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On the completeness of plane cubic curves over finite fields

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We address the problem of determining when a plane algebraic cubic curve is complete as an (n, 3)-arc in \(\mathrm {PG}(2,q)\). Theoretical results are given for absolutely irreducible singular cubic curves, while computer based results are given for \(q\le 81\).

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The authors would like to thank the anonymous referees for their helpful comments that contributed to improve the final version of this paper. Partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) and by the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA-INdAM).

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Correspondence to Fernanda Pambianco.

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Communicated by J. W. P. Hirschfeld.

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Bartoli, D., Marcugini, S. & Pambianco, F. On the completeness of plane cubic curves over finite fields. Des. Codes Cryptogr. 83, 233–267 (2017).

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