Gerold Jäger ; Klas Markström ; Denys Shcherbak ; Lars-Daniel Öhman - Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs

dmtcs:6754 - Discrete Mathematics & Theoretical Computer Science, March 1, 2023, vol. 25:1 - https://doi.org/10.46298/dmtcs.6754
Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column DesignsArticle

Authors: Gerold Jäger ; Klas Markström ; Denys Shcherbak ; Lars-Daniel Öhman

    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.


    Volume: vol. 25:1
    Section: Combinatorics
    Published on: March 1, 2023
    Accepted on: January 5, 2023
    Submitted on: September 2, 2020
    Keywords: Mathematics - Combinatorics,Computer Science - Discrete Mathematics

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