Skip to main content
SpringerLink
Log in

Some New Results on Optimal Codes Over F5

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We present some results on almost maximum distance separable (AMDS) codes and Griesmer codes of dimension 4 over over the field of order 5. We prove that no AMDS code of length 13 and minimum distance 5 exists, and we give a classification of some AMDS codes. Moreover, we classify the projective strongly optimal Griesmer codes over F5 of dimension 4 for some values of the minimum distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. de Boer, Almost MDS codes, Designs, Codes and Cryptography, Vol. (9) (1996) pp. 143–155.

    Google Scholar 

  2. I. Boukliev, Some new optimal linear codes over F 5, In Proc. 25th Spring Conference of the Union of Bulgarian Mathematicians, Kazanlak, April 6- 9 (1996) pp. 81–85.

  3. I. Boukliev and S. Kapralov, The uniqueness of the Grismer [76; 4; 60; 5] code, 27th Spring Conference of the Union of Bulgarian Mathematicians, Pleven, April (1998) pp. 76–80.

  4. I. Boukliev and S. Kapralov, Classification of the Griesmer [49; 4; 36; 4] codes, April, Proceedings of the International Workshop ACCT, Pskov, Russia (1998) pp. 57–60.

  5. I. Boukliev, S. Kapralov, T. Maruta and M. Fukui, Optimal linear codes of dimension 4 over F 5, IEEE Trans. Info. Theory, Vol. (43), No. (1) (1997) pp. 308–313.

    Google Scholar 

  6. I. Bouyukliev and J. Simonis, Some new results for optimal ternary linear codes, in IEEE Trans, Info. Theory, Vol. 48, No. (4) (2002) pp. 981–985.

    Google Scholar 

  7. A. E. Brouwer, Bounds on the size of linear codes, In (V. Pless and W. C. Huffman, eds.) Handbook of Coding Theory, Elsevier, Amsterdam etc., ISBN:0-444-50088-X, 1998. Online version: http://www.win.tue.nl/math/dw/voorlincod.html.

  8. A. E. Brouwer and M. van Eupen, The correspondence between projective codes and 2-weight codes, Designs, Codes and Cryptography, Vol. (11) (1997) pp. 262–266.

    Google Scholar 

  9. A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. Vol. (18) (1986) pp. 97–122.

    Google Scholar 

  10. S. Dodunekov and I. Landgev, On near-MDS codes, J. of Geometry, Vol. (54) (1995) pp. 30–34.

    Google Scholar 

  11. S. Dodunekov and J. Simonis, Codes and projective multisets, Electron. J. Combin. Vol. (5), No. (1) (1998) p. 23 (electronic).

  12. M. van Eupen and P. Lisonek, Classification of some optimal ternary linear codes of small length, Des. Codes Cryptogr., Vol. (10) (1997) pp. 63–84.

    Google Scholar 

  13. M. van Eupen and V. Tonchev, Linear codes and the existence of a reversible Hadamard difference set in Z 2 × Z 2 × Z<pack> 4 5</pack>, Proc. ACCT96, Sozopol, Bulgaria, June 1- 7 (1996) pp. 295–301.

  14. P. Greenough and R. Hill, Optimal linear codes over GF(4), Discrete Math., Vol. (125) (1994) pp. 187–199.

    Google Scholar 

  15. J. H. Griesmer, A bound for error-correcting codes, I. B. M. J Res. Develop, Vol. (4) (1960) pp. 532–542.

    Google Scholar 

  16. N. Hamada and T. Helleseth, On the construction of [q3 − q 2 + 1; 4; q 3 − 2q 2 + q; q]-code meeting the Griesmer bound, In Proc. ACCT, Voneshta Voda, Bulgaria, June 22- 28 (1992) pp. 80–8

  17. N. Hamada, T. Helleseth and O. Ytrehus, There are exactly two nonequivalent [20; 5; 12; 3]-codes, Ars Comb. Vol. (35) (1993) pp. 3–14.

    Google Scholar 

  18. R. Hill, Optimal linear codes: Cryptography and Coding II, (C. Mitchell, ed.), Oxford University Press (1992) pp. 75–104.

  19. S. Kapralov, Classification of some optimal linear codes over GF(5), Proceedings of the International Workshop OCRT, Sozopol, Bulgaria (1998) pp. 151–157.

  20. I. Landgev, The geometry of (n; 3)-arcs in the projective plane of order 5, Proc. ACCT96, Sozopol, Bulgaria, June 1- 7 (1996) pp. 170–175.

  21. I. Landgev, T. Maruta and R. Hill, On the nonexistence of quaternary [51; 4; 37] codes, Finite Fields and their Applications, Vol. (2) (1996) pp. 96–110.

    Google Scholar 

  22. S. Marcugini, A. Milani and F. Pambianco, Existence and classification of NMDS codes over GF(5) and GF(7), Proceedings of the International Workshop ACCT, Bansko, Bulgaria (2000) pp. 232–239.

  23. G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. and Control, Vol. (8) (1965) pp. 170–179.

    Google Scholar 

  24. H. N. Ward, Divisibility of codes meeting the Griesmer bound, J. Comb. Theory Ser. A, Vol. (83) (1998) pp. 79–93.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, P.O. Box 323, 5000, V. Tarnovo, Bulgaria

    Iliya Bouyukliev

  2. Faculty of Information Technology and Systems, Department of Mediamatics, Delft University of Technology, P.O. Box 5031, 2600, GA, Delft, The Netherlands

    Juriaan Simonis

Authors
  1. Iliya Bouyukliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juriaan Simonis
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bouyukliev, I., Simonis, J. Some New Results on Optimal Codes Over F5 . Designs, Codes and Cryptography 30, 97–111 (2003). https://doi.org/10.1023/A:1024763410967

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024763410967

Search

Navigation