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.

翻译：本文首先研究了小阶数的$k \times n$尤登矩形。通过大规模计算机搜索，我们在排除$k = n-1$的近似方形情况后，枚举了所有小参数范围内的尤登矩形。特别地，我们验证了之前对$(n,k) = (7,3)、(7,4)$的计数结果，并扩展到$(11,5)、(11,6)、(13,4)$和$(21,5)$的情况。对于不存在尤登矩形的小参数情况，我们枚举了任意两列共有符号数始终为两个可能值（相差1）之一的矩形，并将其称为"近尤登矩形"。对于生成的所有设计，我们计算了自同构群的阶，并研究了某种变换能在多大程度上产生其他行列设计，即双阵列、三阵列和单半阵列。最后，我们还研究了具有三种可能的列间成对交叠大小的拉丁矩形，并证明这些矩形通过上述变换能够产生无法从尤登矩形获得的三阵列和单半阵列。