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In this paper, we propose a novel approach to enhance spectral efficiency in communication systems by dynamically adjusting the mapping between cyclic permutation coding (CPC) and its injected form. By monitoring channel conditions such as interference levels and impulsive noise strength, the system optimises the coding scheme to maximise data transmission reliability and efficiency. The CPC method employed in this work maps information bits onto non-binary symbols in a cyclic manner, aiming to improve the Hamming distance between mapped symbols. To address challenges such as low data rates inherent in permutation coding, injection techniques are introduced by removing δ column(s) from the CPC codebook. Comparative analyses demonstrate that the proposed dynamic adaptation scheme outperforms conventional permutation coding and injection schemes. Additionally, we present a generalised mathematical expression to describe the relationship between the spectral efficiencies of both coding schemes. This dynamic approach ensures efficient and reliable communication in environments with varying levels of interference and impulsive noise, highlighting its potential applicability to systems like power line communications. [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]

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]

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]

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 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]

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]

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]

Although long polar codes with successive cancellation decoding can asymptotically achieve channel capacity, the performance of short blocklength polar codes is far from optimal. Recently, Arıkan proposed employing a convolutional pre-transformation before the polarization network, called polarization-adjusted convolutional (PAC) codes. In this paper, we focus on improving the performance of short PAC codes concatenated with a cyclic redundancy check (CRC) outer code, CRC-PAC codes, since error detection capability is essential in practical applications, such as the polar coding scheme for the control channel. We propose an enhanced adaptive belief propagation (ABP) decoding algorithm with the assistance of CRC bits for PAC codes. We also derive joint parity-check matrices of CRC-PAC codes suitable for iterative BP decoding. The proposed CRC-aided ABP (CA-ABP) decoding can effectively improve error performance when partial CRC bits are used in the decoding. Meanwhile, the error detection ability can still be guaranteed by the remaining CRC bits and adaptive decoding parameters. Moreover, compared with the conventional CRC-aided list (CA-List) decoding, our proposed scheme can significantly reduce computational complexity, to achieve a better trade-off between the performance and complexity for short PAC codes. [ABSTRACT FROM AUTHOR]

A general framework describing the statistical discrimination of an ensemble of quantum channels is given by the name quantum reading. Several tools can be applied in quantum reading to reduce the error probability in distinguishing the ensemble of channels. Classical and quantum codes can be envisioned for this goal. The aim of this paper is to present a simple but fruitful protocol for this task using classical error-correcting codes. Three families of codes are considered: Reed–Solomon codes, BCH codes, and Reed–Muller codes. In conjunction with the use of codes, we also analyze the role of the receiver. In particular, heterodyne and Dolinar receivers are taken into consideration. The encoding and measurement schemes are connected by the probing step. As probes, we consider coherent states. In such a simple manner, interesting results are obtained. As we show, there is a threshold below which using codes surpass optimal and sophisticated schemes for any fixed rate and code. BCH codes in conjunction with Dolinar receiver turn out to be the optimal strategy for error mitigation in quantum reading. [ABSTRACT FROM AUTHOR]

In this work, the performance of coded continuous phase modulation (CPM) transceivers for orthogonal frequency-division multiplexing (OFDM) in underwater acoustic channels is evaluated. The proposed technique employs the Bose–Chaudhuri–Hocquenghem (BCH) class of cyclic codes, and a CPM-based mapper is being used in place of a traditional OFDM mapper. Bit-error-rates (BERs) for various distances are evaluated. We use the Bellhop acoustic channel simulator that is considered to be very close to a real underwater channel. The performance of the proposed system is evaluated for 23 values of the CPM modulation index h as a function of the distance between the transmitter (Tx) and receiver (Rx). Based upon the error performance, we identified several best-performing CPM indices. We also observed that for Tx–Rx distances of 100 and 250 m, the proposed system gives acceptable performance without the need for equalization. We also compared the out-of-band radiation of the proposed system with PSK-OFDM and observed that the uncoded CPM-OFDM has a better out-of-band (OOB) performance than the traditional OFDM. Moreover, a comparison of BERs with the traditional PSK-OFDM system shows that the proposed system outperforms the traditional OFDM system by a wide margin. [ABSTRACT FROM AUTHOR]

We describe the nonabelian exterior square G ∧ ^ G of a pro-p-group G (with p arbitrary prime) in terms of quotients of free pro-p-groups, providing a new method of construction of G ∧ ^ G and new structural results for G ∧ ^ G . Then, we investigate a generalization of the probability that two randomly chosen elements of G commute: this notion is known as the "complete exterior degree" of a pro-p-group and we will use it to characterize procyclic groups. Among other things, we present a new formula, which simplifies the numerical aspects which are connected with the evaluation of the complete exterior degree. [ABSTRACT FROM AUTHOR]

Accurate direction of arrival (DoA) estimation is paramount in various fields, from surveillance and security to spatial audio processing. This work introduces an innovative approach that refines the DoA estimation process and demonstrates its applicability in diverse and critical domains. We propose a two-stage method that capitalizes on the often-overlooked secondary peaks of the cross-correlation function by introducing a reduced complexity DoA estimation method. In the first stage, a low complexity cost function based on the zero cyclic sum (ZCS) condition is used to allow for an exhaustive search of all combinations of time delays between pairs of microphones, including primary peak and secondary peaks of each cross-correlation. For the second stage, only a subset of the time delay combinations with the lowest ZCS cost function need to be tested using a least-squares (LS) solution, which requires more computational effort. To showcase the versatility and effectiveness of our method, we apply it to the challenging acoustic-based drone DoA estimation scenario using an array of four microphones. Through rigorous experimentation with simulated and actual data, our research underscores the potential of our proposed DoA estimation method as an alternative for handling complex acoustic scenarios. The ZCS method demonstrates an accuracy of 89.4 % ± 2.7 % , whereas the ZCS with the LS method exhibits a notably higher accuracy of 94.0 % ± 3.1 % , showcasing the superior performance of the latter. [ABSTRACT FROM AUTHOR]

In this paper, new construction methods of entanglement-assisted quantum error correction code (EAQECC) from circulant matrices are proposed. We first construct the matrices from two vectors of constraint size, and determine the isotropic subgroup. Then, we also propose a method for calculation of the entanglement subgroup based on standard forms of binary matrices to satisfy the constraint conditions of EAQECC.With isotropic and entanglement subgroups, we determine all the parameters and the minimum distance of the EAQECC. The proposed EAQECC with small lengths are presented to explain the practicality of this construction of EAQECC. Comparison with some earlier constructions of EAQECC shows that the proposed EAQECC is better. [ABSTRACT FROM AUTHOR]

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H = 〈 a , b ∣ 2 a = 2 b = 0 , a 2 = 0 , b 2 = b , a b = b a = 0 〉. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2 n over F 2 is studied. [ABSTRACT FROM AUTHOR]

In this paper, an algebraic structure of a type of double cyclic codes is investigated, which covers some existing codes as special cases. The paper presents generic results about the generating polynomials, minimal generating sets, matrices and dual codes of the proposed codes. [ABSTRACT FROM AUTHOR]

Wind power forecasting is pivotal in promoting a stable and sustainable grid operation by estimating future power outputs from past meteorological and turbine data. The inherent unpredictability in wind patterns poses substantial challenges in synchronizing supply with demand, with inaccuracies potentially destabilizing the grid and potentially causing energy shortages or excesses. This study develops a data-driven approach to forecast wind power from 30 min to 12 h ahead using historical wind power data collected by the Supervisory Control and Data Acquisition (SCADA) system from one wind turbine, the Enercon/E92 2350 kW model, installed at Casa Nova, Bahia, Brazil. Those data were measured from January 2020 to April 2021. Time orientation was embedded using sine/cosine or cyclic encoding, deriving 16 normalized features that encapsulate crucial daily and seasonal trends. The research explores two distinct strategies: error prediction and error correction, both employing a sequential model where initial forecasts via k-Nearest Neighbors (KNN) are rectified by the Extra Trees Regressor. Their primary divergence is the second model's target variable. Evaluations revealed both strategies outperforming the standalone KNN, with error correction excelling in short-term predictions and error prediction showing potential for extended forecasts. This exploration underscores the imperative importance of methodology selection in wind power forecasting. [ABSTRACT FROM AUTHOR]

The soft decision decoding algorithm for cyclic codes, especially the maximum likelihood (ML) decoding algorithm, can obtain significant performance superior to that of algebraic decoding, but the complexity is much higher. To deal with this problem, an improved soft decision decoding algorithm based on a cyclic information set and its efficient implementation architecture are proposed. This algorithm employs the property of the cyclic codes to generate a series of cyclic information sequences by circularly shifting, constructing the cyclic information set. Then, a limited number of candidate information sequences are efficiently generated using an iterative computation method, and the candidate codewords are generated using the very concise encoding method of the cyclic codes. Furthermore, the efficient hardware architecture based on systolic arrays is also proposed to generate candidate information sequences and to select the optimal candidate codewords. An emulation platform is constructed to verify the error correction performance and to determine the optimal decoder parameters. Emulation results indicate that, with appropriate parameter selection, the proposed decoding algorithm can achieve a bit error rate approaching the ML performance while maintaining low complexity. [ABSTRACT FROM AUTHOR]

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds. [ABSTRACT FROM AUTHOR]

Let F q be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of F q . The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R = F q [ u 1 , u 2 ] / 〈 u 1 2 − α 2 , u 2 2 − β 2 , u 1 u 2 − u 2 u 1 〉 . We decompose the ring R by using orthogonal idempotents Δ 1 , Δ 2 , Δ 3 , and Δ 4 as R = Δ 1 R ⊕ Δ 2 R ⊕ Δ 3 R ⊕ Δ 4 R , and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes. [ABSTRACT FROM AUTHOR]

Let R = F 4 + u F 4 , with u 2 = u and S = F 4 + u F 4 + v F 4 , w i t h u 2 = u , v 2 = v , u v = v u = 0 . In this paper, we study F 4 R S -cyclic codes of block length (α , β , γ) and construct cyclic DNA codes from them. F 4 R S -cyclic codes can be viewed as S [ x ] -submodules of F q [ x ] 〈 x α − 1 〉 × R [ x ] 〈 x β − 1 〉 × S [ x ] 〈 x γ − 1 〉 . We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps ψ 1 from R to F 4 2 and ψ 2 from S to F 4 3 , we give a one-to-one correspondence between DNA codons of the alphabets { A , T , G , C } 2 , { A , T , G , C } 3 and the elements of R , S , respectively. Then we discuss necessary and sufficient conditions of cyclic codes over F 4 , R, S and F 4 R S to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-known class of powerful error correction cyclic codes. BCH codes can correct multiple errors with minimal redundancy. Primitive BCH codes only exist for some word lengths, which do not frequently match those employed in digital systems. This paper focuses on double error correction (DEC) codes for word lengths that are in powers of two (8, 16, 32, and 64), which are commonly used in memories. We also focus on hardware implementations of the encoder and decoder circuits for very fast operations. This work proposes new low redundancy and reduced overhead (LRRO) DEC codes, with the same redundancy as the equivalent BCH DEC codes, but whose encoder, and decoder circuits present a lower overhead (in terms of propagation delay, silicon area usage and power consumption). We used a methodology to search parity check matrices, based on error patterns, in order to design the new codes. We implemented and synthesized them, and compared their results with those obtained for the BCH codes. Our implementation of the decoder circuits achieved reductions between 2.8% and 8.7% in the propagation delay, between 1.3% and 3.0% in the silicon area, and between 15.7% and 26.9% in the power consumption. Therefore, we propose LRRO codes as an alternative for protecting information against multiple errors. [ABSTRACT FROM AUTHOR]

The main motivation of this work is to study and obtain some reversible and DNA codes of length n with better parameters. Here, we first investigate the structure of cyclic and skew cyclic codes over the chain ring R : = F 4 [ v ] / 〈 v 3 〉 . We show an association between the codons and the elements of R using a Gray map. Under this Gray map, we study reversible and DNA codes of length n. Finally, several new DNA codes are obtained that have improved parameters than previously known codes. We also determine the Hamming and the Edit distances of these codes. [ABSTRACT FROM AUTHOR]

Optimal demodulation band extraction is a significant step in rolling bearing fault analysis. However, existing methods, primarily based on global indexes and neglecting negative local outliers, cannot identify compound faults in intense noise environments. To address this problem, a novel demodulation band extraction method based on weighted geometric cyclic relative entropy (WGCRE) is proposed. WGCRE is defined on the cyclic sub-bands model of the logarithmic envelope spectrum (LES) to fully consider the bearing characteristic frequency of pseudo-cyclostationarity. In detail, local and global thresholds are separately set by the white noise parameter and harmonic-to-noise ratio to exclude the exogenous noise outliers. On this basis, the WGCRE is defined as a geometrically weighted index of several different fault types to avoid harmonic interference and improve the identification of composite faults. WGCRE–gram, similar to fast kurtogram (FK), is then constructed by replacing kurtosis with WGCRE to extract the optimal demodulation band. Compared with FK and another LES-based method, logarithmic-cycligram, the proposed method is more robust for accurately identifying single and compound faults under external noise. The effectiveness of this method is verified through simulations and actual tests. Simulation experiments of different kinds and intensities of exogenous noise interference preliminarily determine the superior robustness of WGCRE in the face of solid noise. The inner ring, outer ring, and composite fault experiments further confirmed the robust adaptability of WGCRE in the face of complex working conditions. [ABSTRACT FROM AUTHOR]

In this article, we consider a semi-local ring S = F q + u F q , where u 2 = u , q = p s and p is a prime number. We define a multiplication y b = β (b) y + γ (b) , where β is an automorphism and γ is a β -derivation on S so that S [ y ; β , γ ] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S [ y ; β , γ ] and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over S. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software. [ABSTRACT FROM AUTHOR]

In this article, we investigate the formation of reversible cyclic codes (i.e., its codewords forms a symmetry) over the ring S = F 2 + u F 2 + u 2 F 2 , where u 3 = 0 . We find a unique set of generators for cyclic codes over S and classify reversible cyclic codes to their generators. The dual reversible cyclic codes are studied as well. Moreover, we provide some examples of reversible cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we study the structural properties of (α + u 1 β + u 2 γ + u 1 u 2 δ) -constacyclic codes over R = F q [ u 1 , u 2 ] / 〈 u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 〉 where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R. We have given the applications by computing non-binary quantum codes over this ring R. [ABSTRACT FROM AUTHOR]

Secure and reliable information flow is one of the main challenges in social IoT and mobile networks. Information flow and data integrity is still an open research problem. In this paper, we develop new methods of constructing systematic and regular Low-Density Parity-Check Matrices (LDPCM), inspired by the structure of the Sarrus method and geometric designs. Furthermore, these codes have cyclic structure and therefore, are less complex in computation and also require less memory in hardware implementation. Besides, an optimal method of post-processing for deleting girths four is presented. Numerical results show that the codes constructed by these methods perform well over the additive white Gaussian noise (AWGN) channel when decoded with the sum-product LDPC iterative algorithms. The proposed methods can be very efficient in terms of reducing memory consumption and improving the convergence speed of the decoder particularly in IoT and mobile networks. [ABSTRACT FROM AUTHOR]

Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection, error correction, data transmission, and data storage. Codes are studied by various scientific disciplines, such as information theory, electrical engineering, mathematics, linguistics, and computer science, to design efficient and reliable data transmission methods. Many authors in the previous literature have discussed codes over finite fields, Gaussian integers, quaternion integers, etc. In this article, the author defines octonion integers, fundamental theorems related to octonion integers, encoding, and decoding of cyclic codes over the residue class of octonion integers with respect to the octonion Mannheim weight one. The comparison of primes, lengths, cardinality, dimension, and code rate with respect to Quaternion Integers and Octonion Integers will be discussed. [ABSTRACT FROM AUTHOR]

Let R be a Galois ring, G R (p n , r) , of characteristic p n and of order p n r . In this article, we study cyclic codes of arbitrary length, N, over R. We use discrete Fourier transform (DFT) to determine a unique representation of cyclic codes of length, N, in terms of that of length, p s , where s = v p (N) and v p are the p-adic valuation. As a result, Hamming distance and dual codes are obtained. In addition, we compute the exact number of distinct cyclic codes over R when n = 2 . [ABSTRACT FROM AUTHOR]

Bose–Chaudhuri–Hocquenghem (BCH) codes are broadly used to correct errors in flash memory systems and digital communications. These codes are cyclic block codes and have their arithmetic fixed over the splitting field of their generator polynomial. There are many solutions proposed using CPUs, hardware, and Graphical Processing Units (GPUs) for the BCH decoders. The performance of these BCH decoders is of ultimate importance for systems involving flash memory. However, it is essential to have a flexible solution to correct multiple bit errors over the different finite fields (GF(2 m )). In this paper, we propose a pragmatic approach to decode BCH codes over the different finite fields using hardware circuits and GPUs in tandem. We propose to employ hardware design for a modified syndrome generator and GPUs for a key-equation solver and an error corrector. Using the above partition, we have shown the ability to support multiple bit errors across different BCH block codes without compromising on the performance. Furthermore, the proposed method to generate modified syndrome has zero latency for scenarios where there are no errors. When there is an error detected, the GPUs are deployed to correct the errors using the iBM and Chien search algorithm. The results have shown that using the modified syndrome approach, we can support different multiple finite fields with high throughput. [ABSTRACT FROM AUTHOR]

This paper summarizes the details of recently proposed binary locally repairable codes (BLRCs) and their features. The construction of codes over a small alphabet size of symbols is of particular interest for efficient hardware implementation. Therefore, BLRCs are highly noteworthy because no multiplication is required during the encoding, decoding, and repair processes. We explain the various construction approaches of BLRCs such as cyclic code based, bipartite graph based, anticode based, partial spread based, and generalized Hamming code based techniques. We also describe code generation methods based on modifications for linear codes such as extending, shorting, expurgating, and augmenting. Finally, we summarize and compare the parameters of the discussed constructions. [ABSTRACT FROM AUTHOR]

Since the product of complex numbers and rectangular fuzzy complex numbers (RFCN) is not necessarily a RFCN in the former fuzzy complex linear system (FCLS), the scalar multiplication and addition operations of complex numbers and fuzzy complex numbers (FCN) based on a new representation of FCN are proposed. We also introduce a new method for solving FCLS, which can convert FCLS into two distinct linear systems. One is an n × n complex linear system, and the other is an (m n) × (m n) real linear system, where n is the number of unknown variables, and m is the number of substitutional cyclic sets composed of coefficients of FCLS. In particular, using this method to solve one-dimensional fuzzy linear systems, a (2 n) × (2 n) RLS is obtained, which is consistent with Friedman's method. Finally, FCLS based on the RFCN as a special case are also investigated. [ABSTRACT FROM AUTHOR]

The index coding problem consists of a system with a server and multiple receivers with different side information and demand sets, connected by a noiseless broadcast channel. The server knows the side information available to the receivers. The objective is to design an encoding scheme that enables all receivers to decode their demanded messages with a minimum number of transmissions, referred to as an index code length. The problem of finding the minimum length index code that enables all receivers to correct a specific number of errors has also been studied. This work establishes a connection between index coding and error-correcting codes with multiple interpretations from the tree construction of nested cyclic codes. The notion of multiple interpretations using nested codes is as follows: different data packets are independently encoded, and then combined by addition and transmitted as a single codeword, minimizing the number of channel uses and offering error protection. The resulting packet can be decoded and interpreted in different ways, increasing the error correction capability, depending on the amount of side information available at each receiver. Motivating applications are network downlink transmissions, information retrieval from datacenters, cache management, and sensor networks. [ABSTRACT FROM AUTHOR]

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F (4) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical. [ABSTRACT FROM AUTHOR]