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The non-existence of some NMDS codes and the extremal sizes of complete \((n,3)\)-arcs in \(PG(2,16)\)

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Abstract

The non-existence of \([29+h,3+h,26]_{16}\) and \([29+h,4+h,25]_{16}\)-codes, \(h\ge 0\), is proven. These results are obtained using geometrical methods, exploiting the equivalence between NMDS codes of dimension \(3\) and \((n,3)\)-arcs in \(PG(2,q)\). Along the way the packing problem for complete \((n,3)\)-arcs in \(PG(2,16)\) is solved, proving that \(m_{3}(2,16)=28\) and \(t_{3}(2,16)=15\) and that the complete \((28,3)\)-arc and the complete \((15,3)\)-arc are unique up to collineations.

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References

  1. Alderson T.L., Bruen A.A.: Maximal AMDS codes. Appl. Algebra Eng. Commun. Comput. 19, 87–98 (2008).

    Google Scholar 

  2. Ball S.: Multiple blocking sets and arcs in finite planes. J. Lond. Math. Soc. 54, 581–593 (1996).

    Google Scholar 

  3. Bartoli D., Davydov A.A., Marcugini S., Pambianco F.: The minimum order of complete caps in \(PG(4,4)\). Adv. Math. Commun. 5, 37–40 (2011).

    Google Scholar 

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

  5. Bartoli D., Marcugini S., Pambianco F.: New quantum caps in \(PG(4,4)\). J. Comb. Des. 20, 448–466 (2012).

    Google Scholar 

  6. Bartoli D., Bierbrauer J., Edel Y., Faina G., Marcugini S., Pambianco F.: The structure of quaternary quantum caps. Des. Codes Cryptogr. (2013). doi:10.1007/s10623-013-9796-5.

  7. Bartoli D., Davydov A.A., Faina G., Marcugini S., Pambianco F.: A new algorithm and a new type of estimate for the smallest size of complete arcs in \(PG(2,q)\). Electron. Notes Discret. Math. 40, 27–31 (2013).

  8. 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).

    Google Scholar 

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

    Google Scholar 

  10. Bartoli D., Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: Multiple coverings of the farthest-off points with small density from projective geometry (preprint).

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

  12. Bierbrauer J.: Coding Theory, Chapmpan & Hall/CRC, New York (2005).

  13. Bierbrauer J., Yves E.: \(41\) is the largest size of a cap in \(PG(4,4)\). Des. Codes Cryptogr. 16, 151–160 (1999).

  14. Bierbrauer J., Faina G., Marcugini S., Pambianco F.: On the structure of the \((n, r)\)-arcs in \(PG(2, q)\). In: Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory, Zvenigorod, pp. 19–23 (2006).

  15. Bose R.C., Bush K.A.: Orthogonal array of strength two and three. Ann. Math. Stat. 23, 508–524 (1952).

    Google Scholar 

  16. Braun M., Kohnert A., Wassermann A.: Construction of linear codes with prescribed distance. In: OC05 The Fourth International Workshop on Optimal Codes and Related Topics, Pamporovo, pp. 59–63 (2005).

  17. Chao J.M., Kaneta H.: Classical arcs in \(PG(r, q)\) for \(11\le q \le 19\). Discret. Math. 174, 87–94 (1997).

  18. Cook G.: Arcs in a finite projective plane. D.Phil. Thesis, University of Sussex (2011).

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

  20. Davydov A.A., Marcugini S., Pambianco F.: Minimal \(1\)-saturating sets and complete caps in binary projective spaces. J. Comb. Theory A 113, 647–663 (2006).

    Google Scholar 

  21. Davydov A.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).

  22. 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).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

  26. 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.) Proceedings of the Fourth Isle of Thorns Conference on Finite Geometries. Developments in Mathematics 3, pp. 201–246. Kluwer, Boston (2001).

  27. Kiss G., Marcugini S., Pambianco F.: On the spectrum of the sizes of semiovals in \(PG(2, q), q\) odd. Discret. Math. 310, 3188–3193 (2010).

  28. Marcugini S., Milani A., Pambianco F.: Maximal \((n,3)\)-arcs in \(PG(2,11)\). Discret. Math. 208/209, 421–426 (1999).

  29. Marcugini S., Milani A., Pambianco F.: NMDS codes of maximal length over \({\\\mathbf{F}}_{q}, 8\le q\le 11\). IEEE Trans. Inf. Theory 48, 963–966 (2002).

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

  31. Marcugini S., Milani A., Pambianco F.: Maximal \((n,3)\)-arcs in \(PG(2,13)\). Discret. Math. 294, 139–145 (2005).

  32. 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. Discret. Math. 307, 739–747 (2007).

  33. MinT online database. http://mint.sbg.ac.at. Accessed 15 Nov 2012.

  34. Pambianco F., Storme L.: Minimal blocking sets in \(PG(2,9)\). Ars Comb. 89, 223–234 (2008).

    Google Scholar 

  35. Singleton R.C.: Maximum distance \(q\)-nary codes. IEEE Trans. Inf. Theory 10, 116–118 (1964).

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

  37. Thas J.: Some results concerning \(((q+1)(n-1), n)\)-arcs. J. Comb. Theory A 19, 228–232 (1975).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Perugia, Perugia, Italy

    Daniele Bartoli, Stefano Marcugini & Fernanda Pambianco

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  1. Daniele Bartoli
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  2. Stefano Marcugini
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  3. Fernanda Pambianco
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Correspondence to Fernanda Pambianco.

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This is one of several papers published in Designs, Codes and Cryptography comprising the special topic on “Finite Geometries: A special issue in honor of Frank De Clerck”.

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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, 129–134 (2014). https://doi.org/10.1007/s10623-013-9837-0

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