Abstract
Complete \((k,3)\)-arcs in projective planes over finite fields are the geometric counterpart of linear non-extendible Near MDS codes of length \(k\) and dimension \(3\). A class of infinite families of complete \((k,3)\)-arcs in \({\mathrm {PG}}(2,q)\) is constructed, for \(q\) a power of an odd prime \(p\equiv 2 ( { \, \mathrm{mod}\,}3)\). The order of magnitude of \(k\) is smaller than \(q\). This property significantly distinguishes the complete \((k,3)\)-arcs of this paper from the previously known infinite families, whose size differs from \(q\) by at most \(2\sqrt{q}\).
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Acknowledgments
This research was supported by the Italian Ministry MIUR, PRIN 2012 Strutture Geometriche, Combinatoria e loro Applicazioni and by INdAM. The first author acknowledges the support of the European Community under a Marie-Curie Intra-European Fellowship (FACE Project Number 626511).
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.
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Bartoli, D., Giulietti, M. & Zini, G. Complete \((k,3)\)-arcs from quartic curves. Des. Codes Cryptogr. 79, 487–505 (2016). https://doi.org/10.1007/s10623-015-0073-7
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DOI: https://doi.org/10.1007/s10623-015-0073-7