Skip to main content
SpringerLink
Log in

On the Minimum Length of some Linear Codes of Dimension 5

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

Abstract

In this paper, we shall prove that the minimum length n q (5,d) is equal to g q (5,d) +1 for q4−2q2−2q+1≤ d≤ q4 − 2q2 − q and 2q4 − 2q3 − q2 − 2q+1 ≤ d ≤ 2q4−2q3−q2−q, where g q (5,d) means the Griesmer bound \({\sum_{i = 0}^{4}} \lceil {\frac{d}{q^{i}}}\rceil\).

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. Cheon E.J., Kato T., Kim S.J., Nonexistence of [n, 5, d] q codes attaining the Griesmer bound for q4−2q2−2q+1 ≤ d ≤ q4−2q2−q Des. Codes Cryptogr., to appear.

  2. E. J. Cheon, T. Kato and S. J. Kim, Nonexistence of a [g q (5,d), 5, d] q codes for 3q4−4q3−2q+1≤ d≤ 3q4−4q3−q, Discrete Math, to appear.

  3. N. Hamada (1993) ArticleTitleA characterization of some [n, k, d ; q] codes meeting the Griesmer bound using a minihyper in a finite projective geometry Discrete Math 116 229–268 Occurrence Handle10.1016/0012-365X(93)90404-H

    Article  Google Scholar 

  4. R. Hill (1992) Optimal linear codes C. Mitchell (Eds) Cryptography and Coding II Oxford University Press Oxford 75–104

    Google Scholar 

  5. J. W. P. Hirschfeld (1998) Projective Geometries over Finite Fields Clarendon Press Oxford

    Google Scholar 

  6. I. N. Landjev, Nonexistence of [143, 5, 94]3 codes, Proceedings of International Workshop on Optimal Codes and Related Topics, Sozopol, Bulgaria (1995) pp. 108–116.

  7. I. N. Landjev T. Maruta (1999) ArticleTitleOn the minimum length of quaternary linear codes of dimension five Discrete Math. 202 145–161 Occurrence Handle10.1016/S0012-365X(98)00354-9

    Article  Google Scholar 

  8. T. Maruta (2001) ArticleTitleOn the nonexistence of q-ary linear codes of dimension five Des., Codes Cryptogr. 22 165–177

    Google Scholar 

  9. T. Maruta (2001) ArticleTitleThe nonexistence of some quaternary linear codes of dimension 5 Discrete Math. 238 99–113 Occurrence Handle10.1016/S0012-365X(00)00413-1

    Article  Google Scholar 

  10. V. Pless W. C. Huffman (1998) Handbook of Coding Theory Elsevier Science B.V Netherlands

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Gyeongsang National University, Jinju, 660-701, Korea

    E. J. Cheon

  2. Department of Mathematical Sciences, Yamaguchi University, Yamaguchi, 753-8512, Japan

    T. Kato

  3. Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea

    S. J. Kim

Authors
  1. E. J. Cheon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Kato
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. J. Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. J. Cheon.

Additional information

Communicated by: J.D. Key

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheon, E.J., Kato, T. & Kim, S.J. On the Minimum Length of some Linear Codes of Dimension 5. Des Codes Crypt 37, 421–434 (2005). https://doi.org/10.1007/s10623-004-4034-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-004-4034-9

Keywords

MSC 2000

Search

Navigation