Existence of conjugate orthogonal diagonal Latin squares

We shall refer to a diagonal Latin square which is orthogonal to its (3, 2, 1)-conjugate and having its (3, 2, 1)-conjugate also a diagonal Latin square as a (3, 2, 1)-conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers v except 2, 3, 6, and possibly 10. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 449–461, 1997     

Some new row-complete Latin squares

Row-complete Latin squares of orders 9, 15, 21 and 27 are given. The square of order 9 is the smallest possible odd order row-complete Latin square.     

Latin squares with pairwise orthogonal conjugates

Let L¹ denote the set of integers n such that there exists an idempotent Latin square of order n with all of its conjugates distinct and pairwise orthogonal. It is known that L¹ contains all sufficiently large integers. That is, there is a smallest integer n_o such that L¹ contains all integers greater than n_o. However, no upper bound for n_o has been given and the term “sufficiently large” is unspecified. The main purpose of this paper is to establish a concrete upper bound for n_o. In particular it is shown that L¹ contain all integers n>5594, with the possible exception of n=6810.     

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.     

Existence of weakly pandiagonal orthogonal Latin squares

A weakly pandiagonal Latin square of order n over the number set {0, 1, . . . , n-1} is a Latin square having the property that the sum of the n numbers in each of 2n diagonals is the same. In this paper, we shall prove that a pair of orthogonal weakly pandiagonal Latin squares of order n exists if and only if n ≡ 0, 1, 3 (mod 4) and n≠3.     

Holey self-orthogonal Latin squares with symmetric orthogonal mates

We improve the existence results for holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOMs) and show that the necessary conditions for the existence of a HSOLSSOM of typeh ⁿ are also sufficient with at most 28 pairs (h, n) of possible exceptions. Research supported in part by NSERC Grant A-5320 for the first author, NSF Grants CCR-9504205 and CCR-9357851 for the second author, and NSFC Grant 19231060-2 for the third author.     

A remark on doubly diagonalized orthogonal Latin squares

A lower and an upper bound for D(n), the maximum number of mutually orthogonal and doubly diagonalized Latin squares of order n, are given.     

Further results on mutually nearly orthogonal Latin squares

Nearly orthogonal Latin squares are useful for conducting experiments eliminating heterogeneity in two directions and using different interventions each at each level. In this paper, some constructions of mutually nearly orthogonal Latin squares are provided. It is proved that there exist 3 MNOLS(2m) if and only if m ≥ 3 and there exist 4 MNOLS(2m) if and only if m ≥ 4 with some possible exceptions.     

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．     

Some results on construction of orthogonal Latin squares by the method of sum composition

A method of sum composition for construction of orthogona Latin squares was introduced by A. Hedayat and E. Seiden [1]. In this paper we exhibit procedures for constructing a pair of orthogonal Latin squares of size p^α + 4 for primes of the form 4m + 1 or p ≡ 1, 2, 4 mod 7. We also show that for any p > 2n and n even one can construct and orthogonal pair of Latin squares of size p^α + n using the method of sum composition. We observe that the restriction xy = 1 used by Hedayat and Seiden is sometimes necessary.     

Mutually orthogonal Latin squares based on general linear groups

Given a finite group G, how many squares are possible in a set of mutually orthogonal Latin squares based on G? This is a question that has been answered for a few classes of groups only, and for no nonsoluble group. For a nonsoluble group G, we know that there exists a pair of orthogonal Latin squares based on G. We can improve on this lower bound when G is one of GL(2, q) or SL(2, q), q a power of 2, q ≠ 2, or is obtained from these groups using quotient group constructions. For nonsoluble groups, that is the extent of our knowledge. We will extend these results by deriving new lower bounds for the number of squares in a set of mutually orthogonal Latin squares based on the group GL(n, q), q a power of 2, q ≠ 2.     

Five mutually orthogonal Latin squares of orders 24 and 40

Let N(n) denote the maximum number of mutually orthogonal Latin squares of order n. It is proved that N(24) and N(40)5.     

Algebraic representation of affine MDS-codes and mutually orthogonal Latin squares

An algebraic representation of affine MDS-codes and of mutually orthogonal Latin squares (MOLS) is given by introducing the term of a partial ternary. The extension respectively lengthening of partial ternaries, MDS-codes and MOLS is discussed.     

The existence of 3 orthogonal partitioned incomplete Latin squares of type tn

In this paper, we show that there exists a set of 3 orthogonal partitioned incomplete Latin squares of type tⁿ for t a positive integer with a small number of possible exceptions for n.     

New quasi-symmetric designs constructed using mutually orthogonal Latin squares and Hadamard matrices

Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 − (66,30,29) and 2 − (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 − (2u ² − u,u ² − u,u ² − u − 1) and 2 − (2u ² + u,u ²,u ² − u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey–Rankin bound.     

Three new constructions of mutually orthogonal latin squares

Let N(n) denote the maximum number of mutually orthogonal Latin squares of order n. It is shown that N(24)≥ 6, N(48) ≥ 7, N(55) ≥ 6. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 218–220, 2000     

Crisscross Latin squares

Let A be a Latin square of order n. Then the jth right diagonal of A is the set of n cells of A: {(i,j+i):i=0,1…,n?1(modn); and the jth left diagonal of A is the set {(i,j?i):i=0,1…,n?1(modn); A diagonal is said to be complete if every element appears in it exactly once. For n = 2m even, we introduce the concept of a crisscross Latin square which is something in between a diagonal Latin square and a Knut Vik design. A crisscross Latin square is a Latin square such that all the jth right diagonals for even j and all the jth left diagonals for odd j are complete. We show that a necessary and sufficient condition for the existence of a crisscross Latin square of order 2m is that m is even.