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!

  • 2
    $\begingroup$ For what it's worth, this can be done for the 7-point plane: $$\matrix{1&2&3&4&5&6&7\cr2&5&7&1&3&4&6\cr3&7&4&5&6&2&1\cr}$$ $\endgroup$ – Gerry Myerson Mar 17 '15 at 22:59

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.

  • $\begingroup$ Thanks so much for this nice response, @Padraig! Is it known which block designs admit a regular group of automorphisms of their blocks? I'd be very interested to know which latin rectangles one can build through this fashion. Thanks again!! $\endgroup$ – user43928 Mar 18 '15 at 5:11
  • $\begingroup$ Among symmetric designs, these are exactly the ones which correspond to difference sets. You should look at Beth, Jungnickel and Lenz for a good introduction to this topic. For non-symmetric designs, I don't know of a good single reference. But google turns up many results for 'block transitive designs'. $\endgroup$ – Padraig Ó Catháin Mar 18 '15 at 6:22
  • $\begingroup$ Note that taking any subset of {1..n} in your first column and any fixed point free permutation, you can generate a latin rectangle. The block design doesn't really play any part... I guess you could characterise the rectangles arising from designs in terms of how often pairs of elements appear in each column, but that's essentially just a translation of the definition of the design. I can't see that they have any other special properties. $\endgroup$ – Padraig Ó Catháin Mar 18 '15 at 6:29
  • $\begingroup$ Regarding the theorem of Singer you mentioned, I haven't been able to find a reference for that. I have found a weaker version that only applies when $q$ is a power of a prime. Do you have a reference? Thanks again for all your help! $\endgroup$ – user43928 May 26 '15 at 21:40
  • $\begingroup$ I don't have a reference to hand for Singer's result - it is simply about the existence of an element of order $q^{n}-1$ in $\GL(n,q)$. This seems to be what you are referring to - it applies only when $q$ is prime power. If you know of an example of a projective plane where $q$ is not a power of a prime, you should publish it. You'll be quite famous! $\endgroup$ – Padraig Ó Catháin May 31 '15 at 22:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.