Related papers: Roman and Vatican Crossover Designs
We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order $n$ is equivalent to a proper edge-coloring of $K_{n,n}$. A transversal corresponds to a multicolored perfect…
This paper is about the construction of augmented row-column designs for unreplicated trials. The method uses the representation of a $k \times t$ equireplicate incomplete-block design with $t$ treatments in $t$ blocks of size $k$, termed…
A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition when a set of triples is a…
The well-known quadrangle criterion states that a latin square is isotopic to the Cayley table of a group if and only if all quadrangles spanned by the same triple of symbols coincide on the fourth symbol. Gowers and Long (2020)…
Two $n \times n$ Latin squares $L_1, L_2$ are said to be orthogonal if, for every ordered pair $(x,y)$ of symbols, there are coordinates $(i,j)$ such that $L_1(i,j) = x$ and $L_2(i,j) = y$. A $k$-MOLS is a sequence of $k$…
Two Latin squares $L=[l(i,j)]$ and $M=[m(i,j)]$, of even order $n$ with entries $\{0,1,2,\ldots,n-1\}$, are said to be nearly orthogonal if the superimposition of $L$ on $M$ yields an $n\times n$ array $A=[(l(i,j),m(i,j))]$ in which each…
A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders…
For every positive integer $n$ greater than $4$ there is a set of Latin squares of order $n$ such that every permutation of the numbers $1,\ldots,n$ appears exactly once as a row, a column, a reverse row or a reverse column of one of the…
A partial Latin square of order $n$ can be represented by a $3$-dimensional chess-board of size $n\times n\times n$ with at most $n^2$ non-attacking rooks. In Latin squares, a subsystem and its most distant mate together have as many rooks…
This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred to as space-filling designs.
In this article, universally optimal multivariate crossover designs are studied. The multiple response crossover design is motivated by a $3 \times 3$ crossover setup, where the effect of $3$ doses of an oral drug are studied on gene…
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 will investigate the class of \leq\epsilon-dense partial latin squares: partial latin…
This article discusses $A$-, $D$- and $E$-optimality results for multivariate crossover designs, where more than one response is measured from every period for each subject. The motivation for these multivariate designs comes from a $3…
The problem of completing a partially specified n by n Latin square is solved by an alternative proof, based on filling the rows (or diagonals) from 1 to n, using an extended form of Hall's marriage theorem.
We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order $n$…
An $n \times n$ partial Latin square $P$ is called $\alpha$-dense if each row and column has at most $\alpha n$ non-empty cells and each symbol occurs at most $\alpha n$ times in $P$. An $n \times n$ array $A$ where each cell contains a…
A latin hypercuboid of order $N$ is an $N\times...\times N\times k$ array filled with symbols from the set $\{0,...,N-1\} $ in such a way that every symbol occurs at most once in every line. If $k=N$, such an array is a latin hypercube. We…
In crossover design experiments, the proportional model, where the carryover effects are proportional to their direct treatment effects, has draw attentions in recent years. We discover that the universally optimal design under the…
A comprehensive review of the literature on crossover design is needed to highlight its evolution, applications, and methodological advancements across various fields. Given its widespread use in clinical trials and other research domains,…
We introduce a condition on arrays in some way maximally distinct from Latin square condition, as well as some other conditions on algebras, graphs and $0,1$-matrices. We show that these are essentially the same structures, generalising a…