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.     

Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n‐queens solutions

In this article, we show how to construct pairs of orthogonal pandiagonal Latin squares and panmagic squares from certain types of modular n‐queens solutions. We prove that when these modular n‐queens solutions are symmetric, the panmagic squares thus constructed will be associative, where for an n × n associative magic square A = (a_ij), for all i and j it holds that a_ij + an?i?1,n?j?1 = c for a fixed c. We further show how to construct orthogonal Latin squares whose modular difference diagonals are Latin from any modular n‐queens solution. As well, we analyze constructing orthogonal pandiagonal Latin squares from particular classes of non‐linear modular n‐queens solutions. These pandiagonal Latin squares are not row cyclic, giving a partial solution to a problem of Hedayat. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 221–234, 2007     

Sub-latin squares and incomplete orthogonal arrays

The main result of this paper is that for any pair of orthogonal Latin squares of side k, there will exist for all sufficiently large n a pair of orthogonal Latin squares with the first pair as orthogonal sub-squares. The orthogonal array corresponding to a set of pairwise orthogonal Latin squares, minus the sub-array corresponding to orthogonal sub-squares is called an incomplete orthogonal array; this concept is generalized slightly.     

An application of sum composition: A self orthogonal latin square of order ten

A latin square is said to be self orthogonal if it is orthogonal to its own transpose. In this note we utilize the sum composition technique, developed by Hedayat and Seiden, to produce a self orthogonal latin square of order ten, the smallest unsettled order in the published literature.     

Triangulations of orientable surfaces by complete tripartite graphs

Orientable triangular embeddings of the complete tripartite graph K_n,n,n correspond to biembeddings of Latin squares. We show that if n is prime there are at least e^{nlnn-n(1+o(1))} nonisomorphic biembeddings of cyclic Latin squares of order n. If n=kp, where p is a large prime number, then the number of nonisomorphic biembeddings of cyclic Latin squares of order n is at least e^{plnp-p(1+lnk+o(1))}. Moreover, we prove that for every n there is a unique regular triangular embedding of K_n,n,n in an orientable surface.     

A generalization of sum composition: Self orthogonal Latin square design with sub self orthogonal Latin square designs

A generalization of the theory of sum composition of Latin square designs is given. Via this generalized theory it is shown that a self orthogonal Latin square design of order $\text{(3p}{}^{\mathrm{\alpha }}\text{? 1)}\text{2}$ with a subself orthogonal Latin square design of order $\text{(p}{}^{\mathrm{\alpha }}\text{? 1)}\text{2}$ can be constructed for any prime p > 2 and any positive integer α as long as p ≠ 3, 5, 7 and 13 if α = 1. Additional results concerning sets of orthogonal Latin square designs are also provided.     

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.     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Bounds on the number of autotopisms and subsquares of a Latin square

A subsquare of a Latin square L is a submatrix that is also a Latin square. An autotopism of L is a triplet of permutations (α, β, γ) such that L is unchanged after the rows are permuted by α, the columns are permuted by β and the symbols are permuted by γ. Let n!(n?1)!R _n be the number of n×n Latin squares. We show that an n×n Latin square has at most n ^{O(log k)} subsquares of order k and admits at most n ^{O(log n)} autotopisms. This enables us to show that {ie11-1} divides R _n for all primes p. We also extend a theorem by McKay and Wanless that gave a factorial divisor of R _n , and give a new proof that R _p ≠1 (mod p) for prime p.     

Generalized Latin squares of order n with n 2 − 1 distinct elements

Let n ≥ 3 be a positive integer. We show that the number of equivalence classes of generalized Latin squares of order n with n ² ? 1 distinct elements is 4 if n = 3 and 5 if n ≥ 4. It is also shown that all these squares are embeddable in groups. As an application, we obtain a lower bound for the number of isomorphism classes of certain Eulerian graphs with n ² + 2n ? 1 vertices.     

Difference Covering Arrays and Pseudo-Orthogonal Latin Squares

A pair of Latin squares, A and B, of order n, is said to be pseudo-orthogonal if each symbol in A is paired with every symbol in B precisely once, except for one symbol with which it is paired twice and one symbol with which it is not paired at all. A set of t Latin squares, of order n, are said to be mutually pseudo-orthogonal if they are pairwise pseudo-orthogonal. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.     

On the existence of MOLS with equal-sized holes

We consider a pair of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning It is shown that a pair of MOLS withn holes of sizeh exist forh 2 if and only ifn 4 For SOLS (self-orthogonal Latin squares) with holes, we have the same result, with two possible exceptions SOLS with 7 or 13 holes of size 6     

On the maximal number of pairwise orthogonal Steiner triple systems

For some time it has been known that for prime powers p^k = 1 + 3 · 2^st there exists a pair of orthogonal Steiner triple systems of order p^k. In fact, such a pair can be constructed using the method of Mullin and Nemeth for constructing strong starters. We use a generalization of the construction of Mullin and Nemeth to construct sets of mutually orthogonal Steiner triple systems for many of these prime powers. By using other techniques we show that a set of mutually orthogonal Steiner triple systems of any given size can be constructed for all but a finite number of such prime powers.     

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.     

Doubly diagonal orthogonal latin squares

A pair of doubly diagonal orthogonal latin squares of order n, DDOLS(n), is a pair of orthogonal latin squares of order n with the property that each square has a transversal on both the front diagonal (the cells {(i, i):1?i?n}) and the back diagonal (the cells {(i, n + 1?i): 1?i?n}). We show that for all n except n = 2, 3, 6, 10, 12, 14, 15, 18 and 26, there exists a pair of DDOLS(n). Obbviously these do not exist when n = 2, 3 and 6.     

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.     

On large sets of disjoint Steiner triple systems II

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that if n is an odd number, there exist 12 mutually orthogonal Latin squares of order n and D(1 + 2n) = 2n ? 1, then D(1 + 12n) = 12n ? 1.     

On Orthogonal Latin <Emphasis Type="Italic">p</Emphasis>-Dimensional Cubes

We give a construction of p orthogonal Latin p-dimensional cubes (or Latin hypercubes) of order n for every natural number n ≠ 2, 6 and p ≥ 2. Our result generalizes the well known result about orthogonal Latin squares published in 1960 by R. C. Bose, S. S. Shikhande and E. T. Parker.     

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.     

On a version of the Hua problem

We prove that almost all natural numbers n satisfying the congruence n ≡ 3 (mod 24), n ? 0 (mod 5), can be expressed as the sum of three squares of primes, at least one of which can be written as 1 + x ² + y ².