Given a projective plane I'd like to form a latin rectangle from the lines. In particular, I'd like to take each line from the plane, order the elements in some way, and stick them into the matrix as a column.


I liked to know whether this is possible. Better yet, I'd love an algorithm for creating the matrix.


Thanks for your time!



Creating a Latin rectangle from a projective plane
By a theorem of Singer, the automorphism group of $PG(2,q)$ contains a cyclic subgroup $\langle\sigma\rangle$ which acts regularly on points and regularly on lines. Fix a base block, $B$, and list its elements in the first column of your matrix in any order you choose. Then form the rest of the columns by applying $\sigma$ repeatedly to the first row. For the seven point plane, you can take $\sigma = (1, 2, 3, 4, 5, 6, 7)$ and $B = \{1, 2, 4\}$, giving the solution 


$ 1 2 3 4 5 6 7 \\ 2 3 4 5 6 7 1 \\ 4 5 6 7 1 2 3$


The same idea works to produce a latin rectangle from the blocks of a design which admits a regular group of automorphisms. The key phrase here is 'difference set'. You can find all the information you need in the book "Design Theory" by Beth, Jungnickel and Lenz.


