scholarly journals The automorphisms and error orbits of Reed – Solomon codes

Doklady BGUIR ◽  
2020 ◽  
Vol 18 (4) ◽  
pp. 20-27
Author(s):  
S. I. Semyonov ◽  
V. A. Lipnitsky
Keyword(s):  
Galois Field ◽  
Linear Operators ◽  
Theory And Practice ◽  
Bch Codes ◽  
Practical Applications ◽  
Rs Codes ◽  
Reed Solomon Codes ◽  
Field Of Definition ◽  
The Matrix ◽  
Group A

The purpose of this work with its results presented in the article was to develop and transfer to the class of Reed – Solomon codes (RS-codes) the basic provisions of the theory of syndrome norms (TNS), previously developed for the noise-resistant coding of the class of Bose – Chaudhuri – Hocquenghem codes (BCH-codes), which is actively used in theory and practice. To achieve this goal, a transition has been made in the interpretation of the theory of RS-codes from polynomial to matrix language. This approach allows you to fully use the capabilities of Galois field theory. The main difficulty of RS-codes is that they rely on a non-binary alphabet. The same factor is attractive for practical applications of RS-codes. The matrix language allows you to break the syndromes of errors into components that are elements of the Galois field – the field of definition of RS-codes. The TNS for BCH codes is based on the use of automorphisms of these codes – cyclic and cyclotomic substitutions. Automorphisms of RS-codes are studied in detail. The cyclic substitution belongs to the categories of automorphisms of RS-codes and generates a subgroup Г of order N (code length). The cyclotomic substitution does not belong to the class of automorphisms of RS-codes – the power of the alphabet greater than 2 prevents this. When expanding the concept of automorphism of a code beyond substitutions of coordinates of vectors to automorphisms of RS-codes, homotheties or affine substitutions can be attributed, since they also form a cyclic group A of order N. It is shown that cyclic and affine substitutions commute with each other, which, generally speaking, is not typical for linear operators and substitutions. The group Г of cyclic substitutions, the group A of affine substitutions, and the combined AГ group of order N2 generate 3 types of error orbits in RS-codes. The structure of the orbits of errors with respect to the action of groups A, Г and the combined group AГ is studied {231 words}.

scholarly journals Error correction by Reed–Solomon codes using its automorphisms

2021 ◽  
Vol 66 (1) ◽  
pp. 110-116
Author(s):  
V. A. Lipnitsky ◽  
S. I. Semyonov
Keyword(s):  
Error Correction ◽  
Galois Field ◽  
Correction Method ◽  
New Method ◽  
Group Of Automorphisms ◽  
Rs Codes ◽  
Reed Solomon Codes ◽  
Rs Code ◽  
Error Correction Method ◽  
Joint Group

The article explores the syndrome invariants of АГ-group of automorphisms of Reed–Solomon codes (RS-codes) that are a joint group of affine and cyclic permutations. The found real invariants are a set of norms of N Г-orbits that make up one or another АГ-orbit. The norms of Г-orbits are vectors with 2 1 Cδ− coordinates from the Galois field, that are determined by all kinds of pairs of components of the error syndromes. In this form, the invariants of the АГ-orbits were cumbersome and difficult to use. Therefore, their replacement by conditional partial invariants is proposed. These quasi-invariants are called norm-projections. Norm-projection uniquely identifies its АГ-orbit and therefore serves as an adequate way for formulating the error correction method by RS-codes based on АГ-orbits. The power of the АГ-orbits is estimated by the value of N2, equal to the square of the length of the RS-code. The search for error vectors in transmitted messages by a new method is reduced to parsing the АГ‑orbits, but actually their norm-projections, with the subsequent search for these errors within a particular АГ-orbit. Therefore, the proposed method works almost N2 times faster than traditional syndrome methods, operating on the basic of the “syndrome – error” principle, that boils down to parsing the entire set of error vectors until a specific vector is found.

scholarly journals A New Minimize Matrix Computation Coding Method for Distributed Storage Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Chao Yin ◽  
Haitao Lv ◽  
Tongfang Li ◽  
Xiaoping Qu ◽  
Jianzong Wang ◽  
...  
Keyword(s):  
Storage Systems ◽  
Distributed Storage ◽  
Galois Field ◽  
Storage Space ◽  
Matrix Computation ◽  
Rs Codes ◽  
Failure Tolerance ◽  
Xor Operation ◽  
The Matrix ◽  
Distributed Storage Systems

With the number of nodes increasing in scale, the requirements of storage space enlarge sharply in distributed storage systems. Failure-tolerance schemes such as Reed–Solomon codes (RS codes in short) and Cauchy Reed–Solomon codes (CRS codes in short) are used to save storage space. However, these failure-tolerance schemes severely degrade the system performance. In this paper, we propose optimal RS codes (OptRS codes in short) based on RS codes and CRS codes that can offer better performance for encoding and decoding as well as maximizing the utilization of storage space. OptRS codes can speed up the matrix computation which is regarded as the most important factor to impact the efficiency of coding by transferring the matrix computation from the Galois field mapping to the XOR operation. OptRS codes employ an algorithm called row elimination scheme (RE scheme in short), which can eliminate the same XOR operation to minimize the number of XOR operations. We analyze optimal matrices (OM in short) in theory, which prove the optimal performance of OptRS codes over the Galois field. Our method is implemented on the top of the distributed storage system, and code parameters were carefully chosen. The test result shows that OptRS codes can improve the performance in different data block numbers, parity block numbers, block size, normal reading, and degraded reading, compared with RS codes and CRS codes.

ALCHEMI of B2-Ordered Fe50Al45Me5 alloys

1996 ◽  
Vol 54 ◽  
pp. 548-549
Author(s):  
Ian M. Anderson
Keyword(s):  
Room Temperature ◽  
Iron Aluminide ◽  
Low Density ◽  
Intermetallic Alloys ◽  
Practical Applications ◽  
Alloying Additions ◽  
Site Distribution ◽  
Good Corrosion Resistance ◽  
Room Temperature Ductility ◽  
The Matrix

B2-ordered iron aluminide intermetallic alloys exhibit a combination of attractive properties such as low density and good corrosion resistance. However, the practical applications of these alloys are limited by their poor fracture toughness and low room temperature ductility. One current strategy for overcoming these undesirable properties is to attempt to modify the basic chemistry of the materials with alloying additions. These changes in the chemistry of the material cannot be fully understood without a knowledge of the site-distribution of the alloying elements. In this paper, the site-distributions of a series of 3d-transition metal alloying additions in B2-ordered iron aluminides are studied with ALCHEMI.A series of seven alloys of stoichiometry Fe50AL45Me5, with Me = {Ti, V, Cr, Mn, Co, Ni, Cu}, were prepared with identical heating cycles. Microalloying additions of 0.2% B and 0.1% Zr were also incorporated to strengthen the grain boundaries, but these alloying additions have little influence on the matrix chemistry and are incidental to this study.

Heuristic Algorithms for Waste Load Allocation in a River Basin

1989 ◽  
Vol 21 (8-9) ◽  
pp. 1057-1064 ◽  
Author(s):  
Vijay Joshi ◽  
Prasad Modak
Keyword(s):  
Dynamic Programming ◽  
Heuristic Algorithms ◽  
Water Quality Management ◽  
User Preferences ◽  
Theory And Practice ◽  
Waste Load Allocation ◽  
Practical Applications ◽  
Waste Load ◽  
On Line ◽  
The One

Waste load allocation for rivers has been a topic of growing interest. Dynamic programming based algorithms are particularly attractive in this context and are widely reported in the literature. Codes developed for dynamic programming are however complex, require substantial computer resources and importantly do not allow interactions of the user. Further, there is always resistance to utilizing mathematical programming based algorithms for practical applications. There has been therefore always a gap between theory and practice in systems analysis in water quality management. This paper presents various heuristic algorithms to bridge this gap with supporting comparisons with dynamic programming based algorithms. These heuristics make a good use of the insight gained in the system's behaviour through experience, a process akin to the one adopted by field personnel and therefore can readily be understood by a user familiar with the system. Also they allow user preferences in decision making via on-line interaction. Experience has shown that these heuristics are indeed well founded and compare very favourably with the sophisticated dynamic programming algorithms. Two examples have been included which demonstrate such a success of the heuristic algorithms.

The pragmatist influence on managerial ideas and practices

2018 ◽  
Author(s):  
Philippe Lorino
Keyword(s):  
Community Of Inquiry ◽  
Public Institutions ◽  
Theory And Practice ◽  
Direct Influence ◽  
Practical Applications ◽  
Corporate World ◽  
Quality Movement ◽  
Reflection In Action ◽  
Process Perspective ◽  
Exploratory Inquiry

A key idea of pragmatism is the inseparability of theory and practice, thought and action. Pragmatism is said to have had few contacts with the organizational world, and few direct practical applications, except in the domain of education. In particular, the pragmatist direct influence on the managerial world is often undervalued. However, pragmatist ideas have had a significant impact on managerial doctrines and can be traced in today’s debates amongst organization practitioners. This chapter studies three of those channels: Follett’s direct or indirect (for example through Chester Barnard’s work) influence on the corporate world as well as the management of public institutions; the stream of action research and reflection-in-action, in particular Donald Schön’s work; and the development of the quality movement as an anti-Taylorian revolution, deeply influenced by pragmatist thinkers (exploratory inquiry, community of inquiry, instrumental mediations, process perspective), more recently distorted into a Taylorian revival under the “lean management” label.

Enhanced thermal conductivity of polyimide composite film filled with hybrid fillers

2021 ◽  
pp. 095400832110003
Author(s):  
Ruiyi Li ◽  
Chengcheng Ding ◽  
Juan Yu ◽  
Xiaodong Wang ◽  
Pei Huang
Keyword(s):  
Thermal Conductivity ◽  
Composite Film ◽  
Hexagonal Boron Nitride ◽  
Composite Films ◽  
Resistance Index ◽  
Reference Value ◽  
Residual Rate ◽  
Practical Applications ◽  
Hybrid Fillers ◽  
The Matrix

In this article, the polyimide (PI) composite films with synergistically improving thermal conductivity were prepared by adding a few graphene nanoplatelets (GNP) and various hexagonal boron nitride (h-BN) contents into the PI matrix. The thermal conductivity of PI composite film with 1 wt% GNP and 30 wt% h-BN content was 1.21 W(m·k)− 1, which was higher than that of the PI composite film with 30 wt% h-BN content (0.45 W(m·k)− 1), the synergistic efficiency of GNP under various h-BN content (10 wt%, 20 wt%, and 30 wt%) were 1.70, 2.71, and 3.09, respectively. And it was found that the increased h-BN content can suppress the dielectric properties caused by GNP in the matrix. The dielectric permittivity and dielectric loss tangent of 1 wt% GNP/PI composite film were 10.69, 0.661 at 103 Hz, respectively, and that of the 30 wt% h-BN + GNP/PI composite film were 4.29 and 0.1367, respectively. Moreover, the mechanical properties of the PI composite film were suitable for practical applications. And the heat resistance index and the residual rate at 700°C of PI composite film increased to 326.8°C, 74.43%, respectively, and these of PI film were 292.6°C and 59.26%. Thus, it may provide a reference value for applying the filler hybridization/PI film in the electronic packaging materials.

scholarly journals Self-Access and Metacognitive Awareness in Young Learners: How Japanese Sixth Graders Learn How to Learn English

2015 ◽  
pp. 399-412
Author(s):  
Robert J. Werner ◽  
Yu Kobayashi
Keyword(s):  
Sixth Grade ◽  
Sixth Graders ◽  
Theory And Practice ◽  
Metacognitive Awareness ◽  
Motivating Factors ◽  
Practical Applications ◽  
Young Learners ◽  
English Class ◽  
Watching Tv ◽  
English Instruction

This paper discusses theory and practice related to self-access and metacognitive awareness in young learners. While still an emerging field, the paper presents several studies that describe young learners’ self-access through playing online multi-player digital games, watching TV/films, and reading various types of texts. The teaching of metacognitive awareness, or ‘learning how to learn’, is also discussed, and examples illustrate how this knowledge is applied to learning both in class and beyond. The latter part of this paper describes elementary English instruction in Japan and includes practical applications of learning how to learn through examples from a Japanese sixth grade English class. The students discuss motivating factors beyond the classroom, how they access metacognitive knowledge, and strategies they apply to better learn English.

scholarly journals Areal-typological complexity of Lithuanian dialects

2017 ◽  
pp. 1-15
Author(s):  
Vytautas Kardelis
Keyword(s):  
Empirical Data ◽  
Theory And Practice ◽  
Methodological Principle ◽  
Distinctive Features ◽  
Linguistic System ◽  
Theoretical Criterion ◽  
The Matrix ◽  
System 2 ◽  
Phonetic Features ◽  
Stressed Position

This article deals with one of the most promising stages in Lithuanian dialectology discussed in the article “Seven stages of Lithuanian dialectology” published in the 2016 issue of the scientific electronic journal “Lietuvių kalba” (‘The Lithuanian Language’) (see Kardelis 2016). I referred to this stage as typological; however, now I think that the best term for this stage is the term areal-typological complexity (arealtypologische Komplexität) which originates from works by Alfred Lameli (2013). The concept of complexity is not associated with the attempt to classify dialects according to different “distinctive features” but rather with an idea, clearly supported by empirical facts that diatopical variation in language is highly complex. A closer look at the context of Lithuanian dialectology research reveals that the concept of complexity is still not discussed thoroughly; while specific studies are practically non-existent. The most general methodological principle which should be applied in carrying out an areal-typological study of the complexity of Lithuanian dialects could be referred to as the principle of offside. This means that studies of Lithuanian dialectology should offside from the conventionally applied research tradition and from: a) the aim to classify dialects typologically; b) all classifications of Lithuanian dialects published up to date; c) distinctive features described in the contemporary classification (as well as earlier classifications); d) the goal to specify the boundaries of dialects and subdialects. The second general theoretical criterion is related to the approach to the linguistic system. Here I rely on the concept of a diasystem introduced into the field of dialectology by Uriel Weinreich (1954; 1974). The whole area of the Lithuanian language together with its diatopical variants may be interpreted as a diasystem of Lithuanian which consists of separate systems. The most suitable, convenient and universal criteria for the analysis of empirical data established by the long-standing theory and practice of research into phonology are the following: 1) the quality of the elements of a vocalism system; 2) the quantity of the elements of a vocalism system; 3) the interrelationship between the quality and quantity of a vocalism system. Since here we are dealing with the Lithuanian language which features a complex prosodic system, we must introduce an additional criterion, i.e. 4) stress. Empirical data for the present study were collected from modern, phonological “grammars of dialects”. This article does not encompass the whole diasystem of the Lithuanian language since it only tackles the area covered by the Aukštaičiai dialect. The main phonological qualities according to which the basic vocalism model of the diasystem of the Aukštaičiai dialect can be described are the following: 1) the system of long vowels in the stressed position; 2) the shift in the level of rise of low vowels; 3) the system of short vowels in the stressed position; 4) automatic qualitative shifts; 5) vowel reduction (three degrees). The basic vocalism model described in accordance with the above criteria rather clearly indicates that the great differentiation of Lithuanian dialects postulated in the works on Lithuanian dialectology only has a phonetic and not a phonological basis and it can only be based on the differences of phonetic features. A phonological approach to the diasystem of the Aukštaičiai dialect of Lithuanian does not reveal any radical or extreme differentiation. In addition, the model also shows another significant regularity. The more features are taken into account, the lower the occurrence of individual, less significant dialectal elements distinguished on the basis of one feature (in comparison to the classification by A. Girdenis and Z. Zinkevičius). This approach thus allows solving the complexity of the puzzle of Lithuanian dialects (or, rather the complexity of the diasystem of the Aukštaičiai dialect of Lithuanian) which is summarised in Figure 10. As the matrix in Figure 9 illustrates, the diasystem of the Aukštaičiai dialect of Lithuanian so far consists of three zones represented in the matrix by three different colours.

scholarly journals DIFFERENTIATION OF POLYNOMIALS IN SEVERAL VARIABLES OVER GALOIS FIELDS OF FUZZY CARDINALITY AND APPLICATIONS TO REED-MULLER CODES

2018 ◽  
Vol 18 (3) ◽  
pp. 339-348
Author(s):  
V. M. Deundyak ◽  
N. S. Mogilevskaya
Keyword(s):  
Coding Theory ◽  
Communication Systems ◽  
Communication Channels ◽  
Galois Field ◽  
Second Order ◽  
Galois Fields ◽  
Practical Applications ◽  
Several Variables ◽  
Confidential Data ◽  
Partial Distortion

Introduction. Polynomials in several variables over Galois fields provide the basis for the Reed-Muller coding theory, and are also used  in a number of cryptographic problems. The properties of such polynomials specified over the derived Galois fields of fuzzy cardinality are studied. For the results obtained,  two  real-world  applications  are  proposed: partitioning scheme and Reed-Muller code decoder.Materials and Methods. Using linear algebra, theory of Galois fields, and general theory of polynomials in several variables, we have obtained results related to the differentiation and integration  of polynomials  in  several  variables  over  Galois fields of fuzzy cardinality. An analog of the differentiation operator is constructed and studied for vectors.Research Results. On the basis of the obtained results on the differentiation and integration of polynomials, a new decoder for Reed-Muller codes of the second order is given, and a scheme for organizing the partitioned transfer of confidential data is proposed. This is a communication system in which the source data on the sender is divided into several parts and, independently of one  another,  transmitted  through  different communication channels, and then, on the receiver, the initial data is restored of the parts retrieved. The proposed scheme feature is that it enables to protect data, both from the nonlegitimate access, and from unintentional errors; herewith, one  and  the  same  mathematical  apparatus  is  used  in  both cases. The developed decoder for the second-order Reed-Muller codes prescribed over the derived odd Galois field may have a constraint to the recoverable error level; however, its use is advisable for a number of the communication channels.Discussion    and    Conclusions.    The    proposed    practical applications   of   the   results   obtained   are   useful   for   the organization of reliable communication systems. In future, it is planned  to  study  the  restoration  process  of  the  original polynomial by its derivatives, in case of their partial distortion, and the development of appropriate applications.

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