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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., 28 pages, 4 figures

A systematic method for constructing anti-magic squares of every even order (>2) was found. This partially answered an open question asked by Gakuho Abe.

Much contrary to popular belief and even a published result, it is seen that orthogonal array plus one run plans are not necessarily optimal, within the relevant class, for general $s_1 \times \dots \times s_m$ factorials. A broad sufficient condition on $s_1,\dots, s_m$ ensuring the optimality of such plans has been worked out. This condition covers, in particular, all symmetric factorials and thus strengthens some previous results.

We study the construction of orthogonal arrays with three levels and index 1 and the existence of orthogonal arrays with three levels and index 2. For strength greater than or equal to 2, we show that orthogonal arrays with three levels and index 1 are unique, and we establish the maximum number of factors for orthogonal arrays with three levels and index 2.

A Poisson-Voronoi tessellation (PVT) is a tiling of the Euclidean plane in which centers of individual tiles constitute a Poisson field and each tile comprises the locations that are closest to a given center with respect to a prescribed norm. Many spatial systems in which rare, randomly distributed centers compete for space should be well approximated by a PVT. Examples that we can handle rigorously include multitype threshold vote automata, in which $\kappa$ different camps compete for voters stationed on the two-dimensional lattice. According to the deterministic, discrete-time update rule, a voter changes affiliation only to that of a unique opposing camp having more than $\theta$ representatives in the voter's neighborhood. We establish a PVT limit for such dynamics started from completely random configurations, as the number of camps becomes large, so that the density of initial "pockets of consensus" tends to 0. Our methods combine nucleation analysis, Poisson approximation, and shape theory.

A latin square is an $n\times n$ array of $n$ symbols in which each symbol appears exactly once in each row and column. Regarding each symbol as a variable and taking the determinant, we get a degree-$n$ polynomial in $n$ variables. Can two latin squares $L,M$ have the same determinant, up to a renaming of the variables, apart from the obvious cases when $L$ is obtained from $M$ by a sequence of row interchanges, column interchanges, renaming of variables, and transposition? The answer was known to be no if $n\le7$; we show that it is yes for $n=8$. The latin squares for which this situation occurs have interesting special characteristics.

We develop a combinatorial condition necessary for the existence of a saturated asymmetrical orthogonal array of strength 2. This condition limits the choice of integral solutions to the system of equations in the Bose-Bush approach and can thus strengthen considerably the Bose-Bush approach as applied to a symmetrical part of such an array. As a consequence, several nonexistence results follow for saturated and nearly saturated orthogonal arrays of strength 2. One of these leads to a partial settlement of an issue left open in a paper by Wu, Zhang and Wang. Nonexistence of a class of saturated asymmetrical orthogonal arrays of strength 4 is briefly discussed.

Randomized orthogonal arrays provide good sets of input points for exploration of computer programs and for Monte Carlo integration. In 1954, Patterson gave a formula for the randomization variance of the sample mean of a function evaluated at the points of an orthogonal array. That formula is incorrect for most of the arrays that are practical for computer experiments. In this paper we correct Patterson's formula. We also remark on a defect, related to coincidences, in some orthogonal arrays. These are arrays of the form $OA(2q^2, 2q + 1, q, 2)$, where $q$ is a prime power, obtained by constructions due to Bose and Bush and to Addelman and Kempthorne. We conjecture that subarrays of the form $OA(2q^2, 2q, q, 2)$ may be constructed to avoid this defect.

Constructive and nonconstructive techniques are employed to enumerate Latin squares and related objects. It is established that there are (i) 2036029552582883134196099 main classes of Latin squares of order 11; (ii) 6108088657705958932053657 isomorphism classes of one-factorizations of $K_{11,11}$; (iii) 12216177315369229261482540 isotopy classes of Latin squares of order 11; (iv) 1478157455158044452849321016 isomorphism classes of loops of order 11; and (v) 19464657391668924966791023043937578299025 isomorphism classes of quasigroups of order 11. The enumeration is constructive for the 1151666641 main classes with an autoparatopy group of order at least 3., Comment: Minor revision; to appear in Mathematics of Computation

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.

For $n>3$, every $n\times n$ partial Cayley matrix with at most $n-1$ holes can be reconstructed by quadrangle criterion. Moreover, the holes can be filled in given order. Without additional assumptions, this is the best possible result. Reconstruction of other types of multiplication tables is discussed., 7 pages

A $k$-plex of a latin square is a collection of cells representing each row, column, and symbol precisely $k$ times. The classic case of $k=1$ is more commonly known as a transversal. We introduce the concept of a $k$-weight, an integral weight function on the cells of a latin square whose row, column, and symbol sums are all $k$. We then show that several non-existence results about $k$-plexes can been seen as more general facts about $k$-weights and that the weight-analogues of several well-known existence conjectures for plexes actually hold for $k$-weights.

Orthogonal arrays with variable numbers of symbols are shown to be universally optimal as fractional factorial designs. The orthogonality of completely regular Youden hyperrectangles ($F$-hyperrectangles) is defined as a generalization of the orthogonality of Latin squares, Latin hypercubes, and $F$-squares. A set of mutually orthogonal $F$-hyperrectangles is seen to be a special kind of orthogonal array with variable numbers of symbols. Theorems on the existence of complete sets of mutually orthogonal $F$-hyperrectangles are established which unify and generalize earlier results on Latin squares, Latin hypercubes, and $F$-squares.

The elementary constructive methods for BBD's and GYD's which have been used in optimality considerations for a number of years, are listed and illustrated. These are not detailed algebraic or geometric prescriptions for listing each entry, but rather methods for combining LS's and known BIBD's to yield the desired products. Nevertheless, there results a large class of useful and previously unpublished designs.

On developpe une methode pour grouper les 2 k -1 effets factoriels dans un plan factoriel a deux niveaux dans des ensembles disjoints mutuellement (s, t, st) ou st est l'interaction generalisee des effets s et t. Comme application on construit des tableaux orthogonaux OA(2 k ,2 m 4 n , 2) de taille 2 k , m contraintes avec 2 niveaux et n contraintes avec 4 niveaux, satisfaisant m+3n=2 k -1 et de puissance 2. Le nombre maximum de contraintes avec 4 niveaux dans la construction ne peut plus etre ameliore. Dans ce sens le schema de groupement qu'on donne est optimal. On discute les avantages de la presente approche par rapport a d'autres methodes de construction

Rectangular lattice designs are shown to be generally balanced with respect to a particular decomposition of the treatment space. Efficiency factors are calculated, and the analysis, including recovery of interblock information, is outlined. The ideas are extended to rectangular lattice designs with an extra blocking factor.

Finney has used orthogonal partitions in the context of the search for higher order (coarser) partitions of given Latin squares. Hedayat and Seiden use the term $F$-square to denote higher order partitions that are orthogonal to both rows and columns. This note is a short expository treatment of orthogonal partitions in general and is based on the identification of a partition with a vector subspace of Euclidean $N$-space $R^N$. This identification is not new as it is part of the usual vector space approach to analysis of variance. This approach puts the concept of orthogonal partitions in a simple light unencumbered by the language of design of experiments. Another advantage is that certain published bounds on the maximum number of orthogonal partitions of specified type are immediate from the dimensionality restriction imposed by $R^N$. In addition, some counting problems are identified which are of possible interest to researchers in design of experiments and combinatorics.

For the minimum weight in $s$-ary $(n, p)$ linear codes (or for the maximum resolution in $s^{n-p}$ designs), an upper bound has been obtained by Plotkin [4] where $s$ is a prime power. The purpose of this note is to obtain an improvement of the Plotkin's upper bound. Main result is as follows: when $p \geqq 2$, the maximum resolution $R_p(n, s)$ of any $s^{n-p}$ design satisfies the following: \begin{equation*}\begin{split}R_p(n, s) = s^{p-1}q\quad\text{if} m = 0, 1 \\ &\leqq s^{p-1}q + \lbrack s^{p-2}(s - 1)(m - 1)/(s^{p-1} - 1)\rbrack \\ \text{if} m = 2, 3, \cdots, s^{p-1} \\ &\leqq s^{p-1}q + \lbrack(s - 1)m/s\rbrack\quad\text{if} m = s^{p-1} + 1, \cdots, N - 1,\end{split}\end{equation*} where $\lbrack x\rbrack$ is the greatest integer not exceeding $x, n = qN + m$ and $N = (s^p - 1)/(s - 1)$.

Balanced arrays were introduced and first studied by I. M. Chakravarti and called partially balanced arrays. J. N. Srivastava and D. V. Chopra have recently made major contributions to the theory and construction of these arrays. They suggested dropping the adjective "partially" and we have followed their lead. Whereas their work has been concerned mainly with arrays of strength four with two symbols, we are here concerned primarily with arrays of strength two with two symbols. However, when the proofs can be carried out as easily for any strength and any number of symbols, we do state the theorems in full generality. We are concerned here with finding bounds on the maximum possible number of rows and with the problem of constructing balanced arrays for given sets of parameters. Analogous to the problem of constructing other combinatorial configurations, we investigated whether some schemes, i.e. some subsets of columns of balanced arrays, could be extended to full balanced arrays. It is shown, analogously to orthogonal arrays, that balanced arrays of even strength, say $2u$ are extendable to arrays of strength $2u + 1$. A new technique of construction of balanced arrays with the maximum number of constraints is also described. It is shown that BIB designs with $\lambda = 1$ can be utilized for constructing balanced arrays with the number of symbols equal to the block size of the BIB design. For completeness we include an example of the analysis of a partially balanced array of strength two with two symbols when it is used as a fractional factorial design. We exhibit in this way an explicit method of estimating main effects when higher ordered interactions are assumed to be negligible.

It is shown that Bush's bound for maximum number of constraints in an orthogonal array of index unity is uniformly better than Rao's bound. In addition it is shown, using an argument similar to that needed in the proof of the above result, that Noda's characterization of parameters in orthogonal arrays of strength 4 achieving equality in Rao's bound, leads easily to a similar characterization in arrays of strength 5. These results are useful designing experiments for quality control.

A theory is developed following work by Williams (1952) and Kiefer (1960) for exact treatment designs in one dimension in which the errors are a stationary process. It is shown that the designs which achieve the minimax value of any of a wide class of functionals on the information matrix for estimation of treatment differences have a special property. If the process is autoregressive of order $p$ then a random piece of the design of length $p + 1$ exhibits uncorrelated treatment values. Such designs can be formed using full length cyclic error-correcting codes of a suitable order. A new technique is developed for classifying the ergodic combinatorial structure of exact designs of arbitrary or infinite length. It is shown that all designs are, to $p$th order, generated by a finite number of sequences with finite length. The classification is given explicitly up to order 3. The method is used to find asymptotically optimum designs for different processes. It is also shown that the designs can be achieved to within an arbitrarily good approximation as the realization of an ergodic Markov chain of sufficiently high order.

Consider the class of balanced fractional $2^m$ factorial designs of resolution VII. Within this class, optimal designs with respect to the trace criterion are given for any fixed $N$ assemblies, which satisfy (i) $m = 6, 42 \leqq N \leqq 64$, (ii) $m = 7, 64 \leqq N \leqq 90$ and (iii) $m = 8, 93 \leqq N \leqq 128$. The covariance matrices of the estimates of the effects are also given for such designs.

The main purpose of the paper is to obtain the best possible bound on the number of orthogonal F-squares with certain parameters and to give a construction method for some families of orthogonal F-squares which achieve this bound. Also, presented is a set of four mutually orthogonal F-squares of order 6 based on three symbols. This later design is important because there is no orthogonal Latin squares of order 6 which could be used for this purpose as has been pointed out by Hedayat and Seiden (1970). The authors indicate a method of composing orthogonal F-squares. It is shown under what condition a set of orthogonal F-squares can be transformed into an orthogonal array, a structure which is useful for factorial experimentation. (Modified author abstract)

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudoku rectangles that is both necessary and sufficient. In the particular case where $n=pq$, $p|r$, $q|s$, the result is especially simple, as we show that any $r \times s$ partial $(p,q)$-Sudoku rectangle can be completed (no further condition being necessary)., 19 pages, 10 figures

A latin bitrade (T1, T2) is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. A genus may be associated to a latin bitrade by constructing an embedding of the underlying graph in an oriented surface. We report computational enumeration results on the number of spherical (genus 0) latin bitrades up to size 24., Comment: Includes source code