Timeline for Creating a Latin rectangle from a projective plane
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 31, 2015 at 22:59 | comment | added | Padraig Ó Catháin | 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! | |
| May 26, 2015 at 21:40 | comment | added | user43928 | 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! | |
| Mar 18, 2015 at 6:42 | vote | accept | user43928 | ||
| Mar 18, 2015 at 6:29 | comment | added | Padraig Ó Catháin | 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. | |
| Mar 18, 2015 at 6:22 | comment | added | Padraig Ó Catháin | 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'. | |
| Mar 18, 2015 at 5:11 | comment | added | user43928 | 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!! | |
| Mar 18, 2015 at 1:29 | history | answered | Padraig Ó Catháin | CC BY-SA 3.0 |