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New (n, r)-arcs in PG(2, 17), PG(2, 19), and PG(2, 23)

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Abstract

An (n, r)-arc is a set of n points of a projective plane such that some r but no r+1 of them are collinear. The maximum size of an (n, r)-arc in PG(2, q) is denoted by m r (2, q). In this paper a new (95, 7)-arc, (183, 12)-arc, and (205, 13)-arc in PG(2, 17) are constructed, as well as a (243, 14)-arc and (264, 15)-arc in PG(2, 19). Likewise, good large (n, r)-arcs in PG(2, 23) are constructed and a table with bounds on m r (2, 23) is presented. In this way many new 3-dimensional Griesmer codes are obtained. The results are obtained by nonexhaustive local computer search.

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References

  1. Ball, S., Multiple Blocking Sets and Arcs in Finite Planes, J. London Math. Soc., 1996, vol. 54, no. 3, pp. 427–435.

    Google Scholar 

  2. Daskalov, R., On the Maximum Size of Some (k, r)-Arcs in PG(2, q), Discrete Math., 2008, vol. 308, no. 4, pp. 565–570.

    Article  MathSciNet  MATH  Google Scholar 

  3. Hirschfeld, J.W.P. and Storme, L., The Packing Problem in Statistics, Coding Theory, and Finite Projective Spaces: Update 2001, Finite Geometries (Proc. 4th Isle of Thorns Conf.), Blokhuis, A., Hirschfeld, J.W.P., Jungnickel, D., and Thas, J.A., Eds., Dordrecht: Kluwer, 2001, pp. 201–246.

    Google Scholar 

  4. Daskalov, R., On the Existence and the Nonexistence of Some (k, r)-Arcs in PG(2, 17), in Proc. 9th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT-9), Kranevo, Bulgaria, 2004, pp. 95–100.

  5. Daskalov, R. and Metodieva, E., New (k, r)-Arcs in PG(2, 17) and the Related Optimal Linear Codes, Math. Balkanica (N.S.), 2004, vol. 18, no. 1–2, pp. 121–127.

    MathSciNet  MATH  Google Scholar 

  6. Braun, M., Kohnert, A., and Wassermann, A., Construction of (n, r)-Arcs in PG(2, q), Innov. Incidence Geom., 2005, vol. 1, pp. 133–141.

    MathSciNet  MATH  Google Scholar 

  7. Ball, S. and Hirschfeld, J.W.P., Bounds on (n, r)-Arcs and Their Application to Linear Codes, Finite Fields Appl., 2005, vol. 11, no. 3, pp. 326–336.

    Article  MathSciNet  MATH  Google Scholar 

  8. Hill, R., Optimal Linear Codes, Cryptography and Coding II (Proc. 2nd IMA Conf. on Cryptography and Coding, Cirencester, U.K., 1989), Mitchel, C., Ed., Oxford: Oxford Univ. Press, 1992, pp. 75–104.

    Google Scholar 

  9. Bose, R.C., Mathematical Theory of the Symmetrical Factorial Design, Sankhyā, 1947, vol. 8, no. 2, pp. 107–166.

    MATH  Google Scholar 

  10. Barlotti, A., Some Topics in Finite Geometrical Structures, Mimeo Series, no. 439, Chapel Hill: Univ. of North Carolina, Inst. of Statistics, 1965.

    Google Scholar 

  11. Kohnert, A., Arcs in the Projective Planes [on-line tables]. Available at http://www.algorithm.uni-bayreuth.de/en/research/Coding-Theory/PG-arc-table/index.html.

  12. Daskalov, R. and Metodieva, E., Good (n, r)-Arcs in PG(2, 23), in Proc. 6th Int. Workshop on Optimal Codes and Related Topics (OC’2009), Varna, Bulgaria, 2009, pp. 69–74.

  13. Daskalov, R. and Metodieva, E., Multiple Blocking Sets in PG(2, 23), in Proc. 6th Int. Workshop on Optimal Codes and Related Topics (OC’2009), Varna, Bulgaria, 2009, pp. 75–80.

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Authors and Affiliations

  1. Department of Mathematics, Technical University of Gabrovo, Gabrovo, Bulgaria

    R. Daskalov & E. Metodieva

Authors
  1. R. Daskalov
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  2. E. Metodieva
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Correspondence to R. Daskalov.

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Original Russian Text © R. Daskalov, E. Metodieva, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 3, pp. 3–9.

Supported in part by the Bulgarian Ministry of Education and Science under contract in TU-Gabrovo.

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Daskalov, R., Metodieva, E. New (n, r)-arcs in PG(2, 17), PG(2, 19), and PG(2, 23). Probl Inf Transm 47, 217–223 (2011). https://doi.org/10.1134/S003294601103001X

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  • DOI: https://doi.org/10.1134/S003294601103001X

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