Geodesic
A Necessary and Sufficient Condition for the Existence of an (n,r)-arc in PG(2,q) and Its Applications
Serdica Journal of Computing, Tome 6 (2012) no. 3, pp. 253-266.

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Let q be a prime or a prime power ≥ 3. The purpose of this paper is to give a necessary and sufficient condition for the existence of an (n, r)-arc in PG(2, q ) for given integers n, r and q using the geometric structure of points and lines in PG(2, q ) for n > r ≥ 3. Using the geometric method and a computer, it is shown that there exists no (34, 3) arc in PG(2, 17), equivalently, there exists no [34, 3, 31] 17 code.
Keywords: (n, r)-arcs, Projective Plane, Linear Codes 
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Hamada, Noboru; Maruta, Tatsuya; Oya, Yusuke. A Necessary and Sufficient Condition for the Existence of an (n,r)-arc in PG(2,q) and Its Applications. Serdica Journal of Computing, Tome 6 (2012) no. 3, pp. 253-266. http://geodesic.mathdoc.fr/item/SJC_2012_6_3_a0/