Hyperovals in Hall planes
A \(k\)-arc in a projective plane \(\pi\) of order \(n\) is a set of \(k\) points, no three of which are collinear. It is well-known that the maximum possible value of \(k\) is \(n+1\) if \(n\) is odd and \(n+2\) if \(n\) is even. An \((n+1)\)-arc is called an oval and an \((n+2)\)-arc is called a hyperoval. A translation \(n\)-arc of \(\pi\) is an \(n\)-arc of \(\pi\) which admits a group \(G\) of elations (inherited from the automorphism group of \(\pi)\) acting transitively on its points. A translation hyperoval of \(\pi\) is a hyperoval \(H\) of \(\pi\) (necessarily implying \(n\) is even) which contains a translation \(n\)-arc \(K\). If \(H\backslash K=\{P,Q\}\), it is not hard to see that \(PQ\) is the common axis of \(G\). The translation hyperovals of the even order desarguesian planes were determined by \textit{S. E. Payne} [Atti. Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. fis. mat. natur. 51(1971), 328-331 (1972; Zbl 0238.50018)]. In the paper under review the authors construct two families of translation hyperovals in the Hall planes Hall\((q^ 2)\) of even order \(q^ 2\geq 16\). For a given \(q\) any two hyperovals constructed from the same family are equivalent while any two hyperovals constructed from different families are inequivalent. One of the constructions includes the examples of \textit{M. Crismale} [Note Mat. 1, No. 1, 127-136 (1981; Zbl 0511.51012)] and \textit{G. Korchmáros} [J. Comb. Theory, Ser. A 42, 140-143 (1986; Zbl 0587.51009)].
scientific article; zbMATH DE number 64844
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