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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Schemes of a finite projective plane and their extensions


Authors: S. Evdokimov and I. Ponomarenko
Translated by: the authors
Original publication: Algebra i Analiz, tom 21 (2009), nomer 1.
Journal: St. Petersburg Math. J. 21 (2010), 65-93
MSC (2000): Primary 05C25, 51A05
Published electronically: November 4, 2009
MathSciNet review: 2553053
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Abstract: There are several schemes (coherent configurations) associated with a finite projective plane  $ \mathcal{P}$. In the paper, a new scheme is constructed, which, in a sense, contains all of them. It turns out that this scheme coincides with the $ 2$-extension of the nonhomogeneous scheme of  $ \mathcal{P}$ and is uniquely determined up to similarity by the order $ q$ of  $ \mathcal{P}$. Moreover, for $ q\ge 3$, the rank of the scheme does not depend on $ q$ and equals $ 416$. The results obtained have interesting applications in the theory of multidimensional extensions of schemes and similarities.


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Additional Information

S. Evdokimov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
Email: evdokim@pdmi.ras.ru

I. Ponomarenko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
Email: inp@pdmi.ras.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01086-3
PII: S 1061-0022(09)01086-3
Keywords: Projective plane, Galois plane, scheme, graph
Received by editor(s): April 18, 2008
Published electronically: November 4, 2009
Additional Notes: The first author was partially supported by RFBR (grants 07-01-00485 and 06-01-00471).
The second author was partially supported by RFBR (grants 07-01-00485 and 05-01-00899) and by the grant NS-4329.2006.1.
Article copyright: © Copyright 2009 American Mathematical Society