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If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub plane. Otherwise $m \geq n^2 + n$. (This is a theorem of Bruck that can be found in Hall's Group Theory book, I believe.) My question is, how nearly can this bound be achieved?

For example, we have cases for $m=9$ and $n=2$, where $\pi_1$ is one of the non-Desaurgian planes of order $9$, and $\pi_2$ is the Fano plane. Is anything known about how close we can come to the bound in the general case? Has the bound been improved?

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    $\begingroup$ We are trying to implement designs constructions in Sage. We did a lot of work for OA/TD/MOLS already but designs are all great and there is too much stuff to implement. If you like combinatorial designs and code send me an email (I didn't find yours in your profile) there is a lot to do ! $\endgroup$ – Nathann Cohen Jul 28 '14 at 15:55
  • $\begingroup$ Not sure about the bound in general but there are some results about existence of sub-planes that are known. For example, combinatorics.org/ojs/index.php/eljc/article/view/v18i1p2. For details on small planes you can also refer to uwyo.edu/moorhouse/pub/planes $\endgroup$ – Anurag Jan 12 '15 at 12:48
  • $\begingroup$ In general an algebraic projective plane of order $p^e$ contains algebraic subplanes of order $p^f$ for all factors $f|e$, so in particular $n=2$ is seen already for $m=8$. I think it's known that the only projective plane of order $7$ is algebraic, and that one does not contain the Fano plane. $\endgroup$ – Noam D. Elkies Apr 23 '19 at 2:38

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