Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated.

The influence of special natural amendments based on zeolite tuff and different lime materials on some soil chemical properties

This paper deals with the changes in soil active acidity, mobile aluminium, base saturation, iron and manganese under the influence of quicklime (QL), mixture of soft lithothamnian limestone (SLL) and dolomite (D), and special natural amendments (SNA) based on zeolite tuff. Investigations were carried out on pseudogley of mesoelevations, dystric. The four-year trial was set up according to the Latin rectangle method with 18 trial treatments in four replications. While SNA based on zeolite tuff had little effect on changes of the studied parameters, traditional lime materials (LM), owing also to the fact that they were applied at several times higher rates, had a very positive effect. Soil acidity, iron and manganese were reduced under their influence, mobile aluminium, particularly under their higher rate, was fully blocked or reduced within tolerable limits, and base saturation was raised to a satisfactory level. Effects of SNA depended on the ratio of zeolite tuff and the lime component in them. It could be presupposed that their main efficiency happened in the domain of ion exchange with a positive impact on soil fertility.

The Many Formulae for the Number of Latin Rectangles

A $k \times n$ Latin rectangle $L$ is a $k \times n$ array, with symbols from a set of cardinality $n$, such that each row and each column contains only distinct symbols. If $k=n$ then $L$ is a Latin square. Let $L_{k,n}$ be the number of $k \times n$ Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on $L_{k,n}$ and approximations for $L_n$, (c) congruences satisfied by $L_{k,n}$ and (d) the many published formulae for $L_{k,n}$ and related numbers. We also describe in detail the method of Sade in finding $L_{7,7}$, an important milestone in the enumeration of Latin squares, but which was privately published in French. Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$. We reproduce the three formulae for $L_{k,n}$ by Fu that were published in Chinese. We give a formula for $L_{k,n}$ that contains, as special cases, formulae of (a) Fu, (b) Shao and Wei and (c) McKay and Wanless. We also introduce a new equation for $L_{k,n}$ whose complexity lies in computing subgraphs of the rook's graph.

Orthogonal Latin Rectangles

We use a greedy probabilistic method to prove that, for every ε > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1 − ε)n. That is, we show the existence of a second Latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.

On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

On the completion of latin rectangles to symmetric latin squares

We find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n−1)-edge colouring of Kn (n even), and for n-edge colouring of Kn (n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.