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New types of estimates for the smallest size of complete arcs in a finite Desarguesian projective plane

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Abstract

New types of upper bounds for the smallest size t 2(2, q) of a complete arc in the projective plane PG(2, q) are proposed. The value \({t_{2}(2, q) = d(q)\sqrt{q} \ln q}\), where d(q) <  1 is a decreasing function of q, is used. The case \({d(q) < \alpha/ \ln{\beta q} + \gamma}\), where \({\alpha,\beta,\gamma}\) are positive constants independent of q, is considered. It is shown that

$$t_{2}(2, q) < (2/\,{\rm ln} \frac{1}{10}q + 0.32)\sqrt{q}\, {\rm ln}\, q\, {\rm if} \, q \leq 67993, q \,{\rm prime}, {\rm and }\, q \in R,$$

where R is a set of 27 values in the region 69997...110017. Also, for \({q \in [9311,67993]}\), q prime, and \({q \in R}\), it is shown that

$$\sqrt{q}({\rm ln}\, q)^{a_1-bq} < t_{2}(2, q) < \sqrt{q}({\rm ln}\, q)^{a_2-bq},$$

\({a_1=0.771, a_2=0.752, b=2.2 \cdot 10^{-7}.}\) In addition, our results allow us to conjecture that these estimates hold for all q. An algorithm FOP using any fixed order of points in PG(2, q) is proposed for constructing complete arcs. The algorithm is based on an intuitive postulate that PG(2, q) contains a sufficient number of relatively small complete arcs. It is shown that the type of order on the points of PG(2, q) is not relevant.

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References

  1. Abatangelo, V.: A class of complete \({[(q+8)/3]-arcs\,of PG(2, q), with\,q = 2^{h} and\,h (\geq6)}\) even. Ars Combin. 16, 103–111 (1983)

  2. Ali, A.H.: Classification of arcs in Galois plane of order thirteen. Ph.D. Thesis, University of Sussex (1993)

  3. Ball S.: On small complete arcs in a finite plane. Discrete Math. 174, 29–34 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bartoli, D., Davydov, A.A., Faina, G., Marcugini, S., Pambianco, F.: On sizes of complete arcs in PG(2, q). Discrete Math. 312, 680–698 (2012)

  5. Bartoli, D., Davydov, A.A., Faina, G., Marcugini, S., Pambianco, F.: New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane. J. Geom. 104, 11–43 (2013)

  6. Bartoli, D., Davydov, A.A., Faina, G., Marcugini, S., Pambianco, F.: Tables of sizes of small complete arcs in the plane \({PG(2, q), q \leq 190027}\), obtained by an algorithm with fixed order of points (FOP) (2014). http://arxiv.org/abs/1404.0469

  7. Bartoli D., Davydov A.A., Marcugini S., Pambianco F.: A 3-cycle construction of complete arcs sharing (q + 3)/2 points with a conic. Adv. Math. Commun. 7(3), 319–334 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bartoli D., Faina G., Marcugini S., Pambianco F., Davydov A.A.: A new algorithm and a new type of estimate for the smallest size of complete arcs in PG2, q. Electron. Notes Discrete Math. 40, 27–31 (2013)

    Article  Google Scholar 

  9. Bartoli D., Faina G., Marcugini S., Pambianco F.: On the minimum size of complete arcs and minimal saturating sets in projective planes. J. Geom. 104, 409–419 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cohen G., Litsyn S., Zemor G.: On greedy algorithms in coding theory. IEEE Trans. Inform. Theory 42, 2053–2057 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Davydov A.A., Faina G., Marcugini S., Pambianco F.: Computer search in projective planes for the sizes of complete arcs. J. Geom. 82, 50–62 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. DavydovA.A. , Faina G., Marcugini S., Pambianco F.: On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q). J. Geom. 94, 31–58 (2009)

    Article  MathSciNet  Google Scholar 

  13. Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: New inductive constructions of complete caps in PG(N, q), q even. J. Comb. Des. 18, 176–201 (2010)

    MathSciNet  Google Scholar 

  14. Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: Linear nonbinary covering codes and saturating sets in projective spaces. Adv. Math. Commun. 5, 119–147 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Davydov A.A., Marcugini S., Pambianco F.: Complete caps in projective spaces PG(n, q). J. Geom. 80, 23–30 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Faina G., Marcugini S., Milani A., Pambianco F.: The spectrum of the values k for which there exists a complete k-arc in PG2, q for q ≤  23. Ars Combin. 47, 3–11 (1997)

    MATH  MathSciNet  Google Scholar 

  17. Faina G., Pambianco F.: On the spectrum of the values k for which a complete k-cap in PG(n,q) exists. J. Geom. 62, 84–98 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Faina, G., Pambianco, F.: On some 10-arcs for deriving the minimum order for complete arcs in small projective planes. Discrete Math. 208–209, 261–271 (1999)

  19. Gács A., Szőnyi T.: Random constructions and density results. Des. Codes Cryptogr. 47, 267–287 (2008)

    Article  MathSciNet  Google Scholar 

  20. Giulietti M., Korchmáros G., Marcugini S., Pambianco F.: Transitive \({\mathbf{A}_{6}}\) -invariant k-arcs in PG2, q. Des. Codes Cryptogr. 68, 73–79 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Giulietti, M., Ughi, E.: A small complete arc in PG(2, q), q =  p 2, \({p \equiv 3 (mod 4)}\). Discrete Math. 208–209, 311–318 (1999)

  22. Gordon C.E.: Orbits of arcs in PG(N, K) under projectivities. Geom. Dedicata 42, 187–203 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hartman A., Raskin L.: Problems and algorithms for covering arrays. Discrete Math. 284, 149–156 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hirschfeld, J.W.P.: Maximum sets in finite projective spaces>. In: Lloyd, E.K. (ed.) Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., vol. 82, pp. 55–76. Cambridge University Press, Cambridge (1983)

  25. Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn. Clarendon Press, Oxford (1998)

    Google Scholar 

  26. Hirschfeld J.W.P., Sadeh A.: The projective plane over the field of eleven elements. Mitt. Math. Sem. Giessen 164, 245–257 (1984)

    MathSciNet  Google Scholar 

  27. Hirschfeld, J.W.P., Storme, L.: The packing problem in statistics, coding theory and finite geometry: update 2001. In: Blokhuis, A., Hirschfeld, J.W.P., Jungnickel, D., Thas, J.A. (eds.) Finite Geometries, Developments of Mathematics, vol. 3, Proc. of the Fourth Isle of Thorns Conf., Chelwood Gate, 2000, pp. 201–246. Kluwer Academic Publisher, Boston (2001)

  28. Keri G.: Types of superregular matrices and the number of n-arcs and complete n-arcs in PG(r, q). J. Comb. Des. 14, 363–390 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Korchmáros G.: New examples of complete k-arcs in PG(2, q). Eur. J. Combin. 4, 329–334 (1983)

    Article  MATH  Google Scholar 

  31. Lisonek, P.: Computer-assisted studies in algebraic combinatorics. Ph.D. thesis, Research Institute for Symbolic Computation, J. Kepler Univ. Linz, 1994, RISC-Linz Report Series No. 94-68

  32. Lombardo-Radice L.: Sul problema dei k-archi completi di S 2, q . Boll. Unione Mat. Ital. 11, 178–181 (1956)

    MATH  MathSciNet  Google Scholar 

  33. Marcugini S., Milani A., Pambianco F.: Minimal complete arcs in PG(2, q), q ≤  29. J. Combin. Math. Combin. Comput. 47, 19–29 (2003)

    MATH  MathSciNet  Google Scholar 

  34. Marcugini S., Milani A., Pambianco F.: Complete arcs in PG(2, 25): The spectrum of the sizes and the classification of the smallest complete arcs. Discrete Math. 307, 739–747 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Monroe L.: Binary greedy codes. Congressus Numerantium 104, 49–63 (1994)

    MATH  MathSciNet  Google Scholar 

  36. Monroe, L., Pless, V.: Greedy generation of non-binary codes. In: Proc. IEEE Int. Symp. Inform. Theory, ISIT 1995

  37. Östergård, P.R.J.: Computer search for small complete caps. J. Geom. 69, 172–179 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pambianco F., Bartoli D., Faina G., Marcugini S.: Classification of the smallest minimal 1-saturating sets in PG(2, q), q ≤  23. Electron. Notes Discrete Math. 40, 229–233 (2013)

    Article  Google Scholar 

  39. Pambianco F., Davydov A.A., Bartoli D., Giulietti M., Marcugini S.: A note on multiple coverings of the farthest-off points. Electron. Notes Discrete Math. 40, 289–293 (2013)

    Article  Google Scholar 

  40. Pellegrino G.: Un’osservazione sul problema dei k-archi completi in S 2, q , con \({q \equiv 1}\) (mod4). Atti Accad. Naz. Lincei Rend. 63, 33–44 (1977)

    MathSciNet  Google Scholar 

  41. Pellegrino G.: Sugli archi completi dei piani PG(2, q), con q dispari, contenenti (q +  3)/2 punti di una conica. Rend. Mat. 12, 649–674 (1992)

    MATH  MathSciNet  Google Scholar 

  42. Pellegrino G.: Archi completi, contenenti (q +  1)/2 punti di una conica, nei piani di Galois di ordine dispari. Rend. Circ. Mat. Palermo 62(2), 273–308 (1993)

    Article  MathSciNet  Google Scholar 

  43. Pless, V.S.: Coding constructions. In: Pless, V.S., Human, W.C., Brualdi, R.A. (eds.) Handbook of Coding Theory, pp. 141–176. Elsevier, Amsterdam (1998). Section 7

  44. Polverino, O.: Small minimal blocking sets and complete k -arcs in PG(2,p 3). Discrete Math. 208–209, 469–476 (1999)

  45. Segre B.: Le geometrie di Galois. Ann. Mat. Pura Appl. 48, 1–97 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  46. Segre B.: Ovali e curve \({\sigma}\) nei piani di Galois di caratteristica due. Atti Accad. Naz. Lincei Rend. 32, 785–790 (1962)

    MATH  MathSciNet  Google Scholar 

  47. Segre B.: Introduction to Galois geometries. Atti Accad. Naz. Lincei Mem. 8, 133–236 (1967)

    MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  49. Szőnyi T.: Note on the order of magnitude of k for complete k-arcs in PG2, q. Discrete Math. 66, 279–282 (1987)

    Article  MathSciNet  Google Scholar 

  50. Szőnyi, T. Complete arcs in Galois planes: survey. In: Quaderni del Seminario di Geometrie Combinatorie, vol. 94. Università degli studi di Roma, La Sapienza (1989)

  51. Szőnyi, T.: Arcs, caps, codes and 3-independent subsets. In: Faina, G., Tallini, G. (eds.) Giornate di Geometrie Combinatorie, pp. 57–80. Università degli studi di Perugia, Perugia (1993)

  52. Szőnyi, T.: Some applications of algebraic curves in finite geometry and combinatorics. In: Bailey, R.A. (ed.) Surveys in Combinatorics, pp. 198–236. Cambridge University Press, Cambridge (1997)

  53. Tallini, G.: Le geometrie di Galois e le loro applicazioni alla statistica e alla teoria delle informazioni. Rend. Mat. Appl. 19, 379–400 (1960)

  54. Thas, J.A.: M.D.S. codes and arcs in projective spaces: a survey. Le Matematiche (Catania) 47, 315–328 (1992)

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

    Article  MATH  MathSciNet  Google Scholar 

  56. van Zanten, A.J., Suparta, I.N.: On the construction of linear q-ary lexicodes. Des. Codes Crypt. 37, 15–29 (2005)

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy

    Daniele Bartoli, Giorgio Faina, Stefano Marcugini & Fernanda Pambianco

  2. Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol’shoi Karetnyiper. 19, GSP-4, Moscow, 127994, Russian Federation

    Alexander A. Davydov

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  1. Daniele Bartoli
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  2. Alexander A. Davydov
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Correspondence to Daniele Bartoli.

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Bartoli, D., Davydov, A.A., Faina, G. et al. New types of estimates for the smallest size of complete arcs in a finite Desarguesian projective plane. J. Geom. 106, 1–17 (2015). https://doi.org/10.1007/s00022-014-0224-4

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