On orthogonal generalized equitable rectangles

In this note, we give a complete solution of the existence of orthogonal generalized equitable rectangles, which was raised as an open problem in by Stinson (Des Codes Cryptogr 45:347–357, 2007).      

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).     

Existence of centre-complementary magic rectangles

A magic rectangle of size $(m_1,m_2)$ is an $m_1\times m_2$ array consisting of $m_1{m}_2$ consecutive integers in which the sum of each row is a constant and the sum of each column is another (different if $m_1\ne m_2$). It is centre-complementary if the sum of any pair of centrally symmetric positions is constant. As a natural generalization of symmetric magic squares, centre-complementary magic rectangles are instrumental in the construction of 3-dimensional rectangles. In this paper, we focus our attention on the existence of centre-complementary magic rectangles and prove that the necessary conditions for the existence of centre-complementary magic rectangles are also sufficient.     

两两正交拉丁方最大数目的新上界

记D（x）是使得TD（x，n）存在的最小的数．本文给出D（x）的一个上界．     

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.     

Latin trades on three or four rows

Latin trades are closely related to the problem of critical sets in Latin squares. We denote the cardinality of the smallest critical set in any Latin square of order n by scs(n). A consideration of Latin trades which consist of just two columns, two rows, or two elements establishes that scs(n)?n-1. We conjecture that a consideration of Latin trades on four rows may establish that scs(n)?2n-4. We look at various attempts to prove a conjecture of Cavenagh about such trades. The conjecture is proven computationally for values of n less than or equal to 9. In particular, we look at Latin squares based on the group table of Z_n for small n and trades in three consecutive rows of such Latin squares.     

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.     

On the orthogonal Latin squares polytope

Since 1782, when Euler addressed the question of existence of a pair of orthogonal Latin squares (OLS) by stating his famous conjecture, these structures have remained an active area of research. In this paper, we examine the polyhedral aspects of OLS. In particular, we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. Two of these classes are shown to induce facet-defining inequalities of Chvátal rank two. For each such class, we provide a polynomial separation algorithm of the lowest possible complexity.     

Recursive construction of non-cyclic pandiagonal Latin squares

The pandiagonal Latin squares constructed using Hedayat’s method are cyclic. During the last decades several authors have described methods for constructing pandiagonal Latin squares which are semi-cyclic. In this paper we propose a recursive method for constructing non-cyclic pandiagonal Latin squares of any given order n$n$, where n$n$ is a positive composite integer not divisible by 2 or 3. We also investigate the orthogonality properties of the constructed squares and extend our method to construct non-cyclic pandiagonal Sudoku.     

On the number of Sudoku squares

We provide an upper bound on the number of $n^2\times n^2$ Sudoku squares, and explain intuitively why there is reason to believe that the bound is tight up to a multiplicative factor of a much smaller order of magnitude. A similar bound is established for Sudoku squares with rectangular regions.     

Refining invariants for computing autotopism groups of partial Latin rectangles

Prior to using computational tools that find the autotopism group of a partial Latin rectangle (its stabilizer group under row, column and symbol permutations), it is beneficial to find partitions of the rows, columns and symbols that are invariant under autotopisms and are as fine as possible. We look at the lattices formed by these partitions and introduce two invariant refining maps on these lattices. The first map generalizes the strong entry invariant in a previous work. The second map utilizes some bipartite graphs, introduced here, whose structure is determined by pairs of rows (or columns, or symbols). Experimental results indicate that in most cases (ordinarily 99%+), the combined use of these invariants gives the theoretical best partition of the rows, columns and symbols, outperforms the strong entry invariant, which only gives the theoretical best partitions in roughly 80% of the cases.     

An LP-based proof for the non-existence of a pair of orthogonal Latin squares of order 6

This paper presents an alternative proof for the non-existence of orthogonal Latin squares of order 6. Our method is algebraic, rather than enumerative, and applies linear programming in order to obtain appropriate dual vectors. The proof is achievable only after extending previously known results for symmetry elimination.     

Four Mutually Orthogonal Latin Squares of Order 14

The paper gives example of orthogonal array OA(6, 14) obtained from a difference matrix . The construction is equivalent to four mutually orthogonal Latin squares (MOLS) of order 14. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 363–367, 2012     

Multimagic rectangles based on large sets of orthogonal arrays

Large sets of orthogonal arrays (LOAs) have been used to construct resilient functions and zigzag functions by Stinson. In this paper, an application of LOAs in constructing multimagic rectangles is given. Further, some recursive constructions for multimagic rectangles are presented, and some infinite families of bimagic rectangles are obtained.     

A new class of facets for the Latin square polytope

Latin squares of order n have a 1-1 correspondence with the feasible solutions of the 3-index planar assignment problem (3PAP_n). In this paper, we present a new class of facets for the associated polytope, induced by odd-hole inequalities.     

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.