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{\large The projective plane of order 13}\\[3mm]
\textbf{J.W.P. Hirschfeld}\\[3mm]
(joint work with M. Giulietti and G. Korchm\'aros)\\[3mm]
MSC2000: 51E21,11G20
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The Desarguesian plane $PG(2,13)$ provides some insight into the
following questions.
\begin{enumerate}[(1)]
\item
What is the second smallest length of a maximal 3-dimensional MDS
code over ${\bf F}_q$?
\item
What is the maximum number of rational points on a plane
algebraic curve of degree $d$ and genus $g$ over ${\bf F}_q$?
\item
Is there a plane curve of degree $d$ over ${\bf F}_q$ such that
every line of $PG(2,q)$ meets the curve in less than $d$ rational
points, when these points are counted by their intersection numbers?
\end{enumerate}
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