Subsquares in orthogonal latin squares as subspaces in affine geometries: A generalization of an equivalence of bose

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 orderm h blocks of subsquares of orderm h(i−1)/i , wherei dividesh, form the hyperplanes of the affine geometryAG (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 thati=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.