Skip to main content

Algebraic Constructions of Complete m-Arcs Daniele Bartoli, Giacomo Micheli

Abstract

Let m be a positive integer, q be a prime power, and PG(2,q) be the projective plane over the finite field \({\mathbb{F}_q}\). Finding complete m-arcs in PG(2,q) of size less than q is a classical problem in finite geometry. In this paper we give a complete answer to this problem when q is relatively large compared with m, explicitly constructing the smallest m-arcs in the literature so far for any m ≥ 8. For any fixed m, our arcs \({{\cal A}_{q,m}}\) satisfy \(\left| {{{\cal A}_{q,m}}} \right| - q \to - \infty \) as q grows. To produce such m-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the m-completeness of the arc.

This is a preview of subscription content, access via your institution.

References

  1. N. Anbar, and M. Giulietti: Bicovering arcs and small complete caps from elliptic curves, Journal of Algebraic Combinatorics 38 (2013), 371–392.

    MathSciNet  Article  Google Scholar 

  2. N. Anbar, D. Bartoli, M. Giulietti and I. Platoni: Small complete caps from singular cubics, Journal of Combinatorial Designs 22 (2014), 409–424.

    MathSciNet  Article  Google Scholar 

  3. N. Anbar, D. Bartoli, M. Giulietti and I. Platoni: Small complete caps from singular cubics, II, Journal of Algebraic Combinatorics 41 (2015), 185–216.

    MathSciNet  Article  Google Scholar 

  4. D. Bartoli, S. Marcugini and F. Pambianco: On the completeness of plane cubic curves over finite fields, Designs, Codes and Cryptography 83 (2017), 233–267.

    MathSciNet  Article  Google Scholar 

  5. D. Bartoli, P. Speziali and G. Zini: Complete (k, 4)-arcs from quintic curves, Journal of Geometry 108 (2017), 985–1011.

    MathSciNet  Article  Google Scholar 

  6. D. Bartoli, M. Giulietti and G. Zini: Complete (k, 3)-arcs from quartic curves, Designs, Codes and Cryptography 79 (2016), 487–505.

    MathSciNet  Article  Google Scholar 

  7. J. W. S. Cassels: An introduction to diophantine approximation, New York: Cambridge Univ. Press, 1957.

    MATH  Google Scholar 

  8. C. Di Comite: Su k-archi deducibili da cubiche piane, Atti dell’Accademia Nazionale dei Lincei. Rendiconti. Serie 8 33 (1962), 429–435.

    MathSciNet  MATH  Google Scholar 

  9. C. Di Comite: Intorno a certi (q + 9)/2-archi de S2,q, Atti dell’Accademia Nazionale dei Lincei. Rendiconti. Serie 8 47 (1967), 240–244.

    MathSciNet  Google Scholar 

  10. A. Ferraguti and G. Micheli: Full classification of permutation rational functions and complete rational functions of degree three over finite fields, Designs, Codes and Cryptography 88 (2020), 867–886.

    MathSciNet  Article  Google Scholar 

  11. A. Garcia and H. Stichtenoth: Elementary abelian p-extensions of algebraic function fields, Manuscripta Mathematica 72 (1991), 67–79.

    MathSciNet  Article  Google Scholar 

  12. M. Giulietti, F. Pambianco, F. Torres and E. Ughi: On complete arcs arising from plane curves, Designs, Codes and Cryptography 25 (2002), 237–246.

    MathSciNet  Article  Google Scholar 

  13. M. Giulietti: On Plane Arcs Contained in Cubic Curves, Finite Fields and Their Applications 8 (2002), 69–90.

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  15. R. M. Guralnick, T. J. Tucker and M. E. Zieve: Exceptional covers and bijections on rational points, International Mathematics Research Notices (2007), art. ID rnm004.

  16. G. Korchmáros: New example of complete k-arcs in PG(2,q). European Journal of Combinatorics 4 (1983), 329–334.

    MathSciNet  Article  Google Scholar 

  17. N. Hamilton and T. Penttila: Sets of type (a, b) from subgroups of TL(1, pR), Journal of Algebraic Combinatorics 13 (2001), 67–76.

    MathSciNet  Article  Google Scholar 

  18. M. Kosters: A short proof of a Chebotarev density theorem for function fields, Mathematical Communications 22 (2017), 227–233.

    MathSciNet  MATH  Google Scholar 

  19. J. W. P. Hirschfeld: Algebraic Curves, Arcs, and Caps over Finite Fields, Quaderni del Dipartimento di Matematica dell’Università del Salento (Lecce), 1986.

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

  21. J. W. P. Hirschfeld, G. Korchmáros and F. Torres: Algebraic Curves over a Finite Field, Princeton Series in Applied Mathematics, 2008.

  22. J. W. P. Hirschfeld and E. V. D. Pichanick: Bounds for arcs of arbitrary degree in finite Desarguesian planes, Journal of Combinatorial Designs 24 (2016), 184–196.

    MathSciNet  Article  Google Scholar 

  23. J. W. P. Hirschfeld and L. Storme: The packing problem in statistics, coding theory, and finite projective spaces, Journal of Statistical Planning and Inference 72 (1998), 355–380.

    MathSciNet  Article  Google Scholar 

  24. J. W. P. Hirschfeld and J. F. Voloch: The characterization of elliptic curves over finite fields, Journal of the Australian Mathematical Society 45 (1988), 275–286.

    MathSciNet  Article  Google Scholar 

  25. J. H. Kim and V. H. Vu: Small complete arcs in projective planes, Combinatorica 23 (2003), 311–363.

    MathSciNet  Article  Google Scholar 

  26. L. Lombardo-Radice: Sul problema dei k-archi completi in S2,q, (q = pt, p primo dispari), Bollettino dell’Unione Matematica Italiana 11 (1956), 178–181.

    MATH  Google Scholar 

  27. G. Micheli: Constructions of locally recoverable codes which are optimal, IEEE Transactions on Information Theory 66 (2020), 167–175.

    MathSciNet  Article  Google Scholar 

  28. G. Micheli: On the selection of polynomials for the dlp quasi-polynomial time algorithm for finite fields of small characteristic, SIAM Journal on Applied Algebra and Geometry 3 (2019), 256–265.

    MathSciNet  Article  Google Scholar 

  29. J. Nagura: On the interval containing at least one prime number, Proceedings of the Japan Academy, Series A 28 (1952), 177–181.

    MathSciNet  MATH  Google Scholar 

  30. R. Schoof: Non-singular plane cubic curves over finite fields, Journal of Combinatorial Theory, Series A 46 (1987), 183–211.

    MathSciNet  Article  Google Scholar 

  31. B. Segre: Le geometrie di Galois, Annali di Matematica Pura ed Applicata 48 (1959), 1–97.

    MathSciNet  Article  Google Scholar 

  32. B. Segre: Introduction to Galois geometry, (ed. J. W. P. Hirschfeld), Mem. Accad. Naz. Lincei, 8, 133–63, 1967.

  33. B. Segre: Ovali e curve σ nei piani di Galois di caratteristica due, Atti della Accademia nazionale dei Lincei. Rendiconti. Classe di scienze fisiche, matematiche e naturali 32 (1962), 785–790.

    Google Scholar 

  34. H. Stichtenoth: Algebraic Function Fields and Codes, 2nd edn., Springer, 2009, Berlin.

  35. T. Szőnyi: Complete arcs in Galois planes: a survey, Quaderni del Seminario di Geometrie Combinatorie 94, Dipartimento di Matematica “G. Castelnuovo”, Università degli Studi di Roma “La Sapienza”, 1989.

  36. T. Szőnyi: Small complete arcs in Galois planes, Geometriae Dedicata 18 (1985), 161–172.

    MathSciNet  Article  Google Scholar 

  37. T. Szőnyi: Arcs, caps, codes and 3-independent subsets, Giornate di Geometrie Combinatorie, (edits G. Faina, G. Tallini) Univ. Perugia (1993), 57–80

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

    MATH  Google Scholar 

  39. S. Tafazolian: A family of maximal hyperelliptic curves, Journal of Pure and Applied Algebra 216 (2021), 1528–1532.

    MathSciNet  Article  Google Scholar 

  40. S. Tafazolian and J. Top: On certain maximal hyperelliptic curves related to Chebyshev polynomials, Journal of Number Theory 203 (2019), 276–293.

    MathSciNet  Article  Google Scholar 

  41. B. L van der Waerden: Die Zerlegungs-und Trägheitsgruppe als Permutationsgruppen, Math. Ann. 111 (1935), 731–733.

    MathSciNet  Article  Google Scholar 

  42. J. F. Voloch: On the completeness of certain plane arcs II, European Journal of Combinatorics 11 (1990), 491–496.

    MathSciNet  Article  Google Scholar 

  43. J. F. Voloch: Complete arcs in Galois planes of non-square order, in: Advance in Finite Geometries and Designs (eds J. W. P. Hirschfeld, D. R. Huges, J. A. Thas), Oxford University Press 401–406, 1991.

  44. M. Tallini Scafati: Graphic Curves on a Galois Plane, Atti del Convegno di Geometria Combinatoria e sue Applicazioni, Perugia (1970).

  45. H. Wielandt: Finite Permutation Groups, Academic Press, New York, 1964.

    MATH  Google Scholar 

  46. F. Zirilli: Su una classe di k-archi di un piano di Galois. Rendiconti dell’Accademia Nazionale dei Lincei 54 (1973), 393–397.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The research of D. Bartoli was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA — INdAM). In addition, this material is based upon work supported by the National Science Foundation under Grant No. 2127742 (PI: G. Micheli). Part of this work was done while the first author was visiting the University of South Florida. We thank the referees for their valuable comments and suggestions to improve the paper’s quality.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Daniele Bartoli or Giacomo Micheli.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bartoli, D., Micheli, G. Algebraic Constructions of Complete m-Arcs Daniele Bartoli, Giacomo Micheli. Combinatorica (2022). https://doi.org/10.1007/s00493-021-4712-5

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00493-021-4712-5

Mathematics Subject Classification (2010)

  • 05B25
  • 51E21; 11R45
  • 51E20