Abstract

In $\mathsf{PG}\left(2,q\right)$ a point set $K$ is sharply transitive if the collineation group preserving $K$ has a subgroup acting on $K$ as a sharply transitive permutation group. By a result of Korchmáros, sharply transitive hyperovals only exist for a few values of $q$, namely $q=2,4$ and $16$. In general, sharply transitive complete arcs of even size in $\mathsf{PG}\left(2,q\right)$ with $q$ even seem to be sporadic. In this paper, we construct sharply transitive complete $6\left(\sqrt{q}-1\right)$-arcs for $q=4^{2h+1}$, $h\le 4$. As far as we are concerned, these are the smallest known complete arcs in $\mathsf{PG}\left(2,4^7\right)$ and in $\mathsf{PG}\left(2,4^9\right)$; also, $42$ seems to be a new value of the spectrum of the sizes of complete arcs in $\mathsf{PG}\left(2,4^3\right)$. Our construction applies to any $q$ which is an odd power of $4$, but the problem of the completeness of the resulting sharply transitive arc remains open for $q\ge 4^{11}$. In the second part of this paper, sharply transitive subsets arising as orbits under a Singer subgroup are considered and their characters, that is the possible intersection numbers with lines, are investigated. Subsets of $\mathsf{PG}\left(2,q\right)$ and certain linear codes are strongly related and the above results from the point of view of coding theory will also be discussed.

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Mathematical Subject Classification 2000

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