Related papers: Uniform semi-Latin squares and their pairwise-vari…

Ever since E. T. Parker constructed an orthogonal pair of $10\times10$ Latin squares in 1959, an orthogonal triple of $10\times10$ Latin squares has been one of the most sought-after combinatorial designs. Despite extensive work, the…

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely…

A \emph{Latin square} is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square $L$ is \emph{row-Hamiltonian} if the permutation induced by each pair of distinct rows of $L$ is a full cycle…

A Latin square of order $n$ is an $n \times n$ matrix of $n$ symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power $q$ let $\mathbb{F}_q$ denote the finite field of order $q$. A quadratic Latin…

Two Latin squares of order $n$ are $r$-orthogonal if, when superimposed, there are exactly $r$ distinct ordered pairs. The spectrum of all values of $r$ for Latin squares of order $n$ is known. A Latin square $A$ of order $n$ is…

A $k$-plane of a $d$-dimensional array is a subarray formed by fixing $d-k$ coordinates and allowing the remaining $k$ coordinates to vary freely. A Latin hypercube of dimension $d$ and order $n$ is an $n\times n\times\cdots\times n$ array…

We show there is a natural connection between Latin squares and commutative sets of monomials defining geometric structures in finite phase-space of prime power dimensions. A complete set of such monomials defines a mutually unbiased basis…

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 investigate the class of \textbf{$\epsilon$-dense partial Latin squares}:\ partial…

It is known that $N(n)$, the maximum number of mutually orthogonal latin squares of order $n$, satisfies the lower bound $N(n) \ge n^{1/14.8}$ for large $n$. For $h\ge 2$, relatively little is known about the quantity $N(h^n)$, which…

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…

Let $L$ be an order-$n$ Latin square. For $X, Y, Z \subseteq \{1, ... ,n\}$, let $L(X, Y. Z)$ be the number of triples $i\in X, j\in Y, k\in Z$ such that $L(i,j) = k$. We conjecture that asymptotically almost every Latin square satisfies…

A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots,h_k$ such that $h_1+\dots+h_k = n$ is known as a realization. The existence of realizations is a partially solved problem with a few general results for an…

We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We…

In this paper we propose a variant of the linear least squares model allowing practitioners to partition the input features into groups of variables that they require to contribute similarly to the final result. The output allows…

We prove that for any $K$ and $d$, there exist, for all sufficiently large admissible $v$, a pairwise balanced design PBD$(v,K)$ of dimension $d$ for which all $d$-point-generated flats are bounded by a constant independent of $v$. We also…

A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The…

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…

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…

Uniform random generation of Latin squares is a classical problem. In this paper we prove that both Latin squares and Sudoku designs are maximum cliques of properly defined graphs. We have developed a simple algorithm for uniform random…

Nested Steiner quadruple systems are designs derived from Steiner quadruple systems (SQSs) by partitioning each block into pairs. A nested SQS is completely uniform if every possible pair appears with equal multiplicity, and completely…