Embedding a Latin square with transversal into a projective space

Details Embedding a Latin square with transversal into a projective space Embedding a Latin square with transversal into a projective space

2010-05-19

arxiv.org/abs/1005.3460v2

Journal of Combinatorial Theory, Ser. A 118 (2011), 1674--1683

Mathematics

Combinatorics

11 pages. This is the revised final preprint version. Added journal reference and doi

Scientific paper

10.1016/j.jcta.2011.01.013

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n^2 lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n^2-n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n^2 lines of size k. Extending work of Bruen and Colbourn (J. Combin. Th. Ser. A 92 (2000), 88-94), we characterise embeddings of these finite geometries into projective spaces over skew fields.

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