Construction of Asymmetric Orthogonal Arrays With High Strength

Orthogonal Arrays (OAs) have been playing important roles in the field of experimental design. It has been known that OAs are closely related to error-correcting codes. Therefore, many OAs can be constructed from error-correcting codes. But these OAs are suitable for only cases that equal interaction effects can be assumed, for example, all two-factor interaction effects. Since these cases are rare in experimental design, we cannot say that OAs from error-correcting codes are practical. In this paper, we define OAs with unequal strength. In terms of our terminology, OAs from error-correcting codes are OAs with equal strength. We show that OAs with unequal strength are closer to practical OAs than OAs with equal strength. And we clarify the relation between OAs with unequal strength and unequal error-correcting codes. Finally, we propose some construction methods of OAs with unequal strength from unequal error-correcting codes.

Asymmetric orthogonal arrays are very useful in theoretical studies and practical applications. However, it is difficult to construct such designs of high strength. In this article, we use the generator matrix method to obtain some asymmetric orthogonal arrays of strength three. Then we combine it with the general (further) replacement procedure to get more general asymmetric orthogonal arrays of high strength.

In this chapter we present some of the most important families of codes and the orthogonal arrays that are derived from them.KeywordsGenerator MatrixOrthogonal ArrayLinear CodeCyclic CodeGenerator PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider two related problems of extracting correlation from a given multipartite mixed quantum state: the first is the distillation of a conference key when the state is shared between a number of legal players and an eavesdropper; the eavesdropper, apart from starting off with this quantum side information, also observes the public communication between the players. The second is the distillation of Greenberger-Horne-Zeilinger (GHZ) states by means of LOCC from the given mixed state. These problem settings extend our previous paper [FS & AW, IEEE Trans. Inf. Theory 68(2):976-988, 2022], and we generalise its results: using a quantum version of the task of communication for omniscience, we derive a novel lower bound on the distillable secret key from any multipartite quantum state by means of a so-called non-interacting communication protocol. Secondly, by making the secret key distillation protocol coherent, we derive novel lower bounds on the distillation rate of GHZ states. Full details in the long version [1].

General techniques for the construction of asymmetrical orthogonal arrays of strength 2 are presented. These are then applied to special cases to obtain new families of such arrays. Among these are saturated main-effect plans based on $s^m$ runs with factors at $s^{\nu_i}$ levels, $i = 0, 1, \ldots, r,$ where $m \geq v_r, v_0 = 1, v_{i-1}$ divides $v_i, i = 1,2,\ldots, r$, and $s$ is a prime power.

The purpose of this paper is to survey the construction of orthogonal arrays of strength two by using difference sets. Some methods for constructing difference setD(2p,2p,p,2), wherep is a prime or a prime power, are given. It is shown that the Kronecker sum of a difference setD(λ1 p,k 1,p,2) and an orthogonal array (λ2 p 2,k 2,p,2) leads to another orthogonal array (λ1λ2 p 3,k 1 k 2+1,p,2). This enables us to construct orthogonal arrays [2p n+1, 1+2(p+p 2+...+p n ,p),2], [4p n+2, 1+2p+4(p 2+p 3+...+p n+1),p,2], and [8p n+3, 1+2p+4p 2+8(p 3+p 4+...+p n+2),p,2] wherep is a prime or a prime power.

Efficient verification of multipartite quantum states is crucial to many applications in quantum information processing. By virtue of Schmidt decomposition and mutually unbiased bases, here we propose a universal protocol to verify arbitrary multipartite pure quantum states using adaptive local projective measurements. Moreover, we establish a universal upper bound on the sample complexity that is independent of the local dimensions. Numerical calculations further indicate that Haar-random pure states can be verified with a constant sample cost, irrespective of the qudit number and local dimensions, even in the adversarial scenario in which the source cannot be trusted. As alternatives, we provide several simpler variants that can achieve similar high efficiencies without using Schmidt decomposition. The simplest variant consists of only two distinct tests.

We consider a partial trace transformation which maps a multipartite quantum state to collection of local density matrices. We call this collection a mean field state. For the Hilbert spaces $(\CC^2)^{\otimes n}$ and $\CC^2\otimes\CC^2\otimes\CC^4$ the necessary and sufficient conditions under which a mean field state is compatible with at least one multipartite pure state are found. Compatibility of mean field states with more general classes of multipartite quantum states is discussed.

Рассмотрены способы ускоренной оценки технического состояния изделий электронной техники на этапе эксплуатации методами теории планирования активного эксперимента. Предложен конкретный способ, сокращающий общую продолжительность испытаний для получения регрессионновременной модели прогнозирования качества изделий электронной техники. Introduction. The increase of requirements for reliability of electronic products results in development of methods, which allow estimating their technical state during the operation stage quickly and with a high confidence level. The development of such methods is possible, when the dependence of the technical state of electronic products on influencing factors and operating time is plotted. The mathematical expression, approximating such dependence, is called a regressiontime model. Therefore, the most efficient method of its construction is the theory of active experiment design. For obtaining a regressiontime model, specially planned tests are carried out. Meanwhile, there are two methods of including time in the model: to conduct each test in the experiment design during the time t to consider the time t as an additional factor along with factors electronic products operating conditions. The purpose of the work is to study the methods of the construction of a regressiontime model in accordance with the total duration of electronic products tests for reducing financial and time costs. Results. When constructing regressiontime models, using the method of the theory of active experiment design, experiment designs of the second order found the most common application in electronic instrument making: an orthogonal central composite design a rotatable central composite design a Box design a Hartley design a BoxBehnken design. These designs were studied in accordance with the duration of products tests. When comparing methods of the construction of the regressiontime model, the number of factors к, determining the numerical value of electronic products quality indicator was taken as equal to: 2 3 4 5 6. It was obtained that including time in the experiment design as an additional factor reduces the general duration of tests with any number of factors к. The experiment design of the orthogonal central composite order when к2 and к4, the Hartley design when к3 and к5, the BoxBehnken design when к6 have the maximum gain in the duration of tests. Conclusion. When constructing the regressiontime model of forecasting the dependence of electronic products quality indicator on conditions and operating time using the method of the theory of active experiment design, it is necessary to consider time as an additional factor in the experiment design. The inclusion of time into the experiment design as an additional factor reduces the general duration of electronic products reliability tests from 10 to 48 depending on the experiment design type and the number of factors, determining the numerical value of the electronic product quality indicator.

In this article, we propose a new general approach to constructing asymmetrical orthogonal arrays, namely the Kronecker sum. It is interesting since a lot of new mixed-level orthogonal arrays can be obtained by this method.

Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions -- such as Pauli's exclusion principle or the monogamy of quantum entanglement -- which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part, we focus on the one-body marginals of multipartite quantum states; in the second part, we study general quantum marginals from the perspective of entropy.

In this chapter, we present some inequalities related to \(2\times 2\) block PPT matrices and positive matrices partitioned in blocks. When these positive block matrices are PPT, some nice results have been obtained, motivated by the Quantum Information Science. Our study follows a natural thought that conclusions drawn under the PPT assumption should be stronger than those drawn under only the usual positivity assumption. Moreover, we believe the new result presented in this work is of interest in its own right and may serve to understand better the intrinsic properties of PPT matrices in quantum sciences and information theory. Note that multipartite quantum states that have a positive partial transpose concerning all bipartitions of the particles can outperform the separable state in linear interferometers.

In order-of-addition experiments, the full order-of-addition designs are often unaffordable due to their large run sizes. The problem of finding efficient fractional OofA designs arises. The order-of-addition orthogonal arrays are a class of optimal fractional order-of-addition designs for the prevalent pair-wise ordering model, under a variety of widely used design criteria. In the literature, the studies on order-of-addition orthogonal arrays focused on strength 2 while the order-of-addition orthogonal arrays of higher strength have not been investigated yet. In this paper, we focus on order-of-addition orthogonal arrays of strength 3. First, the method of constructing order-of-addition orthogonal arrays of strength 3 is proposed. Second, a theoretical result that states that the order-of-addition orthogonal arrays of strength 3 have better balance properties than those of strength 2 is developed. Third, we provide thorough simulation studies which show that the constructed order-of-addition orthogonal arrays of strength 3 have desirable performance for estimating optimal orders of addition.

The theory of combinatorial designs has been used in widely different areas of computation concerned with the design and analysis of both algorithms and hardware. Combinatorial designs capture a subtle balancing property that is inherent in many difficult problems and hence can provide a sophisticated tool for addressing these problems. The role of combinatorial designs in solving many problems that are basic to the field of computing is explored in this paper. Case studies of many applications of designs to computation are given; these constitute a first survey, which provides a representative sample of uses of designs. More importantly, they suggest paradigms in which designs can be used profitably in algorithm design and analysis.

We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1 ≤ k ≤ n. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes.