Skip to main content
SpringerLink
Log in

The Length of Primitive BCH Codes with Minimal Covering Radius

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

Abstract

It is proved that the covering radius of a primitive binary BCH code of length q-1 and designed distance 2t+1, where EquationSource % MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabg2 % da9iaaikdadaahaaWcbeqaaiaad2gaaaGccqGH+aGpdaWadaqaamaa % bmaabaGaaGOmaiaadshacqGHsislcaaIZaaacaGLOaGaayzkaaWaae % WaaeaacaaIYaGaamiDaiabgkHiTiaaigdaaiaawIcacaGLPaaacaGG % HaaacaGLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaOGaaiilaaaa!48DE! is exactly 2t-1 (the minimum value possible). The bound for q is significantly lower than the one obtained by O. Moreno and C. J. Moreno [9].

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. T. Berger and J. van der Horst, Complete decoding of triple-error-correcting BCH codes, IEEE Trans. Inform. Theory, Vol. 22. (1976) pp. 136–146.

    Google Scholar 

  2. G. D. Cohen, M. G. Karpovsky, H. F. Mattson and J. R. Schatz, Covering radius—survey and recent results, IEEE Trans. Inform. Theory, Vol. 31 (1985) pp. 328–342.

    Google Scholar 

  3. S. D. Cohen, Polynomial factorisation and the construction of regular directed graphs with small diameter, submitted, 1993.

  4. D. C. Gorenstein, W. W. Peterson and N. Zierler, Two error-correcting Bose-Chaudhuri codes are quasi-perfect, Inform. Control, Vol. 3, (1960) pp. 291–294.

    Google Scholar 

  5. R. Göttfert and H. Niederreiter, Hasse-Teichmuller derivatives and products of linear recurring sequences, Contemporary Math., Vol. 168 (1994).

  6. T. Helleseth, On covering radius of cyclic linear codes and arithmetic codes, Discr. Appl. Math., Vol. 11 (1985) pp. 157–173; for the addition of I. E. Shparlinski, see Kiberneticheskii Sbornik., Vol. 25 (1985) pp. 82–84 (Russian).

    Google Scholar 

  7. F. J. McWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977).

  8. H. F. Mattson and J. R. Schatz, A brief survey of covering radius, Ann. Discr. Math., Vol. 18 (1983) pp. 617–624.

    Google Scholar 

  9. C. J. Moreno and O. Moreno, Constructive elementary approach to the covering radius of long BCH codes, conference abstract, (1993).

  10. A. N. Skorobogatov and S. G. Vladuts, The covering radius of long binary BCH codes, Problemy Peredachi Informatsii, Vol. 25 (1989) pp. 38–45 (Russian).

    Google Scholar 

  11. A. Tietäväinen, On the covering radius of long binary BCH codes, Discr. App. Math., Vol. 16 (1987) pp. 75–77.

    Google Scholar 

  12. A. Tietäväinen, An asymptotic bound on the covering radius of binary BCH codes, IEEE Trans. Inform. Theory, Vol. 36 (1990) pp. 211–213.

    Google Scholar 

Download references

Authors
  1. Stephen D. Cohen
    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

Cohen, S.D. The Length of Primitive BCH Codes with Minimal Covering Radius. Designs, Codes and Cryptography 10, 5–16 (1997). https://doi.org/10.1023/A:1008299101833

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008299101833

Search

Navigation