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{\LARGE Open problems in finite projective spaces}
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\textbf{\large J.W.P. Hirschfeld} % your name here
\texttt{jwph@sussex.ac.uk} % your email here
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(\emph{joint work with} J.A. Thas)
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Department of Mathematics\\
University of Sussex\\
Brighton BN1 9QH\\
United Kingdom
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Apart from being an interesting and exciting area in combinatorics with
beautiful results, finite projective spaces or Galois geometries have many
applications to coding theory, algebraic geometry, design theory, graph
theory, cryptology and group theory. As an example, the theory of linear
maximum distance separable codes (MDS codes) is equivalent to the theory of
arcs in $\mathrm{PG}(n,q)$.
Finite projective geometry is also essential for finite algebraic geometry.
See \cite{j1}, \cite{j2}, \cite{j3}.
Unsolved problems in some of the following topics are considered:
\begin{compactenum}[(1)]
\item $k$-arcs;
\item $k$-caps;
\item Hermitian curves and unitals;
\item maximal arcs;
\item blocking sets;
\item flocks;
\item ovoids and spreads;
\item $m$-systems and BLT-sets;
\item algebraic curves over a finite field.
\end{compactenum}
\begin{thebibliography}{99}
\bibitem{j1} J.W.P. Hirschfeld, {\em Projective Geometries over Finite Fields},
Oxford University Press, Oxford, xiv + 555 pp., 1998.
\bibitem{j2} J.W.P. Hirschfeld and J.A. Thas, Open problems in finite projective
spaces, {\em Finite Fields Appl.} {\bf 32} (2015), 44--81.
\bibitem{j3} J.W.P. Hirschfeld and J.A. Thas, {\em General Galois Geometries, Second
Edition}, Springer-Verlag, London, xvi + 409 pp., 2016.
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