scholarly journals Applications of the Projective Plane in Coding Theory

2020 ◽  
Vol 55 (1) ◽  
Author(s):  
Najm Abdulzahra Makhrib Al-Seraji ◽  
Zainab Sadiq Jafar
Keyword(s):  
Projective Plane ◽  
Coding Theory ◽  
Minimum Distance ◽  
Incidence Matrix ◽  
Galois Field ◽  
Programming System ◽  
Combinatorial Structures

The goal of this paper was to study the applications of the projective plane PG (2, q) over a Galois field of order q in the projective linear (n, k, d, q) -code such that the parameters length of code n, the dimension of code k, and the minimum distance d with the error-correcting e according to an incidence matrix have been calculated. Also, this research provides examples and theorems of links between the combinatorial structures and coding theory. The calculations depend on the GAP (groups, algorithms, and programming) system. The method of the research depends on the classification of the points and lines in PG (2, q).

scholarly journals Some Applications of Coding Theory in the Projective Plane of Order Four

2019 ◽  
Vol 30 (1) ◽  
pp. 152
Author(s):  
Najm A. M. AL-Seraji ◽  
Hamza L. M. Ajaj
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Coding Theory ◽  
Minimum Distance ◽  
Incidence Matrix ◽  
Maximum Value ◽  
Order Four ◽  
The Relationship

The main aim of this research is to introduce the relationship between the topic of coding theory and the projective plane of order four. The maximum value of size code M over the finite field of order four and an incidence matrix with the parameters, n (length of code), d (minimum distance of code) and e (error-correcting of code) have been constructed. Some examples and theorems have been given.

scholarly journals A new efficient way based on special stabilizer multiplier permutations to attack the hardness of the minimum weight search problem for large BCH codes

2019 ◽  
Vol 9 (2) ◽  
pp. 1232
Author(s):  
Issam Abderrahman Joundan ◽  
Said Nouh ◽  
Mohamed Azouazi ◽  
Abdelwahed Namir
Keyword(s):  
Coding Theory ◽  
Minimum Distance ◽  
Minimum Weight ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Search Problem ◽  
Public Key Cryptosystems ◽  
True Value ◽  
Efficient Scheme ◽  
Minimum Distances

<span>BCH codes represent an important class of cyclic error-correcting codes; their minimum distances are known only for some cases and remains an open NP-Hard problem in coding theory especially for large lengths. This paper presents an efficient scheme ZSSMP (Zimmermann Special Stabilizer Multiplier Permutation) to find the true value of the minimum distance for many large BCH codes. The proposed method consists in searching a codeword having the minimum weight by Zimmermann algorithm in the sub codes fixed by special stabilizer multiplier permutations. These few sub codes had very small dimensions compared to the dimension of the considered code itself and therefore the search of a codeword of global minimum weight is simplified in terms of run time complexity.  ZSSMP is validated on all BCH codes of length 255 for which it gives the exact value of the minimum distance. For BCH codes of length 511, the proposed technique passes considerably the famous known powerful scheme of Canteaut and Chabaud used to attack the public-key cryptosystems based on codes. ZSSMP is very rapid and allows catching the smallest weight codewords in few seconds. By exploiting the efficiency and the quickness of ZSSMP, the true minimum distances and consequently the error correcting capability of all the set of 165 BCH codes of length up to 1023 are determined except the two cases of the BCH(511,148) and BCH(511,259) codes. The comparison of ZSSMP with other powerful methods proves its quality for attacking the hardness of minimum weight search problem at least for the codes studied in this paper.</span>

scholarly journals A Bound on Permutation Codes

10.37236/2929 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jürgen Bierbrauer ◽  
Klaus Metsch
Keyword(s):  
Projective Plane ◽  
Symmetric Group ◽  
Minimum Distance ◽  
Main Part ◽  
Latin Squares ◽  
Permutation Codes ◽  
Permutation Code ◽  
Mutually Orthogonal Latin Squares ◽  
Hamming Metric ◽  
Completion Theorem

Consider the symmetric group $S_n$ with the Hamming metric. A  permutation code on $n$ symbols is a subset $C\subseteq S_n.$ If $C$ has minimum distance $\geq n-1,$ then $\vert C\vert\leq n^2-n.$ Equality can be reached if and only if a projective plane of order $n$ exists. Call $C$ embeddable if it is contained in a permutation code of minimum distance $n-1$ and cardinality $n^2-n.$ Let $\delta =\delta (C)=n^2-n-\vert C\vert$ be the deficiency of the permutation code $C\subseteq S_n$ of minimum distance $\geq n-1.$We prove that $C$ is embeddable if either $\delta\leq 2$ or if $(\delta^2-1)(\delta +1)^2<27(n+2)/16.$ The main part of the proof is an adaptation of the method used to obtain the famous Bruck completion theorem for mutually orthogonal latin squares.

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.

scholarly journals Cubic Arcs in the Projective Plane Over a Finite Field of Order Twenty Three

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Najm A.M. Al-Seraji ◽  
Asraa A. Monshed
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Coding Theory ◽  
Finite Projective Plane ◽  
Cubic Curves ◽  
The Subject

In this research we are interested in finding all the different cubic curves over a finite projective plane of order twenty-three, learning which of them is complete or not, constructing the stabilizer groups of the cubics in, studying the properties of these groups, and, finally, introducing the relation between the subject of coding theory and the projective plane of order twenty three.

Cohomology of Homotopy 2-types

2019 ◽  
pp. 379-422
Author(s):  
Graham Ellis
Keyword(s):  
Perturbation Theory ◽  
Projective Plane ◽  
Integral Homology ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Crossed Modules ◽  
Homological Perturbation Theory ◽  
Homological Perturbation ◽  
Manual Classification

This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: the fundamental crossed modules of a CW-complex, cat-1-groups, simplicial groups, Moore complexes, the Dold-Kan correspondence, integral homology of simplicial groups, homological perturbation theory. A manual classification of homotopy classes of maps from a surface to the projective plane is also included.

scholarly journals Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Theocharis Theofanidis
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Additional Restriction ◽  
Real Hypersurfaces ◽  
Specific Function ◽  
Complex Projective Plane ◽  
Jacobi Structure ◽  
Complex Projective

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ)have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective planeCP2satisfying a generalization ofϕl=lϕunder an additional restriction on a specific function.

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