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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!

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    $\begingroup$ 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
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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.

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  • $\begingroup$ 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

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