Hokkaido Mathematical Journal

NAKAGAWA Nobuo, SUETAKE Chihiro,

On blocking semiovals with an 8-secant in projective planes of order 9.

Hokkaido Mathematical Journal, 35 (2006) pp.437-456

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Abstract

Let $S$ be a blocking semioval in an arbitrary projective plane $\Pi$ of order 9 which meets some line in 8 points. According to Dover in $[2]$, $20\leq \vert S\vert\leq 24$. In $[8]$ one of the authors showed that if $\Pi$ is desarguesian, then $22\leq\vert S\vert\leq 24$. In this note all blocking semiovals with this property in all non-desarguesian projective planes of order 9 are completely determined. In any non-desarguesian plane $\Pi$ it is shown that $21\leq \vert S\vert \leq 24$ and for each $i\in \{ 21,22,23,24\}$ there exist blocking semiovals of size $i$ which meet some line in 8 points. Therefore, the Dover's bound is not sharp.

MSC(Primary)51E20
MSC(Secondary)
Uncontrolled Keywordsblocking semioval, projective plane, ternary function, finite field, collineation group