We have performed a complete enumeration of nonisotopic triples of mutually orthogonal
k
×
n Latin rectangles for
k
≤
n
≤
7. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this, we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of
k
×
8 rectangles which are formed by extending mutually orthogonal triples with nontrivial autotopisms one row at a time, and requiring that the autotopism group is nontrivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of
4
×
8 rectangles, formed by juxtaposing two selected copies of complete sets of mutually orthogonal Latin squares of order 4.
In this paper we first study $k \times n$ Youden rectangles of small orders.
We have enumerated all Youden rectangles for a range of small parameter values,
excluding the almost square cases where $k = n-1$, in a large scale computer
search. In particular, we verify the previous counts for $(n,k) = (7,3),
(7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For
small parameter values where no Youden rectangles exist, we also enumerate
rectangles where the number of symbols common to two columns is always one of
two possible values, differing by 1, which we call \emph{near Youden
rectangles}. For all the designs we generate, we calculate the order of the
autotopism group and investigate to which degree a certain transformation can
yield other row-column designs, namely double arrays, triple arrays and sesqui
arrays. Finally, we also investigate certain Latin rectangles with three
possible pairwise intersection sizes for the columns and demonstrate that these
can give rise to triple and sesqui arrays which cannot be obtained from Youden
rectangles, using the transformation mentioned above.
In this paper we study Youden rectangles of small orders. We have enumerated all Youden rectangles for all small parameter values, excluding the almost square cases, in a large scale computer search.For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values. We refer to these objects as near Youden rectangles.For all our designs we calculate the size of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays.Finally we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.
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