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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
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