On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

Arrays of strength s on two symbols

This paper gives necessary and sufficient conditions on σ, s, t and on μ, s, t for an array with s+t rows to have strength s and weight σ, or to be balanced and have strength s and weight μ. If a balanced array can exist, the conditions provide a construction. The solutions for t=1,2 are also given in an alternate form useful for the study of trim arrays. The balanced solution for t=1 is more detailed than that known so far, and permits one to determine whether or not a solution exists in possibly fewer steps.     

The lattice of N-run orthogonal arrays

If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the “expansive replacement” construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most ⌊c(N−1)⌋, where c=1.4039…, and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on “mixed spreads”, all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.     

Some new classes of orthogonal Latin hypercube designs

Orthogonal Latin hypercube (OLH) is a good design choice in a polynomial function model for computer experiments, because it ensures uncorrelated estimation of linear effects when a first-order model is fitted. However, when a second-order model is adopted, an OLH also needs to satisfy the additional property that each column is orthogonal to the elementwise square of all columns and orthogonal to the elementwise product of every pair of columns. Such class of OLHs is called OLHs of order two while the former class just possessing two-dimensional orthogonality is called OLHs of order one. In this paper we present a general method for constructing OLHs of orders one and two for n=s^m$n=s^m$ runs, where s and m may be any positive integers greater than one, by rotating a grouped orthogonal array with a column-orthogonal rotation matrix. The Kronecker product and the stacking methods are revisited and combined to construct some new classes of OLHs of orders one and two with other flexible numbers of runs. Some useful OLHs of order one or two with larger factor-to-run ratio and moderate runs are tabulated and discussed.     

Some resolvable orthogonal arrays with two symbols

A method of constructing a resolvable orthogonal array (4λk2,2) which can be partitioned into λ orthogonal arrays (4,k 2,1) is proposed. The number of constraints kfor this type of orthogonal array is at most 3λ. When λ=2 or a multiple of 4, an orthogonal array with the maximum number of constraints of 3λ can be constructed. When λ=4n+2(n≧1) an orthogonal array with 2λ+2 constraints can be constructed. When λ is an odd number, orthogonal arrays can be constructed for λ=3,5,7, and 9 with k=4,8,12, and 13 respectively.     

A Lower Bound to the Measure of Optimality for Main Effect Plans in the Symmetric Factorial Experiments

A design d is called D-optimal if it maximizes det(M _d ), and is called MS-optimal if it maximizes tr(M _d ) and minimizes tr[(M _d )²] among those which maximize tr(M _d ), where M _d stands for the information matrix produced from d under a given model. In this article, we establish a lower bound for tr[(M _d )²] with respect to a main effects model, where d is an s-level symmetric orthogonal array of strength at least one. Non isomorphic two level MS-optimal orthogonal arrays of strength one with N = 10, 14, and 18 runs, non isomorphic three level MS-optimal orthogonal arrays of strength one with N = 6, 12, and 15 runs and non isomorphic four level MS-optimal orthogonal arrays of strength one with N = 12 runs are also presented.     

A note on embedding of orthogonal arrays of strength two

The problem of embedding an orthogonal array of strength 2 into a complete orthogonal array is discussed. It is shown that for any n≠4 any orthogonal array of strength 2 and deficiency 2 can always be embedded into a corresponding complete orthogonal array.     

Construction of orthogonal Latin hypercube designs with flexible run sizes

Latin hypercube designs (LHDs) have recently found wide applications in computer experiments. A number of methods have been proposed to construct LHDs with orthogonality among main-effects. When second-order effects are present, it is desirable that an orthogonal LHD satisfies the property that the sum of elementwise products of any three columns (whether distinct or not) is 0. The orthogonal LHDs constructed by Ye (1998), Cioppa and Lucas (2007), Sun et al. (2009) and Georgiou (2009) all have this property. However, the run size n of any design in the former three references must be a power of two (n=2^c) or a power of two plus one (n=2^c+1), which is a rather severe restriction. In this paper, we construct orthogonal LHDs with more flexible run sizes which also have the property that the sum of elementwise product of any three columns is 0. Further, we compare the proposed designs with some existing orthogonal LHDs, and prove that any orthogonal LHD with this property, including the proposed orthogonal LHD, is optimal in the sense of having the minimum values of ave(|t|), t_max, ave(|q|) and q_max.     

A note on the minimum size of an orthogonal array

It is an elementary fact that the size of an orthogonal array of strength t on k factors must be a multiple of a certain number, say L_t, that depends on the orders of the factors. Thus L_t is a lower bound on the size of arrays of strength t on those factors, and is no larger than L_k, the size of the complete factorial design. We investigate the relationship between the numbers L_t, and two questions in particular: For what t is L_t < L_k? And when L_t = L_k, is the complete factorial design the only array of that size and strength t? Arrays are assumed to be mixed-level.

We refer to an array of size less than L_k as a proper fraction. Guided by our main result, we construct a variety of mixed-level proper fractions of strength k ? 1 that also satisfy a certain group-theoretic condition.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and c_i(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[¹¹¹⁰₀₁₂₁], c₁=[¹₀], c₂=[⁰₂].) Let T_i (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the s^n?r solutions of the equations At=c_i, wheret′=(t₁,…,t_n) is a vector of variables.(Thus, EG(4, 3) consists of the 3⁴=81 points of the form (t₁,t₂,t₃,t₄), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=c_i is s^n?r, where r=Rank(A), and the set of such solutions is said to form an (n?r)-flat, i.e. a flat of (n?r) dimensions. In our example, both T₁ and T₂ are 2-flats consisting of 3^4?2=9 points each. The flats T₁,T₂,…,T_f are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T₁ are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T₂ consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sⁿ symmetric factorial experiment obtained by taking T₁,T₂,…,T_f together. (Thus, in the example, 3⁴=81 treatments of the 3⁴ factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T₁ and T₂ enumerated above.)In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X._j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′._jX._j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s?1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′._jX._j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X._j into the X_ij (i=1,…,f). Furthermore, the X_ij are in turn partitioned into smaller matrices (which we shall here call the) X_ijh. We show that each X_ijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X._j and X′._jX._j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.     

A class of asymmetrical orthogonal resolution-IV designs

The concept of d-resolvability of orthogonal arrays of strength (d+1) is introduced. This is used to construct orthogonal resolution-IV plans of the type $\text{nt·n·2}{}^{\mathrm{n(m?1)}}\text{mn}{}^2\text{t}$. These plans are minimal and a large number of these plans are new.     

Mixed covering arrays of strength three with few factors

Covering arrays with mixed alphabet sizes, or mixed covering arrays, are useful generalizations of covering arrays that are motivated by software and network testing. Suppose that there are k factors, and that the ith factor takes values or levels from a set G_i of size g_i. A run is an assignment of an admissible level to each factor. A mixed covering array, MCA(N;t,k,g₁g₂…g_k), is a collection of N runs such that for any t distinct factors, i₁,i₂,…,i_t, every t-tuple from G_i1×G_i2×?×G_it occurs in factors i₁,i₂,…,i_t in at least one of the N runs. When g=g₁=g₂=?=g_k, an MCA(N;t,k,g₁g₂…g_k) is a CA(N;t,k,g). The mixed covering array number, denoted by MCAN(t,k,g₁g₂…g_k), is the minimum N for which an MCA(N;t,k,g₁g₂…g_k) exists. In this paper, we focus on the constructions of mixed covering arrays of strength three. The numbers MCAN(3,k,g₁g₂…g_k) are determined for all cases with k∈{3,4} and for most cases with k∈{5,6}.     

Tabu search for covering arrays using permutation vectors

A covering array  CA(N;t,k,v)$\mathsf{CA}(N;t,k,v)$ is an N×k$N\times k$ array, in which in every N×t$N\times t$ subarray, each of the v^t$v^t$ possible t  -tuples over v$v$ symbols occurs at least once. The parameter t is the strength   of the array. Covering arrays have a wide range of applications for experimental screening designs, particularly for software interaction testing. A compact representation of certain covering arrays employs “permutation vectors” to encode v^t×1$v^t\times 1$ subarrays of the covering array so that a covering perfect hash family whose entries correspond to permutation vectors yields a covering array. We introduce a method for effective search for covering arrays of this type using tabu search. Using this technique, improved covering arrays of strength 3, 4 and 5 have been found, as well as the first arrays of strength 6 and 7 found by computational search.     

Generalized Bhaskar Rao designs

Generalized Bhaskar Rao designs with non-zero elements from an abelian group G are constructed. In particular this paper shows that the necessary conditions are sufficient for the existence of generalized Bhaskar Rao designs with k=3 for the following groups: ?G? is odd, G=Z^r₂, and G=Z^r₂×H where 3? ?H? and r?1. It also constructs generalized Bhaskar Rao designs with υ=k, which is equivalent to υ rows of a generalized Hadamard matrix of order n where υ?n.     

On the orthogonal designs of order 40

Using a new method we construct all 17 remaining (unresolved for over 20 years) full orthogonal designs of order 40 in three variables. This implies that all full orthogonal designs OD(2^t5;x,y, 2^t5−x−y) exist for all t⩾3. The last two remaining orthogonal designs of order 40 in 2 variables are obtained as a special case of two of these designs.     

Minimal orthogonal main-effect plans and partial replication

This work extends Jacroux's results (Jacroux, Technometrics 34 (1992) 92–96) involving minimal orthogonal main-effect plans (OMEP's). Tables of correct minimal numbers of runs of OMEP's are also provided. In addition, a general method is proposed to construct a class of minimal OMEP's. Moreover, a construction method is provided to obtain partially replicated OMEP's (PROMEP's). Our results provide more duplicate points for some plans than Jacroux's results on PROMEP's (Jacroux Technometrics 35 (1993) 32–36). In addition, a method is derived to construct s₁ × (s₁ − 1)²∥s²₁ minimal PROMEP's with maximal duplicate points.     

Orthogonal arrays and ordered designs

In the first part of this paper, we give a short and direct construction of signed orthogonal array SOA(λ,t,k,v), for any set of parameters λ,t,k,v,t ⩽ k. We also construct a specific basis of the Z-module generated by any SOA(0,t,k,v), for any t,k,v,t ⩽ k. We will then construct an inite family of large set of disjoint ordered designs by applying G.B. Khosrovshahi and S. Ajoodani-Namini's method to Luc. Teirlinck's construction.     

Regular automorphism groups on partial geometries

Recently, Ghinelli (Geom Dedicata (1992) 165–174) had studied the generalized quadrangles which admits automorphism groups acting regularly on the points. In this paper, we generalize her idea to partial geometries, pg(s + 1, t + 1, α). Some examples and basic properties are given. In particular, we prove that under certain conditions on the automorphism group and the lines, such a geometry is a translation net. Applying the results to the case when s = t and the automorphism group G is abelian, we find that either the geometry is a translation net or all the lines of the geometry are generated by a subset of G. Also, for this case, we conjecture that the parameter α is either s or s + 1, except (s, α) = (5, 2), and we have checked that it is true for s ⩽ 500.