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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 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 2n<q9, 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 PG(2,q). If B contains no line then it has at least tq+tq+1 points.

Theorem 2. Let B be a t-fold blocking set in PG(2,p) with p>3 prime. (i) If t<p/2 then |B|(t+1 2)(p+1). (ii) If t>p/2 then |B|(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 2n<q13 containing complete references is given.


MSC:
51E21Blocking sets, ovals, k-arcs