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>

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 Orthogonal Arrays with Parameters $OA(s^3,s^2+s+1,s,2)$ and 3-Dimensional Projective Geometries

10.37236/556 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Kazuaki Ishii
Keyword(s):  
Orthogonal Array ◽  
Prime Power ◽  
Orthogonal Arrays ◽  
3 Dimensional ◽  
Projective Geometries ◽  
Finite Spaces ◽  
Alpha Type

There are many nonisomorphic orthogonal arrays with parameters $OA(s^3,s^2+s+1,s,2)$ although the existence of the arrays yields many restrictions. We denote this by $OA(3,s)$ for simplicity. V. D. Tonchev showed that for even the case of $s=3$, there are at least 68 nonisomorphic orthogonal arrays. The arrays that are constructed by the $n-$dimensional finite spaces have parameters $OA(s^n, (s^n-1)/(s-1),s,2)$. They are called Rao-Hamming type. In this paper we characterize the $OA(3,s)$ of 3-dimensional Rao-Hamming type. We prove several results for a special type of $OA(3,s)$ that satisfies the following condition: For any three rows in the orthogonal array, there exists at least one column, in which the entries of the three rows equal to each other. We call this property $\alpha$-type. We prove the following. (1) An $OA(3,s)$ of $\alpha$-type exists if and only if $s$ is a prime power. (2) $OA(3,s)$s of $\alpha$-type are isomorphic to each other as orthogonal arrays. (3) An $OA(3,s)$ of $\alpha$-type yields $PG(3,s)$. (4) The 3-dimensional Rao-Hamming is an $OA(3,s)$ of $\alpha$-type. (5) A linear $OA(3,s)$ is of $\alpha $-type.

Global Diffusion with Local Dimension: A Study of Iraq War News in South Asia

2004 ◽  
Vol 113 (1) ◽  
pp. 87-97 ◽  
Author(s):  
Akhteruz Zaman
Keyword(s):  
Content Analysis ◽  
South Asia ◽  
Iraq War ◽  
Access To Information ◽  
Local Dimension ◽  
News Production ◽  
War Coverage ◽  
Cultural Perspectives ◽  
War News ◽  
Definition Of

The reporting of war has always been a tough challenge for journalists. Restricted access to information, lack of consensus about journalists' role during wartime and the process of news production make the task daunting. The media's vulnerability is manifested in the pattern of war coverage, which is directly related to a country's historic, social and cultural perspectives. This content analysis of the 2003 Iraq war coverage in the Bangladesh press shows that, despite the publications’ dependence on Western sources for war news, they rejected the Western definition of the war and echoed the opposing version of it harboured by the country's elite.

scholarly journals A geometrical representation theory for orthogonal arrays

1994 ◽  
Vol 49 (2) ◽  
pp. 311-324 ◽  
Author(s):  
David G. Glynn
Keyword(s):  
Representation Theory ◽  
Projective Space ◽  
Orthogonal Array ◽  
Prime Power ◽  
Infinite Order ◽  
Orthogonal Arrays ◽  
Geometrical Representation

Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.

The Spectra for the Conjugate Invariant Subgroups of n2 × 4 Orthogonal Arrays

1980 ◽  
Vol 32 (5) ◽  
pp. 1126-1139 ◽  
Author(s):  
C. C. Lindner ◽  
R. C. Mullin ◽  
D. G. Hoffman
Keyword(s):  
Orthogonal Array ◽  
Orthogonal Arrays

An n2 × k orthogonal array is a pair (P, B) where P = {1, 2, …, n} and B is a collection of k-tuples of elements from P (called rows) such that if i < j ∈ {1, 2, …, k} and x and y are any two elements of P (not necessarily distinct) there is exactly one row in B whose ith coordinate is x and whose jth coordinate is y. We will refer to the ith coordinate of a row r as the ith column of r. The number n is called the order (or size) of the array and k is called the strength.

scholarly journals Bent polynomials over finite fields

1997 ◽  
Vol 56 (3) ◽  
pp. 429-437 ◽  
Author(s):  
Robert S. Coulter ◽  
Rex W. Matthews
Keyword(s):  
Finite Fields ◽  
Linear Functions ◽  
Explicit Method ◽  
Short Discussion ◽  
Large Sets ◽  
Several Variables ◽  
Polynomials Over Finite Fields ◽  
Definition Of ◽  
Orthogonal Systems ◽  
The Relationship

The definition of bent is redefined for any finite field. Our main result is a complete description of the relationship between bent polynomials and perfect non-linear functions over finite fields: we show they are equivalent. This result shows that bent polynomials can also be viewed as the generalisation to several variables of the class of polynomials known as planar polynomials. An explicit method for obtaining large sets of not necessarily distinct maximal orthogonal systems using bent polynomials is given and we end with a short discussion on the existence of bent polynomials over finite fields.

Local dimension and regular points

1992 ◽  
Vol 112 (2) ◽  
pp. 361-368
Author(s):  
Martin T. Barlow ◽  
S. James Taylor
Keyword(s):  
Local Dimension ◽  
Definition Of ◽  
Image Position

The Hausdorff–Besicovitch dimension of a set A ⊆ ℝd, denoted dim (A), relates to the structure of A in the neighbourhood of its thickest point. If A is irregular then one may wish for an index which will describe the size of A in different places. The obvious definition of the local dimension of A near x isand it is not hard to verify that

Sign in / Sign up

Export Citation Format

Share Document