scholarly journals Orthogonal arrays obtainable as solutions to linear equations over finite fields

1990 ◽  
Vol 127 ◽  
pp. 283-300 ◽  
Author(s):  
J.N. Srivastava ◽  
Diane Throop
Keyword(s):  
Finite Fields ◽  
Linear Equations ◽  
Orthogonal Arrays

scholarly journals On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1895 ◽  
Author(s):  
M. Higazy ◽  
A. El-Mesady ◽  
M. S. Mohamed
Keyword(s):  
Computer Science ◽  
Finite Fields ◽  
Orthogonal Array ◽  
Orthogonal Arrays ◽  
Latin Squares ◽  
Error Correcting Codes ◽  
Recursive Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Definition Of

During the last two centuries, after the question asked by Euler concerning mutually orthogonal Latin squares (MOLS), essential advances have been made. MOLS are considered as a construction tool for orthogonal arrays. Although Latin squares have numerous helpful properties, for some factual applications these structures are excessively prohibitive. The more general concepts of graph squares and mutually orthogonal graph squares (MOGS) offer more flexibility. MOGS generalize MOLS in an interesting way. As such, the topic is attractive. Orthogonal arrays are essential in statistics and are related to finite fields, geometry, combinatorics and error-correcting codes. Furthermore, they are used in cryptography and computer science. In this paper, our current efforts have concentrated on the definition of the graph-orthogonal arrays and on proving that if there are k MOGS of order n, then there is a graph-orthogonal array, and we denote this array by G-OA(n2,k,n,2). In addition, several new results for the orthogonal arrays obtained from the MOGS are given. Furthermore, we introduce a recursive construction method for constructing the graph-orthogonal arrays.

scholarly journals A removal lemma for linear configurations in subsets of the circle

2013 ◽  
Vol 56 (3) ◽  
pp. 657-666 ◽  
Author(s):  
Pablo Candela ◽  
Olof Sisask
Keyword(s):  
Finite Fields ◽  
Linear Equations ◽  
Circle Group ◽  
Systems Of Linear Equations ◽  
Removal Lemma

AbstractWe obtain a removal lemma for systems of linear equations over the circle group, using a similar result for finite fields due to Král′, Serra and Vena, and we discuss some applications.

scholarly journals The Mathematical Safe Problem and Its Solution (Part 1)

2020 ◽  
pp. 15-38
Author(s):  
S. Kryvyi ◽  
H. Hoherchak
Keyword(s):  
Mathematical Model ◽  
Finite Fields ◽  
Computer Games ◽  
Time Complexity ◽  
Linear Equations ◽  
Irreducible Polynomial ◽  
Finite Rings ◽  
Systems Of Linear Equations ◽  
Residue Fields ◽  
Mathematical Safe

Introduction. The problem of the mathematical safe arises in the theory of computer games and cryptographic applications. The article considers the formulation of the mathematical safe problem and the approach to its solution using systems of linear equations in finite rings and fields. The purpose of the article is to formulate a mathematical model of the mathematical safe problem and its reduction to systems of linear equations in different domains; to consider solving the corresponding systems in finite rings and fields; to consider the principles of constructing extensions of residue fields and solving systems in the relevant areas. Results. The formulation of the mathematical safe problem is given and the way of its reduction to systems of linear equations is considered. Methods and algorithms for solving this type of systems are considered, where exist methods and algorithms for constructing the basis of a set of solutions of linear equations and derivative methods and algorithms for constructing the basis of a set of solutions of systems of linear equations for residue fields, ghost rings, finite rings and finite fields. Examples are given to illustrate their work. The principles of construction of extensions of residue fields by the module of an irreducible polynomial, and examples of operations tables for them are considered. The peculiarities of solving systems of linear equations in such fields are considered separately. All the above algorithms are accompanied by proofs and estimates of their time complexity. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in many variations of its formulation. The second part of the paper will consider the application of these methods and algorithms to solve the problem of mathematical safe in its various variations. Keywords: mathematical safe, finite rings, finite fields, method, algorithm, solution.

scholarly journals On Coset Coverings of Solutions of Homogeneous Cubic Equations over Finite Fields

10.37236/1566 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Ara Aleksanyan ◽  
Mihran Papikian
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Linear Equations ◽  
Initial Equation ◽  
Arbitrary Size ◽  
Cubic Equation ◽  
Systems Of Linear Equations

Given a cubic equation $x_1y_1z_1+x_2y_2z_2+\cdots +x_ny_nz_n=b$ over a finite field, it is necessary to determine the minimal number of systems of linear equations over the same field such that the union of their solutions exactly coincides with the set of solutions of the initial equation. The problem is solved for arbitrary size of the field. A covering with almost minimum complexity is constructed.

scholarly journals Constructions of irredundant orthogonal arrays

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guangzhou Chen ◽  
Xiaotong Zhang
Keyword(s):  
Finite Fields ◽  
Orthogonal Array ◽  
Orthogonal Arrays ◽  
Physical Review ◽  
Local Dimension ◽  
Difference Matrices ◽  
Uniform State ◽  
Definition Of

<p style='text-indent:20px;'>An <inline-formula><tex-math id="M1">\begin{document}$ N \times k $\end{document}</tex-math></inline-formula> array <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> with entries from <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>-set <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> is said to be an <i>orthogonal array</i> with <inline-formula><tex-math id="M5">\begin{document}$ v $\end{document}</tex-math></inline-formula> levels, strength <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula> and index <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, denoted by OA<inline-formula><tex-math id="M8">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if every <inline-formula><tex-math id="M9">\begin{document}$ N\times t $\end{document}</tex-math></inline-formula> sub-array of <inline-formula><tex-math id="M10">\begin{document}$ A $\end{document}</tex-math></inline-formula> contains each <inline-formula><tex-math id="M11">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple based on <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> exactly <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> times as a row. An OA<inline-formula><tex-math id="M14">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> is called <i>irredundant</i>, denoted by IrOA<inline-formula><tex-math id="M15">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if in any <inline-formula><tex-math id="M16">\begin{document}$ N\times (k-t ) $\end{document}</tex-math></inline-formula> sub-array, all of its rows are different. Goyeneche and <inline-formula><tex-math id="M17">\begin{document}$ \dot{Z} $\end{document}</tex-math></inline-formula>yczkowski firstly introduced the definition of an IrOA and showed that an IrOA<inline-formula><tex-math id="M18">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> corresponds to a <inline-formula><tex-math id="M19">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform state of <inline-formula><tex-math id="M20">\begin{document}$ k $\end{document}</tex-math></inline-formula> subsystems with local dimension <inline-formula><tex-math id="M21">\begin{document}$ v $\end{document}</tex-math></inline-formula> (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of <inline-formula><tex-math id="M22">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform states arise from these irredundant orthogonal arrays.</p>

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