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Problems of Information Transmission

, Volume 55, Issue 3, pp 226–240 | Cite as

Divisible Arcs, Divisible Codes, and the Extension Problem for Arcs and Codes

  • I. LandjevEmail author
  • A. Rousseva
Coding Theory
  • 8 Downloads

Abstract

In an earlier paper we developed a unified approach to the extendability problem for arcs in PG(k - 1, q) and, equivalently, for linear codes over finite fields. We defined a special class of arcs called (t mod q)-arcs and proved that the extendabilty of a given arc depends on the structure of a special dual arc, which turns out to be a (t mod q)-arc. In this paper, we investigate the general structure of (t mod q)-arcs. We prove that every such arc is a sum of complements of hyperplanes. Furthermore, we characterize such arcs for small values of t, which in the case t = 2 gives us an alternative proof of the theorem by Maruta on the extendability of codes. This result is geometrically equivalent to the statement that every 2-quasidivisible arc in PG(k - 1, q), q ≥ 5, q odd, is extendable. Finally, we present an application of our approach to the extendability problem for caps in PG(3, q).

Key words

finite projective geometries arcs blocking sets divisible arcs quasidivisible arcs, Griesmer bound (t mod q)-arcs extendable arcs minihypers caps 

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Notes

Funding

This research has been supported by the Science Research Fund of Sofia University under Contract no. 80-10-81/15.04.2019.

References

  1. 1.
    Hirschfeld, J.W.P., Projective Geometries over Finite Fields, Oxford: Clarendon; New York: Oxford Univ. Press, 1998, 2nd ed.zbMATHGoogle Scholar
  2. 2.
    Landjev I. and Storme L., Linear Codes and Galois Geometries, Current Research Topics in Galois Geometries, Storme L. and De Beule J., Eds., New York: Nova Sci. Publ., 2012, pp. 187–214.zbMATHGoogle Scholar
  3. 3.
    MacWilliams F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz', 1979.zbMATHGoogle Scholar
  4. 4.
    Heise W. and Quattrocchi P., Informations- und Codierungstheorie: mathematische Grundlagen der Daten-Kompression und -Sicherung in diskreten Kommunikationssystemen, Berlin: Springer, 1995, 3rd ed.CrossRefGoogle Scholar
  5. 5.
    Tsfasman M.A., Vladut S.G., and Nogin, D.Yu., Algebraic Geometrie Codes: Basic Notions, Providence, R.I.: Amer. Math. Soc, 2007.CrossRefGoogle Scholar
  6. 6.
    Griesmer J.H., A Bound for Error-Correcting Codes, IBM J. Res. Develop., 1960, vol. 4, no. 5, pp. 532–542.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ward H.N., Divisible Codes-A Survey, Serdica Math. J., 2001, vol. 27, no. 4, pp. 263–278.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hill R. and Lizak P., Extensions of Linear Codes, in Proc. Int. Symp. on Information Theory (ISIT'1995), Whistler, BC, Canada, Sept. 17–22, 1995, p. 345.CrossRefGoogle Scholar
  9. 9.
    Hill R., An Extension Theorem for Linear Codes, Des. Codes Cryptogr., 1999, vol. 17, no. 1–3, pp 151–157.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Maruta T., On the Extendability of Linear Codes, Finite Fields Appl, 2001, vol. 7, no. 2, pp. 350–354.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Maruta T., Extendability of Linear Codes over GF(q) with Minimum Distance d, gcd(d, q) = 1, Discrete Math., 2003, vol. 266, no. 1–3, pp. 377–385.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Maruta T., A New Extension Theorem for Linear Codes, Finite Fields Appl, 2004, vol. 10, no. 4, pp. 674–685.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Maruta T., Extension Theorems for Linear Codes over Finite Fields, J. Geom., 2011, vol. 101, no. 1–2, pp. 173–183.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yoshida Y. and Maruta T., An Extension Theorem for [n, k, d]q Codes with gcd(d, q) = 2, Australas. J. Combin., 2010, vol. 48, pp. 117–131.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Landjev I., Rousseva A., and Storme L., On the Extendability of Quasidivisible Griesmer Arcs, Des. Codes Cryptogr., 2016, vol. 79, no. 3, pp. 535–547.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dodunekov S. and Simonis J., Codes and Projective Multisets, Electron. J. Combin., 1998, vol. 5, no. 1, Research Paper R37.Google Scholar
  17. 17.
    Landjev I., The Geometric Approach to Linear Codes, Finite Geometries (Proc. 4th Isle of Thorns Conf, Chelwood Gate, UK, July 16–21, 2000), Blokhuis A., Hirschfeld, J.W.P., Jungnickel D., and Thas J.A., Eds., Dordrecht: Kluwer, 2001, pp. 247–257.CrossRefGoogle Scholar
  18. 18.
    Hamada N., The Rank of the Incidence Matrix of Points and d-Flats in Finite Geometries, J. Sci. Hiroshima Univ. Ser. A-I Math., 1968, vol. 32, no. 2, pp. 381–396.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ceccherini P.V. and Hirschfeld, J.W.P., The Dimension of Projective Geometry Codes, Discrete Math., 1992, vol. 106/107, pp. 117–126.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Goethals, J.-M. and Delsarte P., On a Class of Majority-Logic Decodable Cyclic Codes, IEEE Trans. Inform. Theory, 1968, vol. 14, no. 2, pp. 182–188.MathSciNetCrossRefGoogle Scholar
  21. 21.
    MacWilliams F.J. and Mann H.B., On the p-Rank of the Design Matrix of a Difference Set, Inform. Control, 1968, vol. 12, no. 5, pp. 474–488.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Smith, K.J.C., On the p-Rank of the Incidence Matrix of Points and Hyperplanes in a Finfite Projective Geometry, J. Combin. Theory, 1969, vol. 7, no. 2, pp. 122–129.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Blokhuis A. and Moorhouse G.E., Some p-Ranks Related to Orthogonal Spaces, J. Algebraic Combin., 1995, vol. 4, no. 4, pp. 295–316.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ball S., Hill R., Landjev I., and Ward H., On (q 2 + q + 2, q + 2)-Arcs in the Projective Plane PG(2, q), Des. Codes Cryptogr., 2001, vol. 24, no. 2, pp. 205–224.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hirschfeld, J.W.P., Finite Projective Spaces in Three Dimensions, Oxford: Oxford Univ. Press, 1985.zbMATHGoogle Scholar
  26. 26.
    Bruen A., Baer Subplanes and Blocking Sets, Bull. Amer. Math. Soc, 1970, vol. 76, no. 2, pp. 342–344.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Polverino, O. Small Blocking Sets in PG(2, p 3), Des. Codes Cryptogr., 2000, vol. 20, no. 3, pp. 319–324.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sziklai P. and Szőnyi T., Blocking Sets and Algebraic Curves, Rend. Circ. Mat. Palermo (2) Suppl., 1998, no. 51, pp. 71–86.MathSciNetzbMATHGoogle Scholar
  29. 29.
    Landjev I. and Vandendriessche P., A Study of (xv t , x-y t-t)-Minihypers in PG(t, q), J. Combin. Theory Ser. A, 2012, pp. 119, no. 6, pp. 1123–1131.Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.New Bulgarian UniversitySofiaBulgaria
  2. 2.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria
  3. 3.Faculty of Mathematics and InformaticsSt. Kliment Ohridski Sofia UniversitySofiaBulgaria

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