Subsquares in orthogonal Latin squares as subspaces in affine geometries: A generalization of an equivalence of Bose
From author's abstract: The combinatorial properties of subsquares in orthogonal Latin squares are examined. Using these properties it is shown that in appropriate orthogonal Latin squares of order \(m^ h\) blocks of subsquares of order \(m^{h(i-1)/i}\), where \(i\) divides \(h\), form the hyperplanes of the affine geometry \(\text{AG}(2i,m^{h/i})\). This means that a given set of mutually orthogonal Latin squares may be equivalent simultaneously to a number of different geometries depending on the order of the subsquares used to form the hyperplanes. In the case that \(i=1\), the subsquares become points, the hyperplanes become lines, and the equivalence reduces to the well-known result of Bose relating orthogonal Latin squares and affine planes.
scientific article; zbMATH DE number 167547