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Multiple blocking sets and arcs in finite planes. (English) Zbl 0904.51002

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.
The main theorems of the paper are the following:
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.
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 \(| B| \geq(t+1)p\).
The latter theorem is a generalization of a result by 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.
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.
Reviewer: C.Zanella (Padova)

MSC:

51E21 Blocking sets, ovals, \(k\)-arcs

Citations:

Zbl 0849.51005
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