# Orthogonal Arrays; Enumeration and Applications

Demirkale, Fatih (2013). Orthogonal Arrays; Enumeration and Applications PhD Thesis, School of Mathematics and Physics, The University of Queensland.

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Author Demirkale, Fatih Orthogonal Arrays; Enumeration and Applications School of Mathematics and Physics The University of Queensland 2013 PhD Thesis Diane DonovanBenjamin A. BurtonDeborah J. Street 157 157 eng 010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)010499 Statistics not elsewhere classified080201 Analysis of Algorithms and Complexity An orthogonal array OA(N, k, s, t) of strength t is an N X k array with elements from an ordered set of s symbols such that any N X t subarray has each t-tuple appearing as a row λ times.    Orthogonal arrays are used in the design of experiments in statistics and in many elds of combinatorics including coding theory and cryptography. They are also important in software testing and have applications in computer science. Orthogonal arrays are also closely related to some combinatorial structures such as mutually orthogonal Latin squares, nite projective planes and Hadamard matrices.    In this thesis, we apply orthogonal arrays to construct D-optimal discrete choice experiments. We give new constructions for discrete choice experiments in which all attributes have the same number of levels. These constructions use several combinatorial structures, such as orthogonal arrays, balanced incomplete block designs and Hadamard matrices. If we assume that only the main eects of the attributes are to be used to explain the results and that all attribute level combinations are equally attractive, we show that the constructed discrete choice experiments are D-optimal.    Later we describe new algorithms to enumerate all orthogonal arrays with given parameters. Key techniques include eliminating signicant space requirements, using previously-computed information on substructures in an efficient way, and using efficient vertex- invariants for nauty. Computational results show these algorithms to be signicantly faster than the prior state-of-the-art, and we use them to give some new enumeration results.    In this thesis, we also study the enumeration of non-isomorphic three column orthogonal arrays on three symbols, and we study an asymptotic formula for the number of OA(N, 3, 3, 2). Later we identify a set of s(s - 1)2 Latin squares and represent these as vectors. We show these vectors can be used to form intercalates which form a basis for a linear space which contains all Latin trades. We also show that (s - 1)3 + 1 of these vectors are linearly independent. We use this theory to study an asymptotic formula for the number of OA(N, 3, s, 2) for s ≥ 3, and we show that for a given s, the number of OA(N, 3, s, 2) is λ(s-1)3(1-o(1)) as λ →∞.    Lastly we study intersection numbers for two simple 2-fold (3s, s, 3) group divisible designs or equivalently two simple OA(2 • s2, 3, s, 2)s for s ≥ 2. More precisely, we develop constructions which show that there exists two simple 2-fold (3s, s, 3) group divisible designs which intersect in precisely K 2 {0, 1, 2,..., 2s2}\ {2s2 -1, 2s2 -2, 2s2 -3, 2s2 -5}triples for s ≥ 5. There are some exceptions for s = 3, 4.    This thesis contributes to the general body of knowledge on existence, enumeration and applications of orthogonal arrays. Orthogonal arraysDiscrete choice experimentsStated preference experimentsD-optimal designsBlock designsLatin squaresCombinatorial enumerationnautyIntersection problemGroup divisible designsQuasigroups

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