Advertisement

Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 841–854 | Cite as

Linear codes close to the Griesmer bound and the related geometric structures

  • Assia RoussevaEmail author
  • Ivan Landjev
Article
  • 58 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

In this paper, we study the behavior of the function \(t_q(k)\) defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k:
$$\begin{aligned} t_q(k)=\max _d(n_q(k,d)-g_q(k,d)), \end{aligned}$$
where the maximum is taken over all minimum distances d. Here \(n_q(k,d)\) is the shortest length of a q-ary linear code of dimension k and minimum distance d, \(g_q(k,d)\) is the Griesmer bound for a code of dimension k and minimum distance d. We give two equivalent geometric versions of this problem in terms of arcs and minihypers. We prove that \(t_q(k)\rightarrow \infty \) when \(k\rightarrow \infty \) which implies that the problem is non-trivial. We prove upper bounds on the function \(t_q(k)\). For codes of even dimension k we show that \(t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)\) which implies that \(t_q(k)\in O(q^{k/2})\) for all k. For three-dimensional codes and q even we prove the stronger estimate \(t_q(3)\le \log q-1\), as well as \(t_q(3)\le \sqrt{q}-1\) for the case q square.

Keywords

Griesmer bound Optimal linear codes Arcs Minihypers 

Mathematics Subject Classification

94B65 51E21 51E22 94B05 94B27 51E15 51E23 

Notes

Acknowledgements

This result was partially supported by the Research Scientific Fund of Sofia University “St. Kl. Ohridski” under Contract No. 80-10-55/19.04.2017.

References

  1. 1.
    Ball S.: Table of bounds on three dimensional linear codes or (n,r) arcs in PG(2,q). https://mat-web.upc.edu/people/simeon.michael.ball/codebounds.html.
  2. 2.
    Ball S., Hill R., Landjev I., Ward H.N.: On \((q^2+q+2, q+2)\)-arcs in the projective plane \(PG(2, q)\). Des. Codes Cryptogr. 24, 205–224 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Belov B.I., Logachev V.N., Sandimirov V.P.: Construction of a class of linear binary codes achieving the Varshamov–Griesmer bound. Probl. Inf. Transm. 10(3), 211–217 (1974).Google Scholar
  4. 4.
    Brouwer A.: Bounds on the minimum distance of linear codes. In: Pless V., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 295–461. Elsevier, New York (1998).Google Scholar
  5. 5.
    Dodunekov S.: Optimal codes, DSc Thesis, Institute of Mathematics, Sofia (1985).Google Scholar
  6. 6.
    Dodunekov S., Simonis J.: Codes and projective multisets. Electron. J. Comb. 5(1), 37 (1998).MathSciNetzbMATHGoogle Scholar
  7. 7.
    Grassl M.: Bounds on the minimum dostance of linear odes and quantum codes. http://www.codetable.de.
  8. 8.
    Griesmer J.H.: A bound for error-correcting codes. IBM J. Res. Dev. 4, 532–542 (1960).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hamada N., Maruta T.: Note on an improvement of the Griesmer bound for \(q\)-ary linear codes. Serdica J. Comput. 5, 199–206 (2011).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hamada N., Maruta T.: A survey of recent results on optimal linear codes and minihypers, manuscript.Google Scholar
  11. 11.
    Hill R.: A First Course in Coding Theory. The Clarendon Press, Oxford University Press, New York (1986).zbMATHGoogle Scholar
  12. 12.
    Hill R.: Optimal Linear Codes, Cryptography and Coding II, pp. 75–104. Oxford University Press, New York (1992).Google Scholar
  13. 13.
    Hill R., Mason J.R.M.: On \((k,n)\)-arcs and the Falsity of the Lunelli-Sce Conjecture, pp. 153–168. Finite Geometries and Designs, London Mathematical Society Lecture Note Series 49Cambridge University Press, Cambridge (1981).zbMATHGoogle Scholar
  14. 14.
    Hirschfeld J., Storme L.: The packing problem in statistics, coding theory and finite projective spaces. In: Blokhuis A., et al. (eds.) Finite Geometries, pp. 201–246. Kluwer, Alphen aan den Rijn (2001).Google Scholar
  15. 15.
    Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Pless, Cambridge (2003).CrossRefzbMATHGoogle Scholar
  16. 16.
    Klein A.: On codes meeeting the Griesmer bound. Des. Codes Cryptogr. 274, 289–297 (2004).zbMATHGoogle Scholar
  17. 17.
    Klein A., Metsch K.: Parameters for which the Griesmer bound is not sharp. Discret. Math. 307, 2695–2703 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Landjev I., Storme L.: Galois geometries and coding theory, chapter 8. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry, pp. 187–214. NOVA Science Publishers, Hauppauge (2012).Google Scholar
  19. 19.
    Landjev I., Vandendriesche P.: A study of \((xv_t, xv_{t-1})\) in \(\text{ PG }(t, q)\). J. Comb. Theory Ser. A 119, 1123–1131 (2012).CrossRefzbMATHGoogle Scholar
  20. 20.
    Ling S., Xing C.: Coding Theory, a First Course. Cambridge University Press, Cambridge (2004).CrossRefGoogle Scholar
  21. 21.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North Holland Publishing Co., Amsterdam (1977).zbMATHGoogle Scholar
  22. 22.
    Maruta T.: On the nonexistence of Griesmer codes attaining the Griesmer bound. Geom. Dedicata 60, 1–7 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Maruta T.: On the achievement of the Griesmer bound. Des. Codes Cryptogr. 12, 83–87 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Maruta T.: Giesmer bound for linear codes over finite fields. http://www.mi.s.osakafu-u.ac.jp/~maruta/griesmer/.
  25. 25.
    Solomon G., Stiffler J.J.: Algebraically punctured cyclic codes. Inf. Control 8, 170–179 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vladut S., Nogin D., Tsfasman M.: Algebro-Geometric Codes. MCNMO, Independent Moscow University (2003) (in Russian).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sofia UniversitySofiaBulgaria
  2. 2.New Bulgarian UniversitySofiaBulgaria
  3. 3.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations