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Zbl 0904.51002
Ball, Simeon
Multiple blocking sets and arcs in finite planes.
(English)
[J] J. Lond. Math. Soc., II. Ser. 54, No.3, 581-593 (1996). ISSN 0024-6107; ISSN 1469-7750/e

A $t$-fold blocking set $B$ in a projective plane is a set of points such that each line contains at least $t$ points of $B$ and some line contains exactly $t$ points of $B$. A $(k,n)$-arc is a set of $k$ points such that some $n$, but no $n+1$ of them, are collinear. Define $m_n(2,q)$ to be the maximum size of a $(k,n)$-arc in $\text{PG}(2,q)$. Note that determining $m_n(2,q)$, or the minimum size of a $(q+1-n)$-fold blocking set, are equivalent problems. In the paper such a difficult question is dealt with. The value of $m_n(2,q)$ was already known for $2\leq n<q\leq 9$, and for $q>9$ only in few particular cases. \par The main theorems of the paper are the following: \par Theorem 1. Let $B$ be a $t$-fold blocking set in $\text{PG}(2,q)$. If $B$ contains no line then it has at least $tq+\sqrt{tq}+1$ points. \par Theorem 2. Let $B$ be a $t$-fold blocking set in $\text{PG}(2,p)$ with $p>3$ prime. (i) If $t<p/2$ then $| B| \geq(t+{1\over 2})(p+1)$. (ii) If $t>p/2$ then $| B| \geq(t+1)p$. \par The latter theorem is a generalization of a result by {\it A. Blokhuis} [Bolyai Soc. Math. Stud. 2, 133-155 (1996; Zbl 0849.51005)] and uses the theory of lacunary polynomials. In some cases the bounds in Theorem 2 are sharp. \par The author finds examples and proves further results that with the help of Theorem 2 yield the exact value of $m_n(2,11)$ and $m_n(2,13)$ for some $n$ and bounds in other cases. A table of all known values of $m_n(2,q)$ for $2\leq n<q\leq 13$ containing complete references is given.
MSC 2000:
*51E21 Blocking sets, ovals, k-arcs

Keywords: multiple blocking sets; arcs; lacunary polynomials

Citations: Zbl 0849.51005

Cited in: Zbl 1142.51004 Zbl 1059.51011

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