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Cryptology is a part of mathematics as encryption and decryption. The purpose of encryption is to make information incomprehensible when it is in the hands of unauthorized people. The receiver can decrypt the message that encrypted by the sender with helping of the key. The important point is that the key cannot be decrypted by other people. One Time Pad method solves this problem. The key is used only once each encryption in this method. So, the key becomes harder to guess. If the key is solved by unauthorized people, the message cannot be solved. Because of with each decryption, many meaningful messages are obtained. Every cyclic shift in a cyclic code constructs a new key and in each encryption is used the new key. Many keys are generated thanks to cyclic codes. In this paper, we improve the new encryption scheme by using the cyclic codes with One Time Pad method. [ABSTRACT FROM AUTHOR]

Quantum error correction lies at the heart of building reliable quantum information processing systems. Stabilizer codes, a fundamental class of quantum errorcorrecting codes, play a pivotal role in mitigating the adverse effects of noise and decoherence in quantum systems. This paper introduces a novel construction of quantum stabilizer codes using Hadamard difference sets, an elegant mathematical concept derived from combinatorial design theory. In this paper, the construction of the quantum stabilizer codes over non- cyclic Hadamard difference sets with parameters (4m²,2m²-m, m²-m), where m is a positive integer is discussed. Firstly, the parity check matrices are constructed from the Circulant permutation matrices with the help of Hadamard difference sets and then, the Symplectic inner product condition for Hadamard difference sets over binary operation for parity check matrices are obtained to affirm the commutative condition for Stabilizer operators which is vital for the error detection. For application, we constructed a Hadamard difference sets with parameters (16,6,2) for m = 2 of ordered pair of the group Z2 Z8 × (non-cyclic group) and quantum stabilizer codes are obtained by parity-check matrix. [ABSTRACT FROM AUTHOR]

Negacyclic BCH codes are an important subclass of negacyclic codes and have good parameters. Inspired by the recent work on cyclic codes published in Wu et al. (Finite Fields Appl 60:101581, 2019), the objective of this paper is to investigate the parameters of the narrow-sense negacyclic BCH codes of length n = q 2 m - 1 q + 1 over GF (q) , where q is an odd prime power. For 2 ≤ δ ≤ q m + 1 - q q + 1 + 2 , the dimension of the narrow-sense negacyclic BCH codes of length n with designed distance δ is determined. For 2 ≤ δ ≤ q 2 m - 1 + 1 q + 1 , a lower bound on the minimum distance of the dual codes of the narrow-sense negacyclic BCH codes of length n with designed distance δ is presented. Compared with cyclic codes, we obtain some negacyclic BCH codes with better parameters. [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]

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight in this code support a 2-design. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined and the parameters of the related 2-designs are also given. Moreover, we prove that the minimum weight codewords in this code support a 3-design when p = 2 , which gives an affirmative answer to Heng's conjecture. [ABSTRACT FROM AUTHOR]

Due to their effective and affordable joining capabilities, resistance spot-welded (RSW) structures are widely used in many industries, including the automotive, aerospace, and manufacturing sectors. Because spot-welded structures are frequently subjected to cyclic stress conditions while in service, fatigue failure is a serious concern. It is essential to comprehend and predict their fatigue behavior in order to guarantee the dependability and durability of the relevant engineering products. The analysis of fatigue failure in spot-welded structures is the main topic of this paper, along with the prediction of fatigue life (Nf) and identification of failure mechanisms. Also, the effects of parameters such as the amount of cyclic load applied, the load ratio, and size of the spot-welding on the Nf were investigated. To achieve this, the fatigue performance of spot-welded joints was simulated using the extended finite element method (XFEM). The XFEM method is particularly suited for capturing intricate crack patterns in spot-welded structures because it allows for the modeling of crack propagation without the need for remeshing. It was observed that when the cycling load was decreased by 20%, Nf increased by around 250%. On the other hand, the fatigue life of the structure, and, hence, the crack propagation rate, was significantly affected by the load ratio and diameter of the spot-welding. This paper presents the details of the novel approach to studying spot-weld fatigue characterization using XFEMs to simulate crack propagation. [ABSTRACT FROM AUTHOR]

In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of [ 2 n , (n - 1) / 2 , d ≥ 2 (n + 1) ] irreducible cyclic codes over GF (q) and a family of [ (q - 1) n , (n - 1) / 2 , d ≥ (q - 1) (n + 1) ] irreducible cyclic codes over GF (q) are presented, where n is a prime such that ord 2 n (q) = (n - 1) / 2 and ord (q - 1) n (q) = (n - 1) / 2 , respectively. The results in this paper complement earlier works on irreducible constacyclic and cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]

Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of ${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection. [ABSTRACT FROM AUTHOR]

In this paper, we introduced double cyclic codes over Rr × Rs, where R = Z2 + uZ2 = {0, 1, u, 1 + u} is the ring with four elements and u² = 0. We first determined the generator polynomials of R-double cyclic codes for odd integers r and s, then gave the generators of duals of free double cyclic codes over Rr × Rs. By defining a linear Gray map, we looked at the binary images of R-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from R-double cyclic codes. Moreover, we studied self-dual R-double cyclic codes and presented an example of a self-dual R-double cyclic code. [ABSTRACT FROM AUTHOR]

In this paper we give three constructions of cyclic self-orthogonal codes over Z 2 k , for k ≥ 3 , from Boolean functions on n variables. The first construction for each k, 3 ≤ k ≤ n , yields a self-orthogonal Z 2 k -code of length 2 n + 2 with all Euclidean weights divisible by 2 k + 1. In the remaining two constructions, for each even n and k ≥ 3 , we generate a self-orthogonal Z 2 k -code of length 2 n + 1. All Euclidean weights in the constructed code are divisible by 2 2 k - 1 or 2 k + 1 , depending on which of the two constructions is used. [ABSTRACT FROM AUTHOR]

This paper is devoted to the research of the isomorphic multidimensional cyclic structure and multidimensional phase structure of the cyclic random process (CRP) and to its formation method, which enables a rigorous formalization of intuitive ideas concerning cyclic stochastic motion. The fundamental properties of the cyclic random process and analytical dependencies between the multidimensional cyclic structure, multidimensional phase structure and rhythm structure of the CRP have been established. This work shows that the CRP is able to take into account the cyclicity of multidimensional distribution functions of cyclic signals as well as the variability in the rhythm of the investigated signals. A subclass of the CRP is the periodic random process, which allows for the use of classical processing methods of cyclic signals with a regular rhythm. Based on a series of experiments, significant advantages of the CRP as a mathematical model of electrocardiographic signals (ECG) compared to the periodic random process are shown. [ABSTRACT FROM AUTHOR]

In recent years, the information requirements of pumping stations have become higher and higher. The prediction of overflow capacity can provide important reference for flood carrying capacity, water resource scheduling and water safety. In order to improve the accuracy, stability and generalization ability of the model, a BiGRU–ARIMA data-driven method based on self-attention mechanism is proposed to predict the flow capacity of the pump station. Bidirectional gated recurrent unit (BiGRU), a variant of cyclic neural network (RNN), can not only deal with nonlinear components well, but also deal with the problem of insufficient dependence over long distances and has a simple structure. Autoregressive integrated moving average (ARIMA) has the advantage of being sensitive to linear components. Firstly, the characteristics of the pre-processed pump station data are selected and screened through Pearson correlation coefficient and a self-attention mechanism. Then, a bi-directional gated recurrent unit (BiGRU) is used to process the nonlinear components of the data, and a dropout layer is added to avoid overfitting phenomena. We extract the linear features of the obtained error terms using the ARIMA model and use them as correction items to correct the prediction results of the BiGRU model. Finally, we obtain the prediction results of the overflow and water level. The variation characteristics of overdischarge are analyzed by the relation of flow and water level. In this paper, the actual production data of a Grade 9 pumping station of Miyun Reservoir is taken as an example to verify the validity of the model. Model performance is evaluated according to mean absolute error (MAE), mean absolute percentage error (MAPE) and linear regression correlation coefficient (R2). The experimental results show that, compared with the single ARIMAX, BiGRU model and BP neural network, the SA–BiGRU–ARIMA hybrid prediction model has a better prediction effect than other data-driven models. [ABSTRACT FROM AUTHOR]

Negacyclic BCH codes are a subclass of neagcyclic codes and are the best linear codes in many cases. However, there have been very few results on negacyclic BCH codes. Let q be an odd prime power and m be a positive integer. The objective of this paper is to study negacyclic BCH codes with length q m - 1 2 and q m + 1 2 over the finite field GF (q) and analyse their parameters. The negacyclic BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes. For certain q and m, compared with cyclic codes with the same dimension and length, the negacyclic BCH codes presented in this paper have a larger minimum distance in some cases. [ABSTRACT FROM AUTHOR]

Let be the finite field with elements, and be two distinct prime numbers different from . In this paper, we first calculate all the -cyclotomic cosets modulo as a preparation for the following parts. Then we give the explicit generator polynomials of all the constacyclic codes of length over and their dual codes. In the rest of this paper, we determine all self-dual cyclic codes of length and their enumeration. This answers a question recently asked by B. Chen, H.Q.Dinh and Liu. In the last section, we calculate the case of length as an example. [ABSTRACT FROM AUTHOR]

Recently, new studies on the decoding of cyclic codes have been developed. They place themselves under the umbrella of Cooper Philosophy and they consist in using (sparse) locator polynomials, which, once evaluated at the syndromes, return the error locations. In particular, it has been recently shown that it is not necessary to use Gröbner bases to compute such kind of polynomials, and that some sparse versions can be found (at least for error correction capability at most 2), using interpolation on the syndrome variety. In this paper, we study the combinatorial structure of the syndrome variety of a cyclic code and some of its variants, for error correction capability 2, by means of standard monomials. Such monomials can be found without computing a Gröbner basis of the syndrome ideal, neither performing any step of Buchberger reduction, that is, in a degröbnerized way. [ABSTRACT FROM AUTHOR]

Binary duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root bound. In this paper, we present several families of binary duadic codes of length 2 m - 1 and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes, which partially solves one case of the open problem proposed in Liu et al (Finite Field Appl 91:102270, 2023). It is shown that the lower bounds on their minimum distances are close to the square root bound. Moreover, the parameters of the dual and extended codes of these binary duadic codes are investigated. [ABSTRACT FROM AUTHOR]

Hulls of linear codes are widely studied due to their good properties and wide applications. Let n = q m - 1 r and C be an [n, k] cyclic code over F q , where r | q - 1 . In this paper, we present several necessary and sufficient conditions for BCH codes of length n that have k - 1 or k ⊥ - 1 dimensional hulls, where k ⊥ is the dimension of C ⊥ . Further, we give the parameters of several families of self-orthogonal codes that arise as hulls of BCH codes. We obtain many optimal codes. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed–Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound. [ABSTRACT FROM AUTHOR]

Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum locality of linear codes. In addition, the minimum locality of many known families of linear codes has not been studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum locality of linear codes, and investigates the minimum locality of a number of families of linear codes, such as q-ary Hamming codes, q-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. To this end, the concepts of linear locality and minimum linear locality are specified. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper. [ABSTRACT FROM AUTHOR]

The algebraic structure of skew cyclic codes over M2(F2), using the F4-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree d(x) is a free code generated by d(x). According to our findings, skew cyclic codes of odd and even lengths are cyclic and 2-quasi-cyclic over M2(F2), respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over M2l(F2) is considered in this paper. [ABSTRACT FROM AUTHOR]

We consider G to be a finite group and p as a prime number. We fix ψ to be an irreducible character of G with its restriction to all p-regular elements of G and ψ 0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of ψ and the defect group of a p-block which contains ψ. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group. [ABSTRACT FROM AUTHOR]

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper. [ABSTRACT FROM AUTHOR]

The fundamental aim of this research is to analyze the configuration of 픽4R submodules, skew cyclic codes over 픽4R and establish their connection with DNA codes, where 픽4 is a field of order 4 and R = 픽4 + u픽4 + v픽4 + w픽4 with u2 = u, v2 = v, w2 = w, uv = vu = 0, vw = wv = 0, wu = uw = 0 is a finite ring. This is achieved by examining particular subclasses like reversible codes. Ultimately, this study aims to utilize Gray maps to derive codes that possess the characteristics of DNA structures. At the end of this paper, we have provided the necessary and sufficient condition for skew cyclic codes to be reversible complement. [ABSTRACT FROM AUTHOR]

• A digital twin-driven (DTD) framework and model are developed for predicting degraded lifting capacity (LC) of aging tower cranes. • A DTD model predicted the degraded LC of a scaled-down prototype, achieving a mean-square error (MSE) of 0.2253 and a coefficient of determination (R2) of 0.9973. • A DTD model is validated using k-5 cross-validation with a prediction accuracy of 0.97 (R2). • Degraded load charts assist operators in placing safe loads and preventing unexpected failures. Aging tower cranes face an elevated risk of failure, primarily due to structural fatigue and deterioration. Surprisingly, the degradation of aging-induced lifting capacity (LC) remains an unexplored domain. In response to this research gap, this paper introduces a digital twin-driven (DTD) framework and model to predict the degraded LC of aging tower cranes. This framework combines theoretical and numerical analysis of fatigue and degradation behavior in tower cranes with real-time vibration data obtained during cyclic load scenarios on the actual cranes. Machine learning (ML) techniques are employed to develop a model that accurately predicts the degraded LC caused by aging. A scaled-down tower crane prototype is adopted as a demonstrative case to illustrate the feasibility and effectiveness of the DTD framework. The DTD model predicts the degraded LC of the prototype with high accuracy, achieving a mean-square error (MSE) of 0.2253 and a coefficient of determination (R2) of 0.9973. The predicted degraded load charts of the tested tower crane for each decade of usage from 0 to 70 years are also presented to assist crane operators in applying safe loads, preventing unexpected failures and damages, and enhancing workplace monitoring and safety. This study helps monitor the safety conditions of tower cranes that are aging and susceptible to structural fatigue and deterioration, facilitates the prediction of the deterioration of complex machines and systems in the construction industry with real-time data, and highlights the potential of DTD approaches in improving efficiency, safety, and decision-making. [ABSTRACT FROM AUTHOR]

The inverse analysis of the intercepted signals to reconstruct the communication scheme used by the transmitter is a key problem in the non‐cooperative context. In this paper, the problem of blind reconstruction of cyclic codes and self‐synchronous scramblers is considered. There is no existing solution to this problem. To solve this problem, a joint blind reconstruction method for cyclic codes and self‐synchronous scramblers is proposed in this paper. A lemma is proposed and proved theoretically that the maximum common factor of the inverse polynomial of the polynomial form of the dual vectors of the self‐synchronous scrambled bit matrix is the product of the parity polynomial of the cyclic code and the feedback polynomial of the self‐synchronous scrambler. Using this lemma, the joint blind reconstruction of the cyclic code and the self‐synchronous scrambler is completed. Finally, a Gaussian elimination based on random column exchange and soft information (GERCESI) method is proposed, which can apply the proposed method to a noisy environment. Simulations prove that the fault‐tolerance performance of the GERCESI method proposed in this paper is at least 0.5 dB higher than that of the GJETP method. [ABSTRACT FROM AUTHOR]

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of F q P Q , where P = F q [ v ] 〈 v 3 − α 2 2 v 〉 and Q = F q [ u , v ] 〈 u 2 − α 1 2 , v 3 − α 2 2 v 〉 are nonchain finite rings and α i is in F q / { 0 } for i ∈ { 1 , 2 } , where q = p m with m ≥ 1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes. [ABSTRACT FROM AUTHOR]

Up to a new invariant μ (b) , the complete b-symbol weight distribution of a particular kind of two-weight irreducible cyclic codes, was recently obtained by Zhu et al. (Des Codes Cryptogr 90(5):1113–1125, 2022). The purpose of this paper is to simplify and generalize the results of Zhu et al., and obtain the b-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes. [ABSTRACT FROM AUTHOR]

A code is said to have locality r if any coordinate value in a codeword of that code can be recovered by at most r other coordinates. In this paper, we have studied the locality of quasi-cyclic codes over finite fields. The generator matrix of a quasi-cyclic code can be represented in the form of circulant matrices. We have obtained a bound on the locality of the code in terms of the weights of the associated polynomials to these circulant matrices. We have further analyzed the bounds on the locality, particularly in the case of 1-generator quasi-cyclic codes. An algorithm to find the locality of a quasi-cyclic code is also presented. We have given a construction of 1-generator quasi-cyclic codes with locality at most r using the zeros of its generator polynomial. Some examples have been given to illustrate the results presented in the paper. [ABSTRACT FROM AUTHOR]

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper. [ABSTRACT FROM AUTHOR]

Let s ≥ 1 be a fixed integer. In this paper, we focus on generating cyclic codes over the ring R (α 1 , α 2 , ... , α s) , where α i ∈ F q ∖ { 0 } , 1 ≤ i ≤ s , by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing F q reversibility. The aim to introduce the F q reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases (A , T , G , C) . Let be a fixed integer. In this paper, we focus on generating cyclic codes over the ring , where , , by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing reversibility. The aim to introduce the reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases . [ABSTRACT FROM AUTHOR]

In the modern world, the security of the digital image is vital due to the frequent communication of digital products over the open network. Accelerated advancement of digital data exchange, the importance of information security in the transmission of data, and its storage has emerged. Multiple uses of the images in the security agencies and the industries and the security of the confidential image data from unauthorized access are emergent and vital. In this paper, Bose Chaudhary Hocquenghem (BCH) codes over the Galois field are used for image encryption. The BCH codes over the Galois field construct MDS (maximum distance separable) matrices and secret keys for image encryption techniques. The encrypted image is calculated, by contrast, correlation, energy, homogeneity, and entropy. Histogram analysis of the encrypted image is also assured in this paper. The proposed image encryption scheme's security analysis results are improved compared to the original AES algorithm. Further, security agencies can utilize this work for their confidential image data. [ABSTRACT FROM AUTHOR]

Let p be a prime with 5 | (p - 1) . Let S be a set of all repeated-root cyclic codes C = ⟨ g (x) ⟩ , (x 5 - 1) | g (x) , of length 5p over a field field F p , whose Hamming distances are at most 7. In this paper, we present a method to find all maximum distance separable (MDS) symbol-pair codes in S. By this method we can easily obtain the results in Ma and Luo (Des Codes Cryptogr 90:121–137, 2022) and new MDS symbol-pair codes, so we remain two possible MDS symbol-pair codes for readers. [ABSTRACT FROM AUTHOR]

Subspace codes are of great use in noncoherent linear network coding (Katti et al in ACM SIGCOMM Comput Commun Rev 38(4):401–412, 2008; Kötter and Kschischang in IEEE Trans Inform Theory 54(8):3579–3591, 2008; Silva et al. in IEEE Trans Inform Theory 54(9):3951–3967, 2008). As a particular subclass of subspace codes, cyclic constant dimension subspace codes can be encoded and decoded more efficiently. There is an increased interest in constructing cyclic constant dimension subspace codes whose sizes and minimum distances are as large as possible. Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017) constructed several cyclic constant dimension subspace codes using Sidon spaces. In this paper, we present a criterion which can be used to determine whether or not the sum of some distinct Sidon spaces is again a Sidon space. Based on this result, we obtain cyclic constant dimension subspace codes via the sum of several Sidon spaces. Our results generalize some results in Niu et al. (Discret Math 343(5):111788, 2020) and Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017). [ABSTRACT FROM AUTHOR]

Let k , m be positive integers and F 2 m be a finite field of order 2 m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring S k = F 2 m [ v 1 , v 2 , ... , v k ] 〈 v i 2 − α i v i , v i v j − v j v i 〉 , for i , j = 1 , 2 , 3 , ... , k , where α i is the non-zero element of F 2 m . As an application, we obtain better quantum error correcting codes over the ring S 1 (for k = 1 ). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes. [ABSTRACT FROM AUTHOR]

Quasi-2-dimensional cyclic (Q2DC) codes over finite fields are important linear codes. They are asymptotically good, and have deep connection with convolutional codes. In this paper, using the trace representation and Gauss sums, we determine the Hamming weight distribution of Q2DC codes. As an application, we construct some classes of constant weight Q2DC codes achieving the Griesmer bound and some classes of two-weight, three-weight Q2DC codes. Moreover, by two-weight Q2DC codes, we obtain secret sharing schemes and association schemes under some special cases. [ABSTRACT FROM AUTHOR]

This paper considers three kinds of quasi-cyclic codes of index two with one generator or two generators and their applications in quantum code construction. In accordance with the algebraic structure of linear codes, we determine the lower bounds of the symplectic weights of these quasi-cyclic codes. Quasi-cyclic codes with the dual-containing property enable the construction of quantum codes. Defining the coefficient symmetric polynomials of the generator polynomials gives a concise condition for the dual-containing of the quasi-cyclic codes. The lower bound results can significantly reduce the scope of the search for a larger minimum distance of quasi-cyclic codes. With these algebraic results and computer supports, we obtain classical quasi-cyclic codes with better parameters and some new quantum codes under the symplectic construction. In particular, two examples of the new quantum codes [ [ 63 , 42 , 6 ] ] 2 , [ [ 51 , 35 , 5 ] ] 2 improve the corresponding codes in Grassl's code table. [ABSTRACT FROM AUTHOR]

In this paper, we give construction of quantum codes over 픽q from cyclic codes over the 픽q + u픽q + v픽q + w픽q + uv픽q, where u² = u,v² = v,w² = w,uv = vu, uw = wu = vw = wv = 0, q = pm and p is an odd prime. We determine the parameters of quantum error correcting codes which constructed from cyclic codes over the 픽q + u픽q + v픽q + w픽q + uv픽q. Finally, we give some examples of quantum codes. [ABSTRACT FROM AUTHOR]

A length ml, index l quasi-cyclic code can be viewed as a cyclic code of length m over the field Fql via a basis of the extension Fql/Fq. This cyclic code is an additive cyclic code. In [C. Güneri, F. Ozdemir, P. Sol'e, On the additive cyclic structure of quasi-cyclic codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize the (l,m) values for one-generator quasi-cyclic codes for which it is impossible to have an Fql -linear image for any choice of the polynomial basis of Fql/Fq. But this characterization for some (l,m) values is very intricate. In this paper, by the use of this characterization, we give a more simple characterization. [ABSTRACT FROM AUTHOR]

The binary quadratic-residue codes and the punctured Reed-Muller codes ${\mathcal {R}}_{2}((m-1)/2, m))$ are two families of binary cyclic codes with parameters $[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length $2^{m}-1$ and dimension near $2^{m-1}$ with good minimum distances. When $m \geq 3$ is odd, the codes become a family of duadic codes with parameters $[2^{m}-1, 2^{m-1}, d]$ , where $d \geq 2^{(m-1)/2}+1$ if $m \equiv 3 \pmod {4}$ and $d \geq 2^{(m-1)/2}+3$ if $m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we determine the weight distribution of several classes of double cyclic codes over Galois rings by Gauss sums. [ABSTRACT FROM AUTHOR]

Modern video object segmentation (VOS) algorithms have achieved remarkably high performance in a sequential processing order, while most of currently prevailing pipelines still show some obvious inadequacy like accumulative error, unknown robustness or lack of proper interpretation tools. In this paper, we place the semi-supervised video object segmentation problem into a cyclic workflow and find the defects above can be collectively addressed via the inherent cyclic property of semi-supervised VOS systems. Firstly, a cyclic mechanism incorporated to the standard sequential flow can produce more consistent representations for pixel-wise correspondance. Relying on the accurate reference mask in the starting frame, we show that the error propagation problem can be mitigated. Next, a simple gradient correction module, which naturally extends the offline cyclic pipeline to an online manner, can highlight the high-frequent and detailed part of results to further improve the segmentation quality while keeping feasible computation cost. Meanwhile such correction can protect the network from severe performance degration resulted from interference signals. Finally we develop cycle effective receptive field (cycle-ERF) based on gradient correction process to provide a new perspective into analyzing object-specific regions of interests. We conduct comprehensive comparison and detailed analysis on challenging benchmarks of DAVIS16, DAVIS17 and Youtube-VOS, demonstrating that the cyclic mechanism is helpful to enhance segmentation quality, improve the robustness of VOS systems, and further provide qualitative comparison and interpretation on how different VOS algorithms work. The code of this project can be found at https://github.com/lyxok1/STM-Training. [ABSTRACT FROM AUTHOR]

A Z 2 s -additive code C of length n is a subgroup of Z 2 s n . Carlet (IEEE Trans Inf Theory 44:1543–1547, 1998) introduced a Gray map Φ (Carlet's Gray map) on Z 2 s and Tapia-Recillas and Vega (SIAM J Discret Math 17:103–113, 2003) have been shown that Φ (C) is a binary linear code if and only if for every u , v ∈ C , 2 (u ⊙ v) ∈ C , which their number is ∣ C ∣ 2 - ∣ C ∣ 2 . Let C be a linear code over Z 8 of type { δ 0 , δ 1 , δ 2 } . In this paper, by using the rows of the generator matrix (in standard form) of C , we introduce a set S with ∣ S ∣ ≤ δ 0 (δ 0 + 3) 2 + δ 1 and show that Φ (C) is linear if and only if for every u , v ∈ S , 2 (u ⊙ v) ∈ C . The results show that for a linear code C over Z 8 calculations to check the linearity of Φ (C) are less than Tapia-Recillas and Vega's method. In addition, for a cyclic code of odd length over Z 8 , a condition on the generator polynomial of this code is given which its Carlet's Gray image is linear. Also, as a result, we show that the Carlet's Gray image of a quadratic residue code over Z 8 is not linear. [ABSTRACT FROM AUTHOR]

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance plays a vital role in determining the error-correcting capability and the constructions of symbol-pair codes with largest possible minimum symbol-pair distance is of great importance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that such codes can acheive the Singleton bound. In this paper, for length 5p, two new classes of MDS symbol-pair codes with minimum symbol-pair distance seven or eight are constructed by utilizing repeated-root cyclic codes over F p , where p is a prime. In addition, we derive a class of MDS symbol-pair codes with minimum symbol-pair distance seven and length 4p. [ABSTRACT FROM AUTHOR]

This paper explores new methods to increase the level of safety of data transfer between sensors and electronic control units (ECUs) in automotive communication. A new model of basic sensors to be used in automotive electronics is proposed. This model contains hardware modules that implement the end-to-end communication protection (E2E) mechanism, as defined by the Automotive Open System Architecture (AUTOSAR) standard. By adding this feature inside the sensors, it is possible that, in addition to increasing the safety level, these sensors can be directly connected to the network ECUs via standard communication buses (e.g., Local Interconnect Network (LIN), Controller Area Network (CAN), Flexray, etc.). This paper describes the model, design, and mapping (in a Field Programmable Gate Array device (FPGA)) of the hardware E2E module capable of generating the Cyclic Redundancy Code (CRC) and counter signal for a customized message. This message represents the output of the new sensor E2E module used in a safety communication as requested by the automotive E2E standard. The model is validated also by comparing the data output of the E2E hardware with the data output of the AUTOSAR software E2E library. Finally, future needs and directions are suggested in this area. [ABSTRACT FROM AUTHOR]

In this paper we introduce Ouroboros, a new family of Key Exchange protocols based on coding theory. The protocols propose a middle ground between the cryptosystems based on $\mathsf {QC}$ - $\mathsf {MDPC}$ codes, which feature small parameter sizes, but have a security reduction to two problems: the syndrome decoding problem and the indistinguishability of the code, and the HQC protocol, which features bigger parameters but has a security reduction to the syndrome decoding problem only. Ouroboros features a reduction to the syndrome decoding problem with only a small overhead compared to the $\mathsf {QC}$ - $\mathsf {MDPC}$ based cryptosystems. The approach is based on an ideal structure and also works for the rank metric. This yields a simple, secure and efficient approach for key exchange, the Ouroboros family of protocols. For the Hamming metric we obtain the same type of parameters (and almost the same simple decoding) as for $\mathsf {MDPC}$ based cryptosystems, but with a security reduction to decoding random quasi-cyclic codes in the Random Oracle Model. This represents a reduction of up to 38% on the public key size compared to HQC, for the most secure parameters. For the rank metric, we obtain better parameters than for RQC, saving up to 31% on the public key for the most secure set of parameters, using non homogeneous errors in Ouroboros. In this full version, the protocol and decoding algorithm have been slightly improved, additional details are given in the security proof, and the protocol is fully described for the rank metric. [ABSTRACT FROM AUTHOR]