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A necessary and sufficient condition for the existence of an $$(n, r)$$-arc in $$\mathrm{PG}(2, q)$$ and its applications. (English) Zbl 1294.94112
Summary: Let $$q$$ be a prime or a prime power $$\geq 3$$. The purpose of this paper is to give a necessary and sufficient condition for the existence of an $$(n, r)$$-arc in $$\mathrm{PG}(2, q)$$ for given integers $$n, r$$ and $$q$$ using the geometric structure of points and lines in $$\mathrm{PG}(2, q)$$ for $$n > r \geq 3$$. Using the geometric method and a computer, it is shown that there exists no $$(34, 3)$$ arc in $$\mathrm{PG}(2, 17)$$, equivalently, there exists no $$[34, 3, 31]_{17}$$ code.
##### MSC:
 94B25 Combinatorial codes 51E21 Blocking sets, ovals, $$k$$-arcs
##### Keywords:
$$(n, r)$$-arcs; projective plane; linear codes
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