Construction Of Orthogonal Arrays With High Strength And D-optimal Augmented Designs

Orthogonal arrays(OAs)are indispensable in statistical and are mainly used for design of experiments,which means that OAs are important in all fields.Since Rao(1947)introduced the structure and application of OAs,many combinatorial mathematicians and statisticians have devoted themselves to the research of OAs,and obtained many ingenious construction methods.In the experimental design,due to the limitation of cost,we may not use the OAs with large runs.However,along with the deepening of research,OAs of high strength can also be applied in other fields,such as quantum information,computer science and cryptography.It is more challenging to construct OAs of high strength because the orthogonality of multiple columns.So far,existing asymmetric OAs of high strength are scarce,also limiting their applications.Hedayat et al.(1999)proposed Research Problem:develop better methods and tools for the construction of mixed OAs with strength t?3.Therefore,there is a need for the construction of high strength OAs?especially some OAs having factors whose numbers of levels are nonprime powers.Constructing OAs with the least number of runs(tight OAs)and OAs with the maximum numbers of factors is always of high interest.In this thesis,we study two class methods with orthogonal partition and juxtaposition to construct asymmetric OAs of high strength.Not only are the methods straightforward,but also they are useful for constructing OAs of arbitrary strengths,numbers of levels,and various sizes since they do not rely on the difference schemes and finite fields.A large number of new OAs and other families of OAs of high strength are constructed by using these construction methods,including OAs having factors whose numbers of levels are nonprime powers,tight OAs and OAs with the largest possible numbers of factors.Some new constructed OAs in Appendix are tabulated for reader's use.It is well-known that determining the existence of OAs is an important prerequisite for constructing OAs.Their existence,for given values of the parameters,is of both practical and theoretical interest.However,relatively less work on the existence of asymmetric OAs of strength greater than two is available,especially tight OAs.OAs are optimal as a class of fractional factorial designs according to a range of optimality criteria.This makes it tempting to construct fractional factorial designs by adjoining additional runs to an OA when the number of runs available for the experiment is only slightly larger than the number in the OA,hence the D-optimality of OA plus p run designs is studied.In this thesis,combining with the Rao's inequalities,some extended matrices corresponding to OAs of even strength and odd strength are respectively defined.These extended matrices are an important breakthrough to solve the problem.According to the properties of the extended matrix,we not only discuss the necessary conditions for the existence of tight OAs of high strength and illustrate some examples for the nonexistence of tight OAs,but also study the conditions that need to be satisfied when an OA of strength t plus p run design is D-optimal.Specially,when t=2,it gives a positive answer to the question proposed by Bird and Street(2016).