New (k;
r)-arcs in the projective plane of order thirteen |
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Authors: | Rumen Nikolov Daskalov María Estela Jiménez Contreras |
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Institution: | (1) Department of Mathematics, Technical University of Gabrovo, 5300 Gabrovo, Bulgaria;(2) Department of Mathematics, University of Sussex, BN1 9RF Brighton, United Kingdom |
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Abstract: | A (k;r)-arc
$\cal K$ is a set of k points of a projective plane PG(2, q) such that some r, but no
r +1 of them, are collinear. The maximum size of a
(k; r)-arc in PG(2,
q) is denoted by m
r (2, q). In this paper a (35; 4)-arc, seven (48; 5)-arcs, a (63; 6)-arc and two (117; 10)-arcs in PG(2, 13) are given. Some were found by means of computer search, whereas the example of a (63; 6)-arc was found by adding points to those of a sextic curve $\cal C$ that was not complete as a (54; 6)-arc. All these arcs are new and improve the lower bounds for m
r (2, 13) given in 10, Table 5.4]. The last section concerns the nonexistence of (40; 4)-arcs in PG(2, 13). |
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Keywords: | 51E21 |
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