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Complete \((k,3)\)-arcs from quartic curves


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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  1. Anbar N., Giulietti M.: Bicovering arcs and small complete caps from elliptic curves. J. Algebr. Comb. 38, 371–392 (2013). doi:10.1007/s10801-012-0407-8.

  2. Anbar N., Bartoli D., Giulietti M., Platoni I.: Small complete caps from singular cubics. J. Comb. Des. 22(10), 409–424 (2014). doi:10.1002/jcd.21366.

  3. Anbar N., Bartoli D., Giulietti M., Platoni I.: Small complete caps from singular cubics. II. J. Algebr. Comb (to appear). (2014). doi:10.1007/s10801-014-0532-7 (published online May).

  4. Bartoli D., Marcugini S., Pambianco F.: The non-existence of some NMDS codes and the extremal sizes of complete \((n,3)\)-arcs in PG(2,16). Des. Codes Cryptogr. 72(1), 129–134 (2014). doi:10.1007/s10623-013-9837-0.

  5. Coolsaet K., Sticker H.: The complete \((k, 3)\)-arcs of PG\((2, q)\), \(q\le 13\). J. Comb. Des. 20, 89–111 (2012). doi:10.1002/jcd.20293.

  6. Dodunekov S., Landjev I.: Near-MDS codes. J. Geom. 54, 30–43 (1995).

  7. Dodunekov S., Landjev I.: Near-MDS codes over some small fields. Discret. Math. 213, 55–65 (2000).

  8. Garcia A., Stichtenoth H.: Elementary Abelian \(p\)-extensions of algebraic function fields. Manuscr. Math. 72, 67–79 (1991).

  9. Giulietti M.: On plane arcs contained in cubic curves. Finite Fields Appl. 8, 69–90 (2002).

  10. Giulietti M., Pasticci F.: On the completeness of certain n-tracks arising from elliptic curves. Finite Fields Appl. 13(4), 988–1000 (2007).

  11. Giulietti M., Pambianco F., Torres F., Ughi E.: On complete arcs arising from plane curves. Des. Codes Cryptogr. 25, 237–246 (2002).

  12. Hamilton N., Penttila, T.: Sets of Type \((a, b)\) From Subgroups of \(\Gamma L(1, p^R)\). J. Algebr. Comb. 13, 67–76 (2001).

  13. Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998).

  14. Hirschfeld J.W.P.: Algebraic curves, arcs, and caps over finite fields. In: Quaderni del Dipartimento di Matematica dell’ Università del Salento 5, Dipartimento di Matematica, Università del Salento, Lecce, (1986).

  15. Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces. J. Stat. Plan. Inference. 72(1–2), 355–380 (1998). R. C. Bose Memorial Conference (Fort Collins, CO, 1995).

  16. Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory, and finite projective spaces: update 2001. In: Blokhuis A, Hirschfeld J.W.P, Jungnickel D, Thas J.A. (eds.) Finite Geometries. Proceedings of the Fourth Isle of Thorns Conference, Developments in Mathematics, vol. 3, pp. 201–246. Kluwer Academic Publishers, Boston (2001).

  17. Hirschfeld J.W.P., Voloch J.F.: The characterization of elliptic curves over finite fields. J. Austral. Math. Soc. 45, 275–286 (1988).

  18. Lombardo-Radice L.: Sul problema dei \(k\)-archi completi in \(S_{2, q}\) (\(q=p^t\), \(p\) primo dispari). Boll. Unione Mat. Ital. 3(11), 178–181 (1956).

  19. Marcugini S., Milani A., Pambianco F.: Classification of the \((n, 3)\)-arcs in PG\((2, 7)\). J. Geom. 80(1–2), 179–184 (2004).

  20. Segre B.: Ovali e curve \(\sigma \) nei piani di Galois di caratteristica due. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 32(8), 785–790 (1962).

  21. Stichtenoth H.: Algebraic Function Fields and Codes. Graduate Texts in Mathematics, vol. 254, 2nd edn. Springer, Berlin (2009).

  22. Szőnyi T.: Complete arcs in galois planes: a survey. In: Quaderni del Seminario di Geometrie Combinatorie, vol. 94, Dipartimento di Matematica G. Castelnuovo, Università degli Studi di Roma La Sapienza, Roma (1989).

  23. Szőnyi T.: Some applications of algebraic curves in finite geometry and combinatorics. In: Surveys in combinatorics, London, 1997. London Mathematical Society Lecture Note Series, vol. 241, pp. 197–236. Cambridge University Press, Cambridge (1997).

  24. Tallini Scafati M.: Graphic curves on a Galois plane. In: Atti del Convegno di Geometria Combinatoria e sue Applicazioni, Perugia, pp. 413–419 (1970).

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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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Correspondence to Massimo Giulietti.

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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).

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