scholarly journals When is an incomplete 3× n latin rectangle completable?

2013 ◽  
Vol 33 (1) ◽  
pp. 57 ◽  
Author(s):  
Reinharadt Euler ◽  
Paweł Oleksik
Keyword(s):  
Latin Rectangle

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

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

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.

scholarly journals On the completion of latin rectangles to symmetric latin squares

2004 ◽  
Vol 76 (1) ◽  
pp. 109-124 ◽  
Author(s):  
Darryn Bryant ◽  
C. A. Rodger
Keyword(s):  
Sufficient Conditions ◽  
Latin Square ◽  
Latin Squares ◽  
Necessary And Sufficient Conditions ◽  
Latin Rectangle ◽  
Necessary And Sufficient ◽  
Edge Colouring

AbstractWe 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.

On Completing Latin Rectangles

1970 ◽  
Vol 13 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Charles C. Lindner
Keyword(s):  
Latin Rectangle

By an (incomplete) r × s latin rectangle is meant an r × s array such that (in some subset of the rs cells of the array) each of the cells is occupied by an integer from the set 1, 2, …, s and such that no integer from the set 1,2, …, s occurs in any row or column more than once. This definition requires that r≦s. If r=s we will replace the word rectangle by square. It is easy to see that for any n≧2 there is an incomplete n × 2n latin rectangle with 2n cells occupied which cannot be completed to a n × 2n latin rectangle. In this paper we prove the following theorem.Theorem 1. An incomplete n × 2n latin rectangle with 2n—1 cells occupied can be completed to a n × 2n latin rectangle.

scholarly journals Orthogonal Latin Rectangles

2008 ◽  
Vol 17 (4) ◽  
pp. 519-536 ◽  
Author(s):  
ROLAND HÄGGKVIST ◽  
ANDERS JOHANSSON
Keyword(s):  
Probabilistic Method ◽  
Latin Rectangle

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.

scholarly journals Arrays and brooks

1967 ◽  
Vol 7 (1) ◽  
pp. 23-31 ◽  
Author(s):  
B. T. Bennett ◽  
R. B. Potts
Keyword(s):  
Rectangular Array ◽  
Standard Order ◽  
Latin Rectangle ◽  
Image Position

Consider an m × n rectangular array whose m rows are permutations of 1, 2, …, n. Such an array will be called a constant-sum array if the sum of the elements in each column is the same (and equal to ½m (n+1)). An example of a 3×9 constant-sum array is In contrast to a Latin rectangle, elements in the same column of a constantsum array may be equal. It will be convenient to assume arrays normalised in the sense that the columns are arranged so that, as in (1), the first row is in the standard order 1, 2, …, n.

scholarly journals The lattice of stable marriages and permutations

1982 ◽  
Vol 33 (3) ◽  
pp. 401-410 ◽  
Author(s):  
J. S. Hwang
Keyword(s):  
Distributive Lattice ◽  
Stable Marriages ◽  
Latin Rectangle

AbstractRecently, we have introduced the notion of stable permutations in a Latin rectangle L(r×c) of r rows and c columns. In this note, we prove that the set of all stable permutations in L (r×c) forms a distributive lattice which is Boolean if and only if c ≤ 2.

Latin rectangle designs for 2n factorial experiments on 32 plots

1951 ◽  
Vol 41 (4) ◽  
pp. 315-316 ◽  
Author(s):  
M. J. R. Healy
Keyword(s):  
Factorial Experiments ◽  
Five Factors ◽  
Latin Rectangle ◽  
Factor Interactions ◽  
Three Factor

Designs are given for factorial experiments with three, four or five factors at two levels each, using thirty-two plots in a 4 × 8 lay-out on the ground. The effects of both rows and columns can be eliminated from the estimate of error. Provided that three-factor interactions can be ignored, information can be retained on all two-factor interactions.

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