Let $\mathbb F$ be a finite field of odd order and $a,b\in\mathbb F\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where χ is the extended quadratic character on $\mathbb F$. Let $Q_{a,b}$ be the quasigroup over $\mathbb F$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \geqslant 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\}= \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \operatorname{Aut}(\mathbb F)$. We also characterize $\operatorname{Aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results, we also characterize the minimal subquasigroups of $Q_{a,b}$.

We show that for all $m,k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever $L$ is a Latin square of order $m$ and the Cartesian product $L^{n}$ of $n$ copies of $L$ is $r$-coloured, there is a monochrome Latin subsquare of $L^{n}$, isotopic to $L^{k}$. In particular, for every prime $p$ and for all $k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever the multiplication table $L({\mathbb{Z}_{p}}^{n})$ of the group ${\mathbb{Z}_{p}}^{n}$ is $r$-coloured, there is a monochrome Latin subsquare of order $p^{k}$. On the other hand, we show that for every group $G$ of order $\leq 15$, there is a 2-colouring of $L(G)$ without a nontrivial monochrome Latin subsquare.

Akbari and Alipour [1] conjectured that any Latin array of order n with at least n2/2 symbols contains a transversal. For large n, we confirm this conjecture, and moreover, we show that n399/200 symbols suffice.

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions.

We report the results of a computer enumeration that found that there are 3155 perfect 1-factorisations (P1Fs) of the complete graph $K_{16}$. Of these, 89 have a nontrivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa [‘Perfect 1-factorisations of $K_{16}$ with nontrivial automorphism group’, J. Combin. Math. Combin. Comput. 47 (2003), 97–111]). We also (i) describe a new invariant which distinguishes between the P1Fs of $K_{16}$, (ii) observe that the new P1Fs produce no atomic Latin squares of order 15 and (iii) record P1Fs for a number of large orders that exceed prime powers by one.

It is proved that every non-trivial Latin square has an upper embedding in a non-orientable surface and every Latin square of odd order has an upper embedding in an orientable surface. In the latter case, detailed results about the possible automorphisms and their actions are also obtained.

The sign of a Latin square is −1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. Let LEn and Lon be, respectively, the number of Latin squares of order n with sign +1 and −1. The Alon-Tarsi conjecture asserts that LEn ≠ Lon when n is even. Drisko showed that LEp+1 ≢ Lop+1 (mod p3) for prime p ≥ 3 and asked if similar congruences hold for orders of the form pk + 1, p + 3, or pq + 1. In this article we show that if t ≤ n, then LEn+1 ≢ L0n+1 (mod t3) only if t = n and n is an odd prime, thereby showing that Drisko’s method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to n ≤ 9, discuss asymptotics for Lo/LE, and propose a generalization of the Alon-Tarsi conjecture.

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

For each positive integer n ≥ 2, there is a well-known regular orientable Hamiltonian embedding of Kn, n, and this generates a regular face 2-colourable triangular embedding of Kn, n, n. In the case n ≡ 0 (mod 8), and only in this case, there is a second regular orientable Hamiltonian embedding of Kn, n. This paper presents an analysis of the face 2-colourable triangular embedding of Kn, n, n that results from this. The corresponding Latin squares of side n are determined, together with the full automorphism group of the embedding.

Face 2-colourable triangulations of complete tripartite graphs $K_{n,n,n}$ correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for $n=3,4,5$ and 6, and we summarize the corresponding results for $n=7$. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs $K^*_{n,n}$, and we also give computational results for these in the cases $n=3,4,5$ and 6.

Let $G(o)$ and $G(*)$ be two groups of finite order $n$, and suppose that each of the sets $\{u\in G;\ uo v=u*v$ for all $v\in G\}$ and $\{v\in G;\ uo v=u*v$ for all $u\in G\}$ has $n/2$ elements. Then $G(*)$ can be obtained from $G(o)$ by one of the two general constructions that are discussed in the paper.

The Hall–Paige conjecture deals with conditions under which a finite group $G$ will possess a complete mapping, or equivalently a Latin square based on the Cayley table of $G$ will possess a transversal. Two necessary conditions are known to be: (i) that the Sylow 2-subgroups of $G$ are trivial or non-cyclic, and (ii) that there is some ordering of the elements of $G$ which yields a trivial product. These two conditions are known to be equivalent, but the first direct, elementary proof that (i) implies (ii) is given here.

It is also shown that the Hall–Paige conjecture implies the existence of a duplex in every group table, thereby proving a special case of Rodney's conjecture that every Latin square contains a duplex. A duplex is a ‘double transversal’, that is, a set of $2n$ entries in a Latin square of order $n$ such that each row, column and symbol is represented exactly twice.

In this paper, we provide a generalization of the classical Rao bound for orthogonal arrays, which can be applied to ordered orthogonal arrays and $\left( t,\,m,\,s \right)$-nets. Application of our new bound leads to improvements in many parameter situations to the strongest bounds (i.e., necessary conditions) for existence of these objects.

In an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and $(T,\,M,\,S)$-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the $\text{LP}$ bound is always at least as strong as the generalized Rao bound.

Our main result is showing the asymptotic existence of tight $\text{OMEPs}$. More precisely, for each fixed number $k$ of rows, and with the exception of $\text{OMEPs}$ of the form $2\times 2\times \cdot \cdot \cdot 2\times 2s\,//\,4s\,\,$ with $s$ odd and with more than three rows, there are only a finite number of tight $\text{OMEP}$ parameters for which the tight $\text{OMEP}$ does not exist.

In 1960, Trevor Evans gave a best possible embedding of a partial latin square of order n in a latin square of order t, for any t ≥ 2n. A latin square of order n is equivalent to a 3-cycle system of Kn, n, n, the complete tripartite graph. Here we consider a small embedding of partial 3k-cycle systems of Kn, n, n of a certain type which generalizes Evans' Theorem, and discuss how this relates to the embedding of patterned holes, another recent generalization of Evans' Theorem.

Let n = 4t +- 2, where the integer t ≧ 2. A necessary condition is given for a particular Latin square L of order n to have a complete set of n — 2 mutually orthogonal Latin squares, each orthogonal to L. This condition extends constraints due to Mann concerning the existence of a Latin square orthogonal to a given Latin square.

A general embedding technique for graph designs and block designs is developed, which transforms the embedding problem for partial designs with ƛ > 1 into the embedding problem for partial designs with ƛ = 1. Given an embedding technique for n-element partial block designs with ƛ = 1 into block designs with f(n) elements, the transformation produces a technique which embeds an «-element partial design with ƛ > 1 and block size k into a design with at most /(3k-1ƛn2) elements. For graph designs and block designs with k > 3, a finite embedding method results. For triple systems, a quadratic embedding technique is obtained immediately; the best previous result here was exponential. Finally, for partial triple systems, Mendelsohn triple systems, and directed triple systems, these quadratic embeddings are improved to linear using a colouring technique.

Let A = (aij) be an idempotent latin square of order n, n ≥ 3, in which aii = i, 1 ≤ i ≤ nc. A set S ⊆ N = {1, 2, …, n} is a cover of A if (N × N)\{(i, i):i ∉ S} = {(i, j): i ∊ S, j ∊ N} ∪ {(j, i): i ∊ S, j ∊ N} ∪ {(i, j): aij ∊ S}. A cover S is minimum for A if |S| < |T| for every cover T of A and we write c(A) = |S|. We denote by c(n) the maximum value of c(A) over all idempotent latin squares A of order n and in this paper show that (7n/10)-3.8 ≤ c (n) < n - n1/3 + 1 for all n ≥ 15. The problem of determining c(n) was first raised by J. Schönheim.