-gates come at a considerable computational cost. One of the key contributions of our work is the optimisation of the quantum gates used within these circuits. We articulate the proposed circuits employing Clifford+T gates, facilitating error correction code implementation. Additionally, we minimise T-gate usage, thereby reducing computational costs and fortifying circuit robustness against errors and environmental disturbances-essential for mitigating the effects of internal and external noise. Our methodology employs a bottom-up examination of comparator circuits, initiating with a detailed study of their gates. Subsequently, we systematically dissect the functions of these gates, thereby advancing towards a comprehensive understanding of the circuit's overall functionality. This meticulous examination forms the foundation of our research, enabling us to identify areas where optimisations can be made to improve their performance. [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]

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]

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]

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]

This paper focuses on studying the properties of constacyclic codes and quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring S, which is defined as S = Iq[u, v]/⟨u² − α², v² − α², uv − vu⟩, where Iq is a finite field of q elements, α be the nonzero elements of the field Iq, and q is a power of an odd prime p such that q = pm, for m≥ 1. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring S into a direct sum of constacyclic codes over Iq. We construct new and better quantum error-correcting codes over the ring S (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4). [ABSTRACT FROM AUTHOR]

Offset quadrature amplitude modulation–based filter bank multicarrier (FBMC-OQAM) method is a promising technology for future wireless communication systems. It offers several advantages over traditional orthogonal frequency-division multiplexing (OFDM) modulation, including higher spectral efficiency, lower out-of-band emission, and improved robustness to time-frequency selective channels. Polar codes, a new class of error-correcting codes, have received much attention recently due to their ability to achieve the Shannon limit with practical decoding complexity. This paper analyzed and investigated the error rate performance of polar-coded FBMC-OQAM systems. Our results show that applying polar codes to FBMC-OQAM systems significantly improves the error rate. In addition, we found that employing random code interleavers can yield additional coding gains of up to 0.75 dB in additive white Gaussian noise (AWGN) and 2 dB in Nakagami-m fading channels. Our findings suggest that polar-coded FBMC-OQAM is a promising combination for future wireless communication systems. We also compared turbo-coded FBMC-OQAM for short code lengths, and our simulations showed that polar codes exhibit comparable error-correcting capabilities. These results will be of interest to researchers and engineers working on the advancement of future wireless communication systems. [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]

The Reed–Muller codes are a family of error-correcting codes that have been widely studied in coding theory. In 2020, Yan and Lin introduced a variant of Reed–Muller codes called symmetric Reed–Muller codes. We investigate linear maps of the automorphism group of symmetric Reed–Muller codes and show that the set of these linear maps forms a subgroup of the general linear group, which is the automorphism group of punctured Reed–Muller codes.We provide a method to determine all the automorphisms in this subgroup explicitly for some special cases. [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]

Polar codes are the popular error-correcting codes and increased their attention after being adopted for the control channel in fifth-generation new radio (5G NR) standards. An efficient hardware architecture for polar code is often required with minimal encoding and decoding complexity. This work proposes a Multi-folded pipelined architecture and analyzes the performance in terms of latency, hardware utilization, and throughput. The designed architecture has two folded architectures interconnected in parallel to output 4-bits simultaneously. Folding transformations are used to reduce the number of idle processing elements (PEs) in every stage leading to the effective utilization of PE. Precomputation is effectively utilized in the PE to reduce the critical path delay, which improves the maximum operating frequency. A Loop-based shifting register (LSR) is employed to reduce the number of registers used. The analytical model for latency and utilization rate has been derived from the scheduling of the proposed architecture. The proposed design shows 63–71% higher hardware utilization than conventional semi-parallel design for code length N = 512 suitable for the physical downlink control channel (PDCCH) in 5G NR. The architecture is also implemented in Virtex-6, ZYNQ-Ultrascale+ MPSoC device for maximum supported code length of 5G NR, i.e., up to 2 10 , compared with the existing decoders. The proposed design also has the benefit of lesser look-up-table (LUT) consumption and zero random-access-memory (RAM) usage with some additional registers, making it suitable for resource-constraint applications. [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]

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]