scholarly journals Results on the Projective Plane over a Finite Field of Order Seventeen

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Najm A. M. AL-Seraji ◽  
Hussam. H. Jawad
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Projective Line ◽  
Projective Mapping ◽  
Cubic Curve ◽  
Inflection Points ◽  
Stabilizer Group

The main goal of this research is to find the projective mapping that transforms a geometric formation called an i -set onto an arc such that the domain of the mapping is a subset of the projective line PG (1,q), q=17 , such that a5-set is called a pentad, a6-set is a hexad, a7-set is a heptad, a8-set is an octad, and a9 -set is a nonad, mapped onto a conicY2-XZ. The research also aims to find the stabilizer group of points on a non-singular cubic curve, with or without rational inflection points, on the projective plane over a finite field of order seventeen, and to give some examples.

scholarly journals Partitions on the Projective Plane Over Galois Field of Order 11^m, m=1, 2, 3

2020 ◽  
pp. 204-214
Author(s):  
Sura M.A. Al-subahawi ◽  
Najm Abdulzahra Makhrib Al-seraji
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Group Actions ◽  
Galois Field ◽  
Projective Line ◽  
Cross Ratio ◽  
Arc Length ◽  
The Cross

This research is concerned with the study of the projective plane over a finite field . The main purpose is finding partitions of the projective line PG( ) and the projective plane PG( ) , in addition to embedding PG(1, ) into PG( ) and PG( ) into PG( ). Clearly, the orbits of PG( ) are found, along with the cross-ratio for each orbit. As for PG( ), 13 partitions were found on PG( ) each partition being classified in terms of the degree of its arc, length, its own code, as well as its error correcting. The last main aim is to classify the group actions on PG( ).

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.

Direct Product Decompositions of Elation Groups

1977 ◽  
Vol 20 (2) ◽  
pp. 173-182
Author(s):  
Julia M. Nowlin Brown
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Direct Product ◽  
Collineation Group

Let G be a collineation group of a projective plane π. Let E be the subgroup generated by all elations in G. In the case that π is finite and G fixes no point or line, F. Piper [6; 7] has proved that if G contains certain combinations of perspectivities, then E is isomorphic to for some finite field g.

scholarly journals Construction of New S-Box Using Action of Quotient of the Modular Group for Multimedia Security

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Imran Shahzad ◽  
Qaiser Mushtaq ◽  
Abdul Razaq
Keyword(s):  
Finite Field ◽  
Modular Group ◽  
Fibonacci Sequence ◽  
Projective Line ◽  
Multimedia Security ◽  
New Technique ◽  
Substitution Box ◽  
Nonlinear Component ◽  
Coset Diagrams ◽  
A New Technique

Substitution box (S-box) is a vital nonlinear component for the security of cryptographic schemes. In this paper, a new technique which involves coset diagrams for the action of a quotient of the modular group on the projective line over the finite field is proposed for construction of an S-box. It is constructed by selecting vertices of the coset diagram in a special manner. A useful transformation involving Fibonacci sequence is also used in selecting the vertices of the coset diagram. Finally, all the analyses to examine the security strength are performed. The outcomes of the analyses are encouraging and show that the generated S-box is highly secure.

scholarly journals Topological Realizations of Line Arrangements

2017 ◽  
Vol 2019 (8) ◽  
pp. 2295-2331
Author(s):  
Daniel Ruberman ◽  
Laura Starkston
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Contact Structure ◽  
Complex Projective Plane ◽  
Real Projective Plane ◽  
Incidence Relation ◽  
Combinatorial Configuration ◽  
Important Variation ◽  
Graph Manifolds ◽  
Complex Projective

Abstract A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to $\mathbb R\rm{P}^1$) in the real projective plane. In this article we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres.

scholarly journals The Number of Points on an Elliptic Cubic Curve over a Finite Field

1980 ◽  
Vol 1 (4) ◽  
pp. 327-333 ◽  
Author(s):  
R. De Groote ◽  
J.W.P. Hirschfeld
Keyword(s):  
Finite Field ◽  
Cubic Curve

scholarly journals Counting the number of trigonal curves of genus 5 over finite fields

2020 ◽  
Vol 208 (1) ◽  
pp. 31-48
Author(s):  
Thomas Wennink
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Euler Characteristic ◽  
Finite Fields ◽  
Moduli Space ◽  
Plane Curves ◽  
Sieve Method ◽  
Main Application ◽  
Trigonal Curves

AbstractThe trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

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