Equivalence classes of Latin squares and nets in P 2
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 2, p. 335-351

The fundamental combinatorial structure of a net in P 2 is its associated set of mutually orthogonal Latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in P 2 . Then we count these equivalence classes for small cases. Finally we prove that the realization spaces of these classes in P 2 are empty to show some non-existence results for 4-nets in P 2 .

La structure combinatoire fondamentale d’un filet dans P 2 est donnée par l’ensemble des carrés latins orthogonaux associé. Nous définissons des classes d’équivalence de carrés latins orthogonaux a l’aide de classes d’équivalence des lignes apparaisant dans le filet de P 2 . Nous comptons le nombre de classes d’équivalence pour certains exemples de carrés petits. Finalement, nous montrons que les espaces de réalisations de ces classes pour n=4 et k=4,5,6 sont vides et nous en déduisons que les filets correspondants n’existent pas.

@article{AFST_2014_6_23_2_335_0,
     author = {Dunn, Corey and Miller, Matthew and Wakefield, Max and Zwicknagl, Sebastian},
     title = {Equivalence classes of Latin squares and nets in ${\mathbb{C}P}^2$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {2},
     year = {2014},
     pages = {335-351},
     doi = {10.5802/afst.1409},
     mrnumber = {3205596},
     zbl = {1296.05030},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2014_6_23_2_335_0}
}
Dunn, Corey; Miller, Matthew; Wakefield, Max; Zwicknagl, Sebastian. Equivalence classes of Latin squares and nets in ${\mathbb{C}P}^2$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 2, pp. 335-351. doi : 10.5802/afst.1409. http://www.numdam.org/item/AFST_2014_6_23_2_335_0/

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