Journal articles: 'Algebraic Coding Theory'

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SAKATA, Shojiro. "Algebraic Coding Theory." IEICE ESS Fundamentals Review 1, no. 3 (2008): 3_44–3_57. http://dx.doi.org/10.1587/essfr.1.3_44.

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MATSUI, Hajime. "Algebraic Methods in Coding Theory." IEICE ESS Fundamentals Review 8, no. 3 (2015): 151–61. http://dx.doi.org/10.1587/essfr.8.151.

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Wood, Jay A. "Spinor groups and algebraic coding theory." Journal of Combinatorial Theory, Series A 51, no. 2 (July 1989): 277–313. http://dx.doi.org/10.1016/0097-3165(89)90053-8.

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Campillo, Antonio, Patrick Fitzpatrick, Edgar Martínez-Moro, and Ruud Pellikaan. "Special issue algebraic coding theory and applications." Journal of Symbolic Computation 45, no. 7 (July 2010): 721–22. http://dx.doi.org/10.1016/j.jsc.2010.03.006.

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Hirschfeld, J. W. P. "INTRODUCTION TO CODING THEORY AND ALGEBRAIC GEOMETRY." Bulletin of the London Mathematical Society 23, no. 5 (September 1991): 498–500. http://dx.doi.org/10.1112/blms/23.5.498.

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Yang, Guomin, Chik How Tan, Yi Mu, Willy Susilo, and Duncan S. Wong. "Identity based identification from algebraic coding theory." Theoretical Computer Science 520 (February 2014): 51–61. http://dx.doi.org/10.1016/j.tcs.2013.09.008.

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Nagaraj, S. V. "Review of Algebraic Coding Theory Revised Edition by Elwyn Berlekamp." ACM SIGACT News 48, no. 1 (March 10, 2017): 23–26. http://dx.doi.org/10.1145/3061640.3061645.

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Astola, Jaakko. "Digital signal processing, applications to communications and algebraic coding theory." Signal Processing 23, no. 2 (May 1991): 215–16. http://dx.doi.org/10.1016/0165-1684(91)90076-u.

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Chen, Fuxing, Hui Li, Xuesong Tan, and Shuo-Yen Robert Li. "Multicast Switching Fabric Based on Network Coding and Algebraic Switching Theory." IEEE Transactions on Communications 64, no. 7 (July 2016): 2999–3010. http://dx.doi.org/10.1109/tcomm.2016.2577679.

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Couvreur, Alain. "Sums of residues on algebraic surfaces and application to coding theory." Journal of Pure and Applied Algebra 213, no. 12 (December 2009): 2201–23. http://dx.doi.org/10.1016/j.jpaa.2009.03.009.

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Heß, Florian. "Harald Niederreiter and Chaoping Xing: “Algebraic Geometry in Coding Theory and Cryptography”." Jahresbericht der Deutschen Mathematiker-Vereinigung 113, no. 2 (May 3, 2011): 117–19. http://dx.doi.org/10.1365/s13291-011-0019-6.

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Nguemo, Miradain Atontsa, and Calvin Tcheka. "Sheaf cohomology on network codings: maxflow-mincut theorem." Applied General Topology 18, no. 2 (October 2, 2017): 219. http://dx.doi.org/10.4995/agt.2017.3371.

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Surveying briefly a novel algebraic topological application sheaf theory into directed network coding problems, we obtain the weak duality in multiple source scenario by using the idea of modified graph. Furthermore,we establish the maxflow-mincut theorem with network coding sheaves in the case of multiple source.

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Yuanxing, Li, Cheng Jian, and Wang Xinmei. "A joint signature encryption and error correction public-key cryptosystem based on algebraic coding theory." Journal of Electronics (China) 9, no. 1 (January 1992): 33–39. http://dx.doi.org/10.1007/bf02778590.

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Liu, Ce Lun, Kai Zhang, Sen Quan Fan, and Yan Dong Huang. "Research on Synchronous Parallel Coding Algorithm of Medium Length BCH and FPGA Implementation." Advanced Materials Research 846-847 (November 2013): 1084–87. http://dx.doi.org/10.4028/www.scientific.net/amr.846-847.1084.

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The error correction capability of BCH is very strong, especially for the short and medium length of codes, which performance reaches the theory value. Besides, it has a strict algebraic structure, which plays an important role in the encode theory. But traditional serial BCH has a low throughput, so it cannot satisfy high speed communication. This paper designed a parallel synchronization encode and decode algorithm, and used FPGA to simulate. Experiment result shows that the proposed algorithm can satisfy the requirement of code rate and hardware resources consumption, so the algorithm has a very useful practical value.

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HE, MATTHEW, and SERGEY PETOUKHOV. "THE GENETIC CODE, HADAMARD MATRICES AND ALGEBRAIC BIOLOGY." Journal of Biological Systems 18, spec01 (October 2010): 159–75. http://dx.doi.org/10.1142/s0218339010003688.

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Algebraic theory of coding is one of modern fields of applications of algebra. This theory uses matrix algebra intensively. This paper is devoted to an application of Kronecker's matrix forms of presentations of the genetic code for algebraic analysis of a basic scheme of degeneracy of the genetic code. Similar matrix forms are utilized in the theory of signal processing and encoding. The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. This matrix in the third Kronecker power is the (8*8)-matrix, which contains all 64 genetic triplets in a strict order with a natural binary numeration of the triplets by numbers from 0 to 63. Peculiarities of the basic scheme of degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix. This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing and encoding, spectral analysis, quantum mechanics and quantum computers. Furthermore, many kinds of cyclic permutations of genetic elements lead to reconstruction of initial Hadamard matrices into new Hadamard matrices unexpectedly. This demonstrates that matrix algebra is one of promising instruments and of adequate languages in bioinformatics and algebraic biology.

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Muhiuddin, G., D. Al-Kadi, W. A. Khan, and C. Jana. "Hybrid Structures Applied to Subalgebras of BCH-Algebras." Security and Communication Networks 2021 (July 16, 2021): 1–8. http://dx.doi.org/10.1155/2021/8960437.

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The algebraic structures have many applications in coding theory, cryptography, and security networks. In this paper, the notion of hybrid subalgebras of BCH -algebras is introduced and related properties are investigated. Moreover, some characterizations of hybrid subalgebras of BCH -algebras are given. Furthermore, we state and prove some theorems in hybrid subalgebras of BCH -algebras. The homomorphic images and inverse images of fuzzy BCH -subalgebras are studied and discussed.

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LI, ZHE, SHUGONG ZHANG, and TIAN DONG. "FINITE SETS OF AFFINE POINTS WITH UNIQUE ASSOCIATED MONOMIAL ORDER QUOTIENT BASES." Journal of Algebra and Its Applications 11, no. 02 (April 2012): 1250025. http://dx.doi.org/10.1142/s021949881100549x.

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The quotient bases for zero-dimensional ideals are often of interest in the investigation of multivariate polynomial interpolation, algebraic coding theory, and computational molecular biology, etc. In this paper, we discuss the properties of zero-dimensional ideals with unique monomial order quotient bases, and verify that the vanishing ideals of Cartesian sets have unique monomial order quotient bases. Furthermore, we reveal the relation between Cartesian sets and the point sets with unique associated monomial order quotient bases.

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Reyes, Alex D. "Mathematical framework for place coding in the auditory system." PLOS Computational Biology 17, no. 8 (August 2, 2021): e1009251. http://dx.doi.org/10.1371/journal.pcbi.1009251.

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In the auditory system, tonotopy is postulated to be the substrate for a place code, where sound frequency is encoded by the location of the neurons that fire during the stimulus. Though conceptually simple, the computations that allow for the representation of intensity and complex sounds are poorly understood. Here, a mathematical framework is developed in order to define clearly the conditions that support a place code. To accommodate both frequency and intensity information, the neural network is described as a space with elements that represent individual neurons and clusters of neurons. A mapping is then constructed from acoustic space to neural space so that frequency and intensity are encoded, respectively, by the location and size of the clusters. Algebraic operations -addition and multiplication- are derived to elucidate the rules for representing, assembling, and modulating multi-frequency sound in networks. The resulting outcomes of these operations are consistent with network simulations as well as with electrophysiological and psychophysical data. The analyses show how both frequency and intensity can be encoded with a purely place code, without the need for rate or temporal coding schemes. The algebraic operations are used to describe loudness summation and suggest a mechanism for the critical band. The mathematical approach complements experimental and computational approaches and provides a foundation for interpreting data and constructing models.

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Fujita, Hachiro. "Quantum McEliece public-key cryptosystem." Quantum Information and Computation 12, no. 3&4 (March 2012): 181–203. http://dx.doi.org/10.26421/qic12.3-4-1.

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The McEliece cryptosystem is one of the best-known (classical) public-key cryptosystems, which is based on algebraic coding theory. In this paper, we present a quantum analogue of the classical McEliece cryptosystem. Our quantum McEliece public-key cryptosystem is based on the theory of stabilizer codes and has the key generation, encryption and decryption algorithms similar to those in the classical McEliece cryptosystem. We present an explicit construction of the quantum McEliece public-key cryptosystem using Calderbank-Shor-Steane codes based on generalized Reed-Solomon codes. We examine the security of our quantum McEliece cryptosystem and compare it with alternative systems.

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Semerenko, Vasyl, and Oleksandr Voinalovich. "The simplification of computationals in error correction coding." Technology audit and production reserves 3, no. 2(59) (June 30, 2021): 24–28. http://dx.doi.org/10.15587/2706-5448.2021.233656.

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The object of research is the processes of error correction transformation of information in automated systems. The research is aimed at reducing the complexity of decoding cyclic codes by combining modern mathematical models and practical tools. The main prerequisite for the complication of computations in deterministic linear error-correcting codes is the use of the algebraic representation as the main mathematical apparatus for these types of codes. Despite the universalism of the algebraic approach, its main drawback is the impossibility of taking into account the characteristic features of all subclasses of linear codes. In particular, the cyclic property is not taken into account at all for cyclic codes. Taking this property into account, one can go to a fundamentally different mathematical representation of cyclic codes – the theory of linear automata in Galois fields (linear finite-state machine). For the automaton representation of cyclic codes, it is proved that the problem of syndromic decoding of these codes in the general case is an NP-complete problem. However, if to use the proposed hierarchical approach to problems of complexity, then on its basis it is possible to carry out a more accurate analysis of the growth of computational complexity. Correction of single errors during one time interval (one iteration) of decoding has a linear decoding complexity on the length of the codeword, and error correction during m iterations of permutations of codeword bits has a polynomial complexity. According to three subclasses of cyclic codes, depending on the complexity of their decoding: easy decoding (linear complexity), iteratively decoded (polynomial complexity), complicate decoding (exponential complexity). Practical ways to reduce the complexity of computations are considered: alternate use of probabilistic and deterministic linear codes, simplification of software and hardware implementation by increasing the decoding time, use of interleaving. A method of interleaving is proposed, which makes it possible to simultaneously generate the burst errors and replace them with single errors. The mathematical apparatus of linear automata allows solving together the indicated problems of error correction coding.

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Arora, Sanjeev, Yuanzhi Li, Yingyu Liang, Tengyu Ma, and Andrej Risteski. "Linear Algebraic Structure of Word Senses, with Applications to Polysemy." Transactions of the Association for Computational Linguistics 6 (December 2018): 483–95. http://dx.doi.org/10.1162/tacl_a_00034.

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Word embeddings are ubiquitous in NLP and information retrieval, but it is unclear what they represent when the word is polysemous. Here it is shown that multiple word senses reside in linear superposition within the word embedding and simple sparse coding can recover vectors that approximately capture the senses. The success of our approach, which applies to several embedding methods, is mathematically explained using a variant of the random walk on discourses model (Arora et al., 2016). A novel aspect of our technique is that each extracted word sense is accompanied by one of about 2000 “discourse atoms” that gives a succinct description of which other words co-occur with that word sense. Discourse atoms can be of independent interest, and make the method potentially more useful. Empirical tests are used to verify and support the theory.

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Köck, Bernhard, and Joseph Tait. "Faithfulness of Actions on Riemann-Roch Spaces." Canadian Journal of Mathematics 67, no. 4 (August 1, 2015): 848–69. http://dx.doi.org/10.4153/cjm-2014-015-2.

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AbstractGiven a faithful action of a finite groupGon an algebraic curveXof genusgX≥ 2, we giveexplicit criteria for the induced action ofGon the Riemann–Roch spaceH0(X,OX(D)) to be faithful,whereDis aG-invariant divisor on X of degree at least 2gX− 2. This leads to a concise answer to the question of when the action ofGon the spaceH0(X,Ωx⊗m) of global holomorphic polydifferentials of order m is faithful. IfXis hyperelliptic, we provide an explicit basis of H0(X,Ωx⊗m). Finally, we giveapplications in deformation theory and in coding theory and discuss the analogous problem for theaction ofGon the first homologyH1(X,ℤ/mℤ) ifXis a Riemann surface.

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Çalışkan, Basri, and Kemal Balıkçı. "Counting Z2 Z4 Z8-additive Codes." European Journal of Pure and Applied Mathematics 12, no. 2 (April 29, 2019): 668–79. http://dx.doi.org/10.29020/nybg.ejpam.v12i2.3419.

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In Algebraic Coding Theory, all linear codes are described by generator matrices. Any linear code has many generator matrices which are equivalent. It is important to find the number of the generator matrices for constructing of these codes. In this paper, we study Z_2 Z_4 Z_8-additive codes, which are the extension of recently introduced Z_2 Z_4-additive codes. We count the number of arbitrary Z_2 Z_4 Z_8-additive codes. Then we investigate connections to Z_2 Z_4 and Z_2 Z_8-additive codes with Z_2 Z_4 Z_8, and give some illustrative examples.

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Liu, Qian, and Yujuan Sun. "Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3." Journal of Algebra and Its Applications 18, no. 04 (March 25, 2019): 1950069. http://dx.doi.org/10.1142/s0219498819500695.

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Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

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Günlü, Onur, and Rafael Schaefer. "An Optimality Summary: Secret Key Agreement with Physical Unclonable Functions." Entropy 23, no. 1 (December 24, 2020): 16. http://dx.doi.org/10.3390/e23010016.

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We address security and privacy problems for digital devices and biometrics from an information-theoretic optimality perspective to conduct authentication, message encryption/decryption, identification or secure and private computations by using a secret key. A physical unclonable function (PUF) provides local security to digital devices and this review gives the most relevant summary for information theorists, coding theorists, and signal processing community members who are interested in optimal PUF constructions. Low-complexity signal processing methods are applied to simplify information-theoretic analyses. The best trade-offs between the privacy-leakage, secret-key, and storage rates are discussed. Proposed optimal constructions that jointly design the vector quantizer and error-correction code parameters are listed. These constructions include modern and algebraic codes such as polar codes and convolutional codes, both of which can achieve small block-error probabilities at short block lengths, corresponding to a small number of PUF circuits. Open problems in the PUF literature from signal processing, information theory, coding theory, and hardware complexity perspectives and their combinations are listed to stimulate further advancements in the research on local privacy and security.

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Dömösi, Pál, Carolin Hannusch, and Géza Horváth. "A Cryptographic System Based on a New Class of Binary Error-Correcting Codes." Tatra Mountains Mathematical Publications 73, no. 1 (August 1, 2019): 83–96. http://dx.doi.org/10.2478/tmmp-2019-0007.

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Abstract In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. A public-key cryptosystem based on algebraic coding theory, DSN Progress Report. 44, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: Construction of self-dual binary 22k, 22k−1, 2k-codes, Algebra and Discrete Math. 21 (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.

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SIRA-RAMÍREZ, HEBERTT, and MICHEL FLIESS. "AN ALGEBRAIC STATE ESTIMATION APPROACH FOR THE RECOVERY OF CHAOTICALLY ENCRYPTED MESSAGES." International Journal of Bifurcation and Chaos 16, no. 02 (February 2006): 295–309. http://dx.doi.org/10.1142/s0218127406014812.

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In this article, we use a variant of a recently introduced algebraic state estimation method obtained from a fast output signal time derivatives computation process. The fast time derivatives calculations are entirely based on the consequences of using the "algebraic approach" in linear systems description (basically, module theory and non-commutative algebra). Here, we demonstrate, through computer simulations, the effectiveness of the proposed algebraic approach in the accurate and fast (i.e. nonasymptotic) estimation of the chaotic states in some of the most popular chaotic systems. The proposed state estimation method can then be used in a coding–decoding process of a secret message transmission using the message-modulated chaotic system states and the reliable transmission of the chaotic system observable output. Simulation examples, using Chen's chaotic system and the Rossler system, demonstrate the important features of the proposed fast state estimation method in the accurate extraction of a chaotically encrypted messages. In our simulation results, the proposed approach is shown to be quite robust with respect to (computer generated) transmission noise perturbations. We also propose a way to evade computational singularities associated with the local lack of observability of certain chaotic system outputs and still carry out the encrypting and decoding of secret messages in a reliable manner.

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HOSSEINI, A., and A. RAHNAMAI BARGHI. "TABLE ALGEBRAS OF RANK 3 AND ITS APPLICATIONS TO STRONGLY REGULAR GRAPHS." Journal of Algebra and Its Applications 12, no. 05 (May 7, 2013): 1250216. http://dx.doi.org/10.1142/s0219498812502167.

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A table algebra is called quasi self-dual if there exists a permutation on the set of primitive idempotents under which any Krein parameter is equal to its corresponding structure constants. In this paper we investigate the question of when a table algebra of rank 3 is quasi self-dual. As a direct consequence we find necessary and sufficient conditions for the Bose–Mesner algebra of a given strongly regular graph to be quasi self-dual. In fact, our result generalizes the well-known Delsarte's characterization of a self-duality of the Bose–Mesner algebra of a strongly regular graph given in [P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl.10 (1973) 1–97]. Among our results we determine conditions under which the Krein parameters of an integral table algebra of rank 3 are non-negative rational numbers.

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Riznyk, V. V. "FORMALIZATION CODING METHODS OF INFORMATION UNDER TOROIDAL COORDINATE SYSTEMS." Radio Electronics, Computer Science, Control, no. 2 (July 7, 2021): 144–53. http://dx.doi.org/10.15588/1607-3274-2021-2-15.

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Contents. Coding and processing large information content actualizes the problem of formalization of interdependence between information parameters of vector data coding systems on a single mathematical platform. Objective. The formalization of relationships between information parameters of vector data coding systems in the optimized basis of toroidal coordinate systems with the achievement of a favorable compromise between contradictory goals. Method. The method involves the establishing harmonious mutual penetration of symmetry and asymmetry as the remarkable property of real space, which allows use decoded information for forming the mathematical principle relating to the optimal placement of structural elements in spatially or temporally distributed systems, using novel designs based on the concept of Ideal Ring Bundles (IRB)s. IRBs are cyclic sequences of positive integers which dividing a symmetric sphere about center of the symmetry. The sums of connected sub-sequences of an IRB enumerate the set of partitions of a sphere exactly R times. Two-and multidimensional IRBs, namely the “Glory to Ukraine Stars”, are sets of t-dimensional vectors, each of them as well as all modular sums of them enumerate the set node points grid of toroid coordinate system with the corresponding sizes and dimensionality exactly R times. Moreover, we require each indexed vector data “category-attribute” mutually uniquely corresponds to the point with the eponymous set of the coordinate system. Besides, a combination of binary code with vector weight discharges of the database is allowed, and the set of all values of indexed vector data sets are the same that a set of numerical values. The underlying mathematical principle relates to the optimal placement of structural elements in spatially and/or temporally distributed systems, using novel designs based on tdimensional “star” combinatorial configurations, including the appropriate algebraic theory of cyclic groups, number theory, modular arithmetic, and IRB geometric transformations. Results. The relationship of vector code information parameters (capacity, code size, dimensionality, number of encodingvectors) with geometric parameters of the coordinate system (dimension, dimensionality, and grid sizes), and vector data characteristic (number of attributes and number of categories, entity-attribute-value size list) have been formalized. The formula system is derived as a functional dependency between the above parameters, which allows achieving a favorable compromise between the contradictory goals (for example, the performance and reliability of the coding method). Theorem with corresponding corollaries about the maximum vector code size of conversion methods for t-dimensional indexed data sets “category-attribute” proved. Theoretically, the existence of an infinitely large number of minimized basis, which give rise to numerous varieties of multidimensional “star” coordinate systems, which can find practical application in modern and future multidimensional information technologies, substantiated. Conclusions. The formalization provides, essentially, a new conceptual model of information systems for optimal coding and processing of big vector data, using novel design based on the remarkable properties and structural perfection of the “Glory to Ukraine Stars” combinatorial configurations. Moreover, the optimization has been embedded in the underlying combinatorial models. The favorable qualities of the combinatorial structures can be applied to vector data coded design of multidimensional signals, signal compression and reconstruction for communications and radar, and other areas to which the GUS-model can be useful. There are many opportunities to apply them to numerous branches of sciences and advanced systems engineering, including information technologies under the toroidal coordinate systems. A perfection, harmony and beauty exists not only in the abstract models but in the real world also.

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Muhiuddin, G., and Abdulaziz M. Alanazi. "(m, n)-Ideals in Semigroups Based on Int-Soft Sets." Journal of Mathematics 2021 (July 7, 2021): 1–10. http://dx.doi.org/10.1155/2021/5546596.

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Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-soft m , n -ideals, int-soft m , 0 -ideals, and int-soft 0 , n -ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-soft m , n -ideal, int-soft m , 0 -ideal, and int-soft 0 , n -ideal are studied. Also, characterizations of various types of semigroups such as m , n -regular semigroups, m , 0 -regular semigroups, and 0 , n -regular semigroups in terms of their int-soft m , n -ideals, int-soft m , 0 -ideals, and int-soft 0 , n -ideals are provided.

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Nadal, Jean-Pierre, and Nestor Parga. "Redundancy Reduction and Independent Component Analysis: Conditions on Cumulants and Adaptive Approaches." Neural Computation 9, no. 7 (October 1, 1997): 1421–56. http://dx.doi.org/10.1162/neco.1997.9.7.1421.

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In the context of both sensory coding and signal processing, building factorized codes has been shown to be an efficient strategy. In a wide variety of situations, the signal to be processed is a linear mixture of statistically independent sources. Building a factorized code is then equivalent to performing blind source separation. Thanks to the linear structure of the data, this can be done, in the language of signal processing, by finding an appropriate linear filter, or equivalently, in the language of neural modeling, by using a simple feedforward neural network. In this article, we discuss several aspects of the source separation problem. We give simple conditions on the network output that, if satisfied, guarantee that source separation has been obtained. Then we study adaptive approaches, in particular those based on redundancy reduction and maximization of mutual information. We show how the resulting updating rules are related to the BCM theory of synaptic plasticity. Eventually we briefly discuss extensions to the case of nonlinear mixtures. Through out this article, we take care to put into perspective our work with other studies on source separation and redundancy reduction. In particular we review algebraic solutions, pointing out their simplicity but also their drawbacks.

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Barbay, Jérémy. "Optimal Prefix Free Codes with Partial Sorting." Algorithms 13, no. 1 (December 31, 2019): 12. http://dx.doi.org/10.3390/a13010012.

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We describe an algorithm computing an optimal prefix free code for n unsorted positive weights in time within O ( n ( 1 + lg α ) ) ⊆ O ( n lg n ) , where the alternation α ∈ [ 1 . . n − 1 ] approximates the minimal amount of sorting required by the computation. This asymptotical complexity is within a constant factor of the optimal in the algebraic decision tree computational model, in the worst case over all instances of size n and alternation α . Such results refine the state of the art complexity of Θ ( n lg n ) in the worst case over instances of size n in the same computational model, a landmark in compression and coding since 1952. Beside the new analysis technique, such improvement is obtained by combining a new algorithm, inspired by van Leeuwen’s algorithm to compute optimal prefix free codes from sorted weights (known since 1976), with a relatively minor extension of Karp et al.’s deferred data structure to partially sort a multiset accordingly to the queries performed on it (known since 1988). Preliminary experimental results on text compression by words show α to be polynomially smaller than n, which suggests improvements by at most a constant multiplicative factor in the running time for such applications.

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Lipnitskij, V. A., and A. V. Serada. "PROPERTIES OF GROUPS G OF DOUBLE ERRORS AND ITS INVARIANTS IN BCH CODES." «System analysis and applied information science», no. 2 (August 7, 2018): 40–46. http://dx.doi.org/10.21122/2309-4923-2018-2-40-46.

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The goal of the work is the further extending the scope of application of code automorthism in methods and algorithms of error correction by these codes. The effectiveness of such approach was demonstrated by norm of syndrome theory that was developed by Belarusian school of noiseless coding at the turn of the XX and XXI century. The group Г of the cyclical shift of vector component lies at the core of the theory. Under its action The error vectors are divided into disjoint Г-orbits with definite spectrum of syndromes. This allowed to introduce norms of syndrome of a family of BCH codes that are invariant over action of group Г. Norms of syndrome are unique characteristic of error orbit Г of any decoding set, hence it is the basis of permutation norm methods of error decoding. Looking over the Г-orbits of errors not the errors these methods are faster than classic syndrome methods of error decoding, are avoided from the complex process of solving the algebraic equation in Galois field, are simply implemented.A detailed theory for automorphism group G of BCH codes obtained by adding cyclotomic substitution to the group Г develops in the article. The authors held a detailed study of structure of G-orbit of errors as union of orbits Г of error vectors; one-to-one mapping of this structure on the norm structure of group Г. These norms being interconnected by Frobenius automorphism in the Galois field – field of BCH code constitute the complete set of roots of the only irreducible polynomial. It is a polynomial invariant of its orbit G. The main focus of the work is on the description of properties and specific features of groups G of double errors and its polynomial invariants.

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Kang, Kyung Tae, Seok-Zun Song, and Young Bae Jun. "Multipolar Intuitionistic Fuzzy Set with Finite Degree and Its Application in BCK/BCI-Algebras." Mathematics 8, no. 2 (February 2, 2020): 177. http://dx.doi.org/10.3390/math8020177.

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When events occur in everyday life, it is sometimes advantageous to approach them in two directions to find a solution for them. As a mathematical tool to handle these things, we can consider the intuitionistic fuzzy set. However, when events are complex and the key to a solution cannot be easily found, we feel the need to approach them for hours and from various directions. As mathematicians, we wish we had the mathematical tools that apply to these processes. If these mathematical tools were developed, we would be able to apply them to algebra, topology, graph theory, etc., from a close point of view, and we would be able to apply these research results to decision-making and/or coding theory, etc., from a distant point of view. In light of this view, the purpose of this study is to introduce the notion of a multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set), and to apply it to algebraic structure, in particular, a BCK/BCI-algebra. The notions of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra are introduced, and related properties are investigated. Relations between a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal are discussed. Characterizations of a k-polar intuitionistic fuzzy subalgebra/ideal are provided, and conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal are provided. In a BCI-algebra, relations between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal are discussed. A characterization of a closed k-polar intuitionistic fuzzy ideal is considered, and conditions for a k-polar intuitionistic fuzzy ideal to be closed are provided.

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Riznyk, V. V., and D. Yu Skrybaylo-Leskiv. "IMPROVEMENT OF CYCLIC CODES EFFECTIVENESS BY COMBINATORIAL OPTIMIZATION METHODS." Ukrainian Journal of Information Technology 2, no. 1 (2020): 66–72. http://dx.doi.org/10.23939/ujit2020.02.066.

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The methods of improving the cyclic codes efficiency constructed on the basis of combinatorial configurations of the type "ideal ring bundles" (IRB) s by three factors – correction ability, power of coding method and complexity of the decoding procedure are considered. The method is based on the principle of combinatorial optimization, grounded on the algebraic theory of ordered integer sequences with a circular structure, all the numbers, as well as all sums of consecutive numbers exhaust the value sofnatural row numbers. Two theoretically grounded approaches to increase of noise immunity of cyclic codes are offered: implementation of optimized IRB-code, as well as monolithic and group one. Optimized cyclic IRB-code favorably differs from the rest of the codes of this class by the highest correction capacity at the same length of code words. Optimized IRB-codes constitute a large group of cyclic codes designed on a combinatorial models with selection of corresponding relationships between the parameters of the code to achieve its specified technical characteristics. Noise protected monolithic and group codes belong to the group of self-correcting codes with a ring structure and probabilistic assessment of the level of noise protection. This property allow so instant lydetect a particular part or all invalid characters in the code word by the majority principle. Mathematical calculations have been performed to calculate the optimized ratios between the parameters of cyclic IRB-codes, under which they reach maximum correction capacity. The algorithm of constructing and increasing the power of coding methods of optimized noise-resistant IRB-codes is examined and analyzed. The concrete examples of increase efficiency of combinatorial optimization cyclic codes methods with appropriate calculations and tables are given. The comparative analysis of the IRB-codes with the Golay codes and Bose – Chaudhuri – Hocquenghe (BCH) codes with respect to correction ability, power encoding method and computational complexity of decoding procedures is carried out. The advantages and disadvantages of cyclic, and ringmonolithic and group IRB-codes in comparison with classical analogues are determined. The prospect so fusing the research results in the problems of information and communication technologies are outlined.

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Martins Rodrigues, P., and J. Sousa Ramos. "Symbolic Representations and ${\mathbb T}^n$ Automorphisms." International Journal of Bifurcation and Chaos 13, no. 07 (July 2003): 2005–10. http://dx.doi.org/10.1142/s0218127403007849.

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Raigorodskii, A. M. "Combinatorial Geometry and Coding Theory*." Fundamenta Informaticae 145, no. 3 (August 19, 2016): 359–69. http://dx.doi.org/10.3233/fi-2016-1365.

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WADA, MASAAKI. "CODING LINK DIAGRAMS." Journal of Knot Theory and Its Ramifications 02, no. 02 (June 1993): 233–37. http://dx.doi.org/10.1142/s0218216593000143.

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Bouyukliev, Iliya, Stanislav Bulygin, and Edgar Martínez-Moro. "Foreword: computer algebra in coding theory and cryptography." Applicable Algebra in Engineering, Communication and Computing 24, no. 3-4 (June 20, 2013): 157–58. http://dx.doi.org/10.1007/s00200-013-0199-7.

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Laigle-Chapuy, Yann. "Permutation polynomials and applications to coding theory." Finite Fields and Their Applications 13, no. 1 (January 2007): 58–70. http://dx.doi.org/10.1016/j.ffa.2005.08.003.

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Xing, Chaoping. "Special issue on algebra and coding theory." Finite Fields and Their Applications 12, no. 4 (November 2006): 493. http://dx.doi.org/10.1016/j.ffa.2006.09.002.

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Moreno, Carlos J., and Oscar Moreno. "An improved Bombieri-Weil bound and applications to coding theory." Journal of Number Theory 42, no. 1 (September 1992): 32–46. http://dx.doi.org/10.1016/0022-314x(92)90106-y.

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Ding, Cunsheng, and Chaoping Xing. "Special issue on cryptography and coding theory." Journal of Complexity 20, no. 2-3 (April 2004): 133. http://dx.doi.org/10.1016/j.jco.2003.12.002.

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Flaut, Cristina. "Some application of difference equations in Cryptography and Coding Theory." Journal of Difference Equations and Applications 25, no. 7 (May 21, 2019): 905–20. http://dx.doi.org/10.1080/10236198.2019.1619713.

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Puchinger, Sven, and Antonia Wachter-Zeh. "Fast operations on linearized polynomials and their applications in coding theory." Journal of Symbolic Computation 89 (November 2018): 194–215. http://dx.doi.org/10.1016/j.jsc.2017.11.012.

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Boucher, Delphine, and Felix Ulmer. "Coding with skew polynomial rings." Journal of Symbolic Computation 44, no. 12 (December 2009): 1644–56. http://dx.doi.org/10.1016/j.jsc.2007.11.008.

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KENDZIORRA, ANDREAS, and STEFAN E. SCHMIDT. "NETWORK CODING WITH MODULAR LATTICES." Journal of Algebra and Its Applications 10, no. 06 (December 2011): 1319–42. http://dx.doi.org/10.1142/s0219498811005208.

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Kötter and Kschischang presented in 2008 a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this alphabet is the map d : (U, V) ↦ dim (U+V)- dim (U∩V). In this paper we generalize this model to arbitrary modular lattices, i.e. we consider codes, which are subsets of modular lattices. The used metric in this general case is the map d : (u, v) ↦ h(u ∨ v) - h(u ∧ v), where h is the height function of the lattice. We apply this model to submodule lattices. Moreover, we show a method to compute the size of spheres in certain modular lattices and present a sphere packing bound, a sphere covering bound, and a Singleton bound for codes, which are subsets of modular lattices.

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Tzanakis, Nikos, and John Wolfskill. "The diophantine equation x2 = 4qa2 + 4q + 1, with an application to coding theory." Journal of Number Theory 26, no. 1 (May 1987): 96–116. http://dx.doi.org/10.1016/0022-314x(87)90099-0.

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Quistorff, Jörn. "Fundamentals of coding type problems." Journal of Discrete Mathematical Sciences and Cryptography 10, no. 1 (February 2007): 9–39. http://dx.doi.org/10.1080/09720529.2007.10698106.

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Pumplün, Susanne. "Quotients of orders in algebras obtained from skew polynomials with applications to coding theory." Communications in Algebra 46, no. 11 (September 19, 2018): 5053–72. http://dx.doi.org/10.1080/00927872.2018.1461882.

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Journal articles on the topic 'Algebraic Coding Theory'

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