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An $(r, \delta)$ -locally repairable code ($(r, \delta)$ -LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$ -LRCs produced by Gopalan et al.. An $(r, \delta)$ -LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. generalized the construction of cyclic $r$ -LRCs proposed by Tamo et al. , and constructed several classes of optimal $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of , , this paper first characterizes $(r, \delta)$ -locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$ -LRCs proposed in , , and our method seems more convenient to obtain optimal cyclic $(r, \delta)$ -LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively with $(r+\delta -1)\nmid n$ can be obtained from our method. [ABSTRACT FROM AUTHOR]

Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for every positive even integer ℓ. In this paper, we study the case where $ x^m-1 $ x m − 1 has an arbitrary number of irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] ; in the previous studies, only some special cases where $ x^m-1 $ x m − 1 has exactly two or three irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for any even positive integer ℓ, where we require that $ q \equiv 1 \pmod {4} $ q ≡ 1 (mod 4) if the index $ \ell \ge 6 $ ℓ ≥ 6. By implementation of our method, we obtain a new optimal binary self-dual code $ [172, 86, 24] $ [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4. [ABSTRACT FROM AUTHOR]

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]

In [C. Ding, An infinite family of Steiner systems $ S(2, 4, 2^m) $ from cyclic codes, J. Combin. Des. 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 2 \pmod{4} \ge 6 $ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 0 \pmod{4} \ge 4 $ supported by this family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $ S(2, 4, 2^m) $ for all even $ m \geq 4 $. Furthermore, this paper also determines the parameters of other $ 2 $-designs supported by this family of extended cyclic codes. [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]

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]

In this paper, we construct a class of Z 4 Z 4 Z 4 -additive cyclic codes generated by 3-tuples of polynomials. We discuss their algebraic structure and show that generator matrices can be constructed for all codes in this class. We study asymptotic properties of this class of codes by using a Bernoulli random variable. Moreover, let 0 < δ < 1 be a real number such that the entropy h 4 ((k + l + t) δ 6) < 1 4 , we show that the relative minimum distance converges to δ and the rate of the random codes converges to 1 k + l + t , where k, l, and t are pairwise co-prime positive odd integers. Finally, we conclude that the Z 4 Z 4 Z 4 -additive cyclic codes are asymptotically good. [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]

Let p 1 , p 2 , ... , p r , q be distinct primes, q odd. Let m = ∏ i = 1 r p i , where r ≥ 2 be an integer. In this paper, it is observed that the explicit expressions of primitive idempotents from R p i are sufficient to compute the explicit expressions of primitive idempotent in semisimple ring R m = F q [ x ] / (x m − 1). It is also shown that the results obtained in [A. Sahni and P. T. Sehgal, Minimal cyclic codes of length p n q , Finite Fields Appl. 18(5) (2012) 1017–1036; P. Kumar and S. K. Arora, λ -mapping and primitive idempotents in semisimple ring ℜ m , Comm. Algebra 41(10) (2013) 3679–3694] are simple corollaries to the results obtained in the paper. [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]

Let p be an odd prime and m and s positive integers, with m even. Let further F p m be the finite field of p m elements and R = F p m + u F p m ( u 2 = 0 ). Then R is a finite chain ring of p 2 m elements, and there is a Gray map from R N onto F p m 2 N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over F p m by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length p s over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length p s over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. [ABSTRACT FROM AUTHOR]

In this paper, δ -dual codes over finite commutative semi-simple rings are defined as a generalization of dual codes over finite commutative semi-simple rings. Some properties of δ -dual codes are given. We present necessary and sufficient conditions for a λ -constacyclic code of length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. We also study the δ -dual of skew Θ - λ -constacyclic codes over finite commutative semi-simple rings. Among others, we also give necessary and sufficient conditions for a skew Θ - λ -constacyclic code of any length n to be δ -self-dual, δ -self-orthogonal, δ -dual-containing, δ -LCD over finite commutative semi-simple rings. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the behaviour of cyclic codes over a finite ring, where the ring S = Tq[v, w]/⟨v² - α², w³ - β²w, vw - wv⟩ is constructed using a field Tq of order q, with q being an odd prime power, and nonzero elements α and β from Tq. With the help of Gray images of cyclic code over the defined ring S, we construct LCD codes. A condition is also established for cyclic codes that contain their duals. Additionally, by using the self-orthogonal condition of cyclic codes over S, we are able to construct quantum codes. [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 this paper, we introduce F q 2 -double cyclic codes of length n = r + s , where F q 2 is the Galois field of q 2 elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any F q 2 -double cyclic code. For any F q 2 -double cyclic code C , we will define the Euclidean dual code C ⊥ based on the Euclidean inner product and the Hermitian dual code C ⊥ H based on the Hermitian inner product. We will construct a relationship between C ⊥ and C ⊥ H and then find the generator polynomials for the Hermitian dual code C ⊥ H. As an application of our work, we will present examples of optimal parameter linear codes over the finite field F 4 and also examples of optimal quantum codes that were derived from F 4 -double cyclic codes using the Hermitian inner product. [ABSTRACT FROM AUTHOR]

In an interesting paper, Prof. C. Ding provided three constructions of cyclic codes of length being a product of two primes. Numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field. However, not much is known about these codes. In this paper, we explain some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance. We also provide a general construction of cyclic codes of composite length which are related to Ding’s constructions. Numerical data shows that it produces many best cyclic codes as well. Finally, we point out how these cyclic codes can be used to construct convolutional codes with large free distance. [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]

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]

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]

A class of optimal three-weight [ q k - 1 , k + 1 , q k - 1 (q - 1) - 1 ] cyclic codes over I F q , with k ≥ 2 , achieving the Griesmer bound, was presented by Heng and Yue (IEEE Trans Inf Theory 62(8):4501–4513, 2016. https://doi.org/10.1109/TIT.2016.2550029). In this paper we study some of the subfield codes of this class of optimal cyclic codes when k = 2 . The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal, which is a desirable property that is useful in the design of a secret sharing scheme based on a linear code. Moreover, a specific example of a secret sharing scheme based on one of these subfield codes is given. Finally, a class of optimal two-weight linear codes over I F q , achieving the Griesmer bound, whose duals are almost optimal with respect to the sphere-packing bound is presented. Through a different approach, this class of optimal two-weight linear codes was reported very recently by Heng (IEEE Trans Inf Theory 69(2):978–994, 2023. https://doi.org/10.1109/TIT.2022.3203380). Furthermore, it is shown that these optimal codes can be used to construct strongly regular graphs. [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 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]

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]

In this paper, we develop a theory for constructing cyclic codes of odd length over the ring R = ℤ 4 + u ℤ 4 + v ℤ 4 , where u 2 = v 2 = u v = v u = 0 , which plays an important role in DNA computing. A direct link between the elements of R and the 6 4 codons used in the amino acids of the living organisms is established. Then, we investigate reversible cyclic codes and reversible complement cyclic codes of odd length over R. Moreover, we give some properties of binary images of the codons under the Gray map. Finally, two examples of cyclic codes over R with their minimum Hamming distance will be studied as well. [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]

In this paper, the structures of the linear codes over a family of the rings At = Z4 [u1, ..., ut]/i² - ui, uiuj - ujui> are given, where i, j = 1, 2, ..., t, i ≠ j, Z4 = {0, 1, 2, 3}. A map between the elements of the At and the alphabet ... is constructed. The DNA codes are obtained with three different methods, by using the cyclic, skew cyclic codes over a family of the rings At and θi-set, where θi is a non trivial automorphism on Ai, for i = 1, 2, ..., t. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in monomial codes with associated vector a = (a0, a1,..., an-1), introduced in [4], and more generally in linear codes invariant under a monomial matrix M = diag(a0,a1,...,an-1)Pσ where σ is a permutation and Pσ its associated permutation matrix. We discuss some connections between monomial codes and codes invariant under an arbitrary monomial matrix M. Next, we identify monomial codes with associated vector a = (a0,a2,...,an-1) by the ideals of the polynomial ring ..., via a special isomorphism ... which preserves the Hamming weight and differs from the classical isomorphism used in the case of cyclic codes and their generalizations. This correspondence leads to some basic characterizations of monomial codes such as generator polynomials, parity check polynomials, and others. Next, we focus on the structure of l--quasi-monomial (l--QM) codes of length n = ml, where on the one hand, we characterize them by the Rq,m --submodules of Rq,ml . On the other hand, l--QM codes are seen as additive monomial codes over the extension 픽ql/Fq. So, as in the case of quasi-cyclic codes [8], we characterize those codes that have 픽ql--linear images with respect to a basis of the extension 픽ql/Fq, based on the CRT decomposition. Finally, we show that l--QM codes and additive monomial codes are asymptotically good. [ABSTRACT FROM AUTHOR]

The last few decades have seen an increase in the determination of the parameters of the primitive BCH codes. Indeed, BCH codes are powerful in terms of encoding and decoding. They are applied in several fields such as: satellite communications, cryptography, compact disk drives etc, and have good structural properties. Nevertheless, the dimension and the minimum distance of those codes are not known in general. In this paper, we present a class of narrow sense primitive BCH codes of designed distance .... Also, we investigate their Bose distance and dimension. [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]

This paper explores how energy piles interact under imbalanced and balanced daily temperature cycles and a range of monotonic thermal loads, combining field experiments and numerical simulations on two bored energy piles with a spacing of 3.5 m. Monotonic heating and cooling loads were simulated for temperature changes of |ΔT| = 5°C, 10°C, 15°C, and 20°C. Balanced, cooling-oriented imbalanced, and heating-oriented imbalanced thermal cycles were simulated between 0°C and 40°C with heating-to-cooling time ratios of 12:12, 16:8, and 8:16, respectively. One of the two energy piles' axial and radial thermomechanical responses was investigated during single- and dual-pile operations. Soil temperature changes between the piles were greater for dual-pile operation, leading to increased thermal interaction, particularly for higher magnitudes of monotonic thermal loads. However, dual-pile operation did not alter the ground temperatures near the edge of the piles for the pile spacing considered. They remained similar for single- and dual-pile operations for the setting investigated in this paper. As a result, the pile temperatures, axial and radial thermal stresses, and thermal stress rates were similar for all single- and dual-pile operations simulations. Cyclic temperatures, particularly balanced cyclic loads, induced lower thermal effects in the piles and soil than in other cases. Overall, the results from this study provide validated insights into the situations where thermal interaction and different temperatures typical of heat pumps should be considered in designing groups of energy piles. [ABSTRACT FROM AUTHOR]

The Hochschild homology of the ring k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic cyclic homology of k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 over k for k = Z. These calculations have been known for k = Q or fields containing it, but this paper studies the torsion which one gets for k = Z and through it, via the universal coefficient theorem, for general k. The computations are related to homology, cohomology, and Tate cohomology of cyclic groups. [ABSTRACT FROM AUTHOR]

BCH codes are a special subclass of cyclic codes and have many important applications in data storage and communication systems. In this paper, we investigate the structure of binary linear complementary dual (LCD) BCH codes with length $ n = \frac{{{2^m} + 1}}{3} $, where $ m \geq 7 $ is an odd integer. By exploring cyclotomic cosets modulo $ n $, we determine the dimension of LCD BCH codes for designed distance $ \delta $ in the range $ 2 \le \delta \le{2^{\frac{{m + 1}}{2}}} $. Furthermore, we compute the first five largest coset leaders modulo $ n $ and construct some binary LCD BCH codes. We also present two families of optimal binary linear codes from LCD BCH codes. [ABSTRACT FROM AUTHOR]