Mathematics – Algebraic Geometry
Scientific paper
2010-04-15
Preliminary version as Technical Report of Department of Mathematics of University of Perugia- Italy, 4/2002
Mathematics
Algebraic Geometry
7 pages
Scientific paper
A non-singular plane algebraic curve of degree n(n>=4) is called maximally symmetric if it attains the maximum order of the projective automorphism groups for non-singular plane algebraic curves of degree n. Highly symmetric curves give rise to extremely good error-correcting codes and are ideal for the construction of good universal hash families and authentication codes (see \cite{unihash1}, \cite{JBbook}, \cite{unihash3}, \cite{unihash2}). In this work it is proved that the maximally symmetric non-singular plane curves of degree n in P^2 (n not in {4,6}) are projectively equivalent to the Fermat curve x^n+y^n+z^n. For some particular values of n<=20 the result has been obtained in \cite{enr}, \cite{kmp1}, \cite{kmp2}, \cite{sem}. As to the two exceptional cases, {n=4, 6}, it is known that the Klein quartic \cite{har} and the Wiman sextic \cite{doi} are respectively the uniquely determined maximally symmetric curves.
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