Construction of orthogonal arrays of strength three by augmented difference schemes

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Abstract

Difference schemes are a powerful tool for developing orthogonal arrays (OAs). In this study, we define an augmented difference scheme, i.e., a generalization of a difference scheme, and present a general method for constructing such schemes of strength three. As an application of the proposed method, we construct difference schemes with any number of levels, including D3(d2,c,d) with the maximal value c for any prime power d. Furthermore, using the newly constructed difference schemes, a large number of new OAs of strength three can be infinitely obtained, including many tight arrays. Accordingly, we provide a positive answer to two open problems: how to develop new methods for the construction of difference schemes of high strength with the ultimate goal of obtaining better OAs of high strength, and how to develop better methods and tools for the construction of OAs.

Introduction

Orthogonal arrays (OAs) are essential in statistics. They are used primarily for designing experiments and are therefore vital in all areas of human science, including medicine, agriculture, and manufacturing. Moreover, OAs have potential applications in areas such as quantum information theory [9], [10], [23], cryptography [6], wireless sensor networks [8], [19] and computer science. In Tang's paper [33], the constructed Latin hypercube designs based on OAs have attractive space-filling property and space-filling designs have been appropriate designs for computer experiments [7], [35]. In a big data era, OAs of high strength are applied to high-dimensional data processing in computer experiments [1], [25], [32], [37]. Orthogonal arrays have played and continue to play a prominent role. Because of their powerful applications, many researchers devoted themselves to constructing more OAs. The construction of OAs of strength two has been extensively studied such as [16], [18], [24], [40], [41], [42], however, methods for constructing OAs of high strength have not been studied as extensively as those of strength two. A general construction method for OAs with all factors having prime power levels of strength three and four is proposed in [30], [31]. Based on [30], [31], Zhang et al. in [38], [39] explored the replacement procedure to obtain certain new families of OAs of strength three. Another general construction method for the OAs of high strength by using an orthogonal partition is presented in [20], [21], [22]. Although there have been multiple efforts for constructing OAs of high strength, many unsolved problems persist. Therefore, there is an urgent need for better methods for constructing OAs of high strength.

Difference schemes are a simple but powerful tool for the construction of OAs. The most interesting application for schemes with two levels was proposed by Seiden et al. [27], [28]. Other results, mostly for strength two or three, were proposed in [17]. Wang et al. [34] studied a resolvable generalized difference matrix of strength t. Hedayat et al. [14] investigated the existence of difference schemes of high strength and provided certain difference schemes with levels d4. Moreover, for prime power dt, Chen et al. [4] presented a method for obtaining a difference scheme Dt(dt1,d,d) while t-simple matrices are constructed in [5], [15], which are closely related to difference schemes Dt(dt1,d,d) and Dt(dt1,d+1,d). Although difference schemes of high strength are a natural generalization of ordinary difference schemes, it is more difficult to construct useful schemes of high strength. Therefore, they have to date had limited application because of the accompanying challenging problems [13].

In this study, we define augmented difference schemes, which are a generalization of difference schemes, and present a general method for constructing such schemes of strength three. As an application of the proposed method, we construct difference schemes with any number of levels, including D3(d2,c,d) with the maximal value c for any prime power d. Furthermore, a large number of new OAs of strength three can be obtained using the newly constructed difference schemes, including many tight arrays. Accordingly, we provide a positive answer to two open problems in [13]: (1) how to develop new methods for the construction of difference schemes of high strength with the ultimate goal of obtaining better OAs of high strength and (2) how to develop better methods and tools for the construction of OAs.

Section snippets

Preliminaries

Let AT be the transposition of the matrix A and (d)=(0,1,,d1)T. Let 0r and 1r denote the r×1 vectors of 0s and 1s, respectively. Let A be an abelian group of order d. The Kronecker product AB is defined in [13] and Kronecker sum AB is the Kronecker product with multiplication replaced by addition on A. For convenience, we usually denote all elements of Galois field GF(d) by 0,1,,d1. Let Fdm be a dm×m matrix whose rows are all possible m-tuples over GF(d) and Hr denote a Hadamard matrix of

Main results

Theorem 1

For i=0,1,2, let M(m+1)ni=(CmnCi) be an (m+1)×n matrix over GF(d), where C0=1nT, C1=[1n1T,0], and C2=[1n2T,0,0]. If any three columns of each M(m+1)ni are linearly independent, then FdmCmn is an augmented difference scheme D3i(dm,n,d). In particular, if Cmn=[0m,Cm(n1)], then FdmCm(n1) is an OA(dm,dn1,2).

Proof

It is clear thatFdm+1M(m+1)n0=[0dFdm,(d)0dm](Cmn1nT)=(d)[FdmCmn]. Since any three columns of M(m+1)n0 are linearly independent, Fdm+1M(m+1)n0 is an OA(dm+1,dn,3) [30]. Then FdmCmn is a

Conclusion

In this study, we presented a general method for constructing augmented difference schemes of strength three. As an application of the proposed method, we constructed difference schemes with any number of levels. In particular, we included some optimal difference schemes. Furthermore, new infinite families of OAs of strength three can be obtained. Thus, we provided a positive answer to two open problems in [13]. In addition, a Latin hypercube based on the newly constructed orthogonal array will

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11971004).

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