Related papers: Enumeration of Sets of Mutually Orthogonal Latin R…

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$. Here we will present a census of such triples, classified by various properties, including the order…

We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\le9$ we 1. Determine the number of orthogonal mates for each species of latin square of order $n$. 2. Calculate the…

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 $k$…

We investigate MacNeish's conjecture (known to be false in general) in the setting of what we call "transitive" Mutually Orthogonal Latin Squares (MOLS). When we restrict our attention to "simply transitive" MOLS, we find that the…

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…

We show that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. We also show that for any $t\geq 2$, a pair of orthogonal partial Latin squares…

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter $d$ over an alphabet of $q$ elements…

The current paper deals with the enumeration and classification of the set $\mathcal{SOR}_{r,n}$ of self-orthogonal $r\times r$ partial Latin rectangles based on $n$ symbols. These combinatorial objects are identified with the independent…

Computing the autotopism group of a partial Latin rectangle can be performed in a variety of ways. This pilot study has two aims: (a) to compare these methods experimentally, and (b) to identify the design goals one should have in mind for…

An arrangement of s elements in s rows and s columns, such that no element repeats more than once in each row and each column is called a Latin square of order s. If two Latin squares of the same order superimposed one on the other and in…

Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin…

This paper deals with distinct computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this…

We will see that every finite projective plane of order k > 1 gives rise to a complete set of (k-1) MPLS (= mutually projective latin squares) of order k and by reversing the process we can construct a finite projective plane of order k…

Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the…

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols.…

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $m\ge2$, a set of $m+1$ partitions of a set $\Omega$, any $m$ of which are the minimal non-trivial elements of a Cartesian lattice, either form…

A latin square of order $n$ is an $n\times n$ array of $n$ symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of $n$ entries such that no two entries share the same row, column…

We offer a piece of evidence that the problems of finding the number of mutually unbiased bases (MUB) and mutually orthogonal Latin squares (MOLS) might not be equivalent. We study a particular procedure which has been shown to relate the…

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, `triple products' of MOLS can result in a gain of one square. In terms of transversal designs,…

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…