Direct Constructions for General Families of Cyclic Mutually Nearly Orthogonal Latin Squares

Two Latin squares and , of even order n with entries , are said to be nearly orthogonal if the superimposition of L on M yields an array in which each ordered pair , and , occurs at least once and the ordered pair occurs exactly twice. In this paper, we present direct constructions for the existence of general families of three cyclic mutually orthogonal Latin squares of orders , , and . The techniques employed are based on the principle of Methods of Differences and so we also establish infinite classes of “quasi‐difference” sets for these orders.     

正交拉丁方组和泛对角线幻方的一种构造法

利用线性取余变换构造素数阶完备正交拉丁方组,给出泛对角线幻方的一种构造法.     

The Non-Existence of Maximal Sets of Four Mutually Orthogonal Latin Squares of Order 8

We establish the non-existence of a maximal set of four mols (mutually orthogonal Latin squares) of order 8 and the non-existence of (8, 5) projective Hjelmslev planes. We present a maximal set of four mols of order 9.     

Latin Squares without Orthogonal Mates

In 1779 Euler proved that for every even n there exists a latin square of order n that has no orthogonal mate, and in 1944 Mann proved that for every n of the form 4k + 1, k ≥ 1, there exists a latin square of order n that has no orthogonal mate. Except for the two smallest cases, n = 3 and n = 7, it is not known whether a latin square of order n = 4k + 3 with no orthogonal mate exists or not. We complete the determination of all n for which there exists a mate-less latin square of order n by proving that, with the exception of n = 3, for all n = 4k + 3 there exists a latin square of order n with no orthogonal mate. We will also show how the methods used in this paper can be applied more generally by deriving several earlier non-orthogonality results.     

The Search for Pseudo Orthogonal Latin Squares of Order Six

We report on the completecomputer search for a strongly regular graph with parameters(36,15,6,6) and chromatic number six. The resultis that no such graph exists.     

用正交对角拉丁方及幻矩与和阵构作mn阶标准二次幻方

当m和n为同奇或同偶的正整数且m,n≠1,2,3,6时,用m和n阶正交对角拉丁方及{0,1,…,mn-1）上的m×n幻矩与和阵,构作了mn阶标准二次幻方．     

Atomic Latin squares of order eleven

A Latin square is pan‐Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i ≠ j. A Latin square is atomic if all of its conjugates are pan‐Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1‐factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan‐Hamiltonian Latin square of order n describes a perfect 1‐factorization of K_n,n, and vice versa. Perfect 1‐factorizations of K_n,n can be constructed from a perfect 1‐factorization of K_n+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn‐square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self‐orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self‐orthogonal Latin squares in the same main class as a given Latin square. © 2003 Wiley Periodicals, Inc.     

Mutually orthogonal equitable Latin rectangles

Let ab=n². We define an equitable Latin rectangle as an a×b matrix on a set of n symbols where each symbol appears either or times in each row of the matrix and either or times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of ka×b mutually orthogonal equitable Latin rectangles as a k– MOELR (a,b;n). When a≠9,18,36, or 100, then we show that the maximum number of k– MOELR (a,b;n)≥3 for all possible values of (a,b).     

A Three‐Factor Product Construction for Mutually Orthogonal Latin Squares

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, “triple products” of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish.     

A Note on the Construction of Orthogonal Latin Hypercube Designs

Latin hypercube designs have been found very useful for designing computer experiments. In recent years, several methods of constructing orthogonal Latin hypercube designs have been proposed in the literature. In this article, we report some more results on the construction of orthogonal Latin hypercubes which result in several new designs.     

Completions of ‐Dense Partial Latin Squares

A classical question in combinatorics is the following: given a partial Latin square P, when can we complete P to a Latin square L? In this paper, we investigate the class of ε‐dense partial Latin squares: partial Latin squares in which each symbol, row, and column contains no more than ‐many nonblank cells. Based on a conjecture of Nash‐Williams, Daykin and Häggkvist conjectured that all ‐dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a novel technique derived from a paper by Jacobson and Matthews on generating random Latin squares, and use this technique to study ε‐dense partial Latin squares that contain no more than filled cells in total. In this paper, we construct completions for all ε‐dense partial Latin squares containing no more than filled cells in total, given that . In particular, we show that all ‐dense partial Latin squares are completable. These results improve prior work by Gustavsson, which required , as well as Chetwynd and Häggkvist, which required , n even and greater than 10⁷.     

Uniform Semi‐Latin Squares and Their Schur‐Optimality

Let n and k be integers, with and . An semi‐Latin square S is an array, whose entries are k‐subsets of an ‐set, the set of symbols of S, such that each symbol of S is in exactly one entry in each row and exactly one entry in each column of S. Semi‐Latin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that an semi‐Latin square S is uniform if there is a constant μ such that any two entries of S, not in the same row or column, intersect in exactly μ symbols (in which case ). We prove that a uniform semi‐Latin square is Schur‐optimal in the class of semi‐Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria. We give a simple construction to make an semi‐Latin square S from a transitive permutation group G of degree n and order , and show how certain properties of S can be determined from permutation group properties of G. If G is 2‐transitive then S is uniform, and this provides us with Schur‐optimal semi‐Latin squares for many values of n and k for which optimal semi‐Latin squares were previously unknown for any optimality criterion. The existence of a uniform semi‐Latin square for all integers is shown to be equivalent to the existence of mutually orthogonal Latin squares (MOLS) of order n. Although there are not even two MOLS of order 6, we construct uniform, and hence Schur‐optimal, semi‐Latin squares for all integers . & 2012 Wiley Periodicals, Inc. J. Combin. Designs 00: 1–13, 2012     

The Existence of Latin Squares without Orthogonal Mates

A latin square is a bachelor square if it does not possess an orthogonal mate; equivalently, it does not have a decomposition into disjoint transversals. We define a latin square to be a confirmed bachelor square if it contains an entry through which there is no transversal. We prove the existence of confirmed bachelor squares for all orders greater than three. This resolves the existence question for bachelor squares.     

一类试验次数为4p2的非对称正交表的构造

Nowadays orthogonal arrays play important roles in statistics,computer science, coding theory and cryptography.The usual difference matrices are essential for the con- struction of many mixed orthogonal arrays.But there are also many orthogonal arrays, especially mixed-level or asymmetrical which can not be obtained by the usual difference matrices.In order to construct these asymmetrical orthogonal arrays,a class of special matrices,so-called generalized difference matrices,were discovered by Zhang(1989,1990, 1993) by the orthogonal decompositions of projective matrices.In this article,an interesting equivalent relationship between the orthogonal arrays and the generalized difference matri- ces is presented.As an application,a family of orthogonal arrays of run sizes 4p~2,such as L_(36)(6~13~42~(10)),are constructed.     

A Family of Asymmetrical Orthogonal Arrays with Run Sizes 4p^2

Nowadays orthogonal arrays play important roles in statistics, computer science, coding theory and cryptography. The usual difference matrices are essential for the construction of many mixed orthogonal arrays. But there are also many orthogonal arrays, especially mixed-level or asymmetrical which can not be obtained by the usual difference matrices. In order to construct these asymmetrical orthogonal arrays, a class of special matrices, so-called generalized difference matrices, were discovered by Zhang（1989, 1990, 1993） by the orthogonal decompositions of projective matrices. In this article, an interesting equivalent relationship between the orthogonal arrays and the generalized difference matrices is presented. As an application, a family of orthogonal arrays of run sizes 4p2, such as L36（6^13^42^10）, are constructed.     

Some New Maximal Sets of Mutually Orthogonal Latin Squares

Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented. The first construction uses maximal partial spreads in PG(3, 4) \ PG(3, 2) with r lines, where r ∈ {6, 7}, to construct transversal-free translation nets of order 16 and degree r + 3 and hence maximal sets of r + 1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8. The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q ⁺ (2m + 1, q), and yields infinite classes of q ^{2n ? 1} ? 1 MAXMOLS(q ²ⁿ ), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.     

Orthogonal arrays obtained by generalized difference matrices with g levels

Nowadays orthogonal arrays play important roles in statistics, computer science, coding theory and cryptography. The usual difference matrices are essential for the construction for many mixed orthogonal arrays. But there are also orthogonal arrays which cannot be obtained by the usual difference matrices, such as mixed orthogonal arrays of run size 60. In order to construct these mixed orthogonal arrays, a class of special so-called generalized difference matrices were discovered by Zhang (1989,1990,1993,2...     

Orthogonal arrays constructed by generalized Kronecker product

In this paper, we propose a new general approach to construct asymmetrical orthogonal arrays, namely generalized Kronecker product. The operation is not usual Kronecker product in the theory of matrices, but it is interesting since the interaction of two columns of asymmetrical orthogonal arrays can be often written out by the generalized Kronecker product. As an application of the method, some new mixed-level orthogonal arrays of run sizes 72 and 96 are constructed.