Skip to main content

On a characterization of some minihypers in PG(t,q) (q=3 or 4) and its applications to error-correcting codes

Part of the Lecture Notes in Mathematics book series (LNM,volume 1518)

Abstract

A set F of f points in a finite projective geometry PG(t, q) is an {f, m; t, q}-minihyper if m (≥0) is the largest integer such that all hyperplanes in PG(t, q) contain at least m points in F where t≥2, f≥1 and q is a prime power. Hamada and Deza [9], [11] characterized all {2v α+1+2v β+1, 2v α+2v β;t, q}-minihypers for any integers t,q,α and β such that q≥5 and 0≤α<β<t where v l =(q l−1)/(q−1) for any integer l≥0. Recently, Hamada [5], [6] and Hamada, Helleseth and Ytrehus [18] characterized all {2v1+2v2, 2v0+2v1;t, q}-minihypers for the case t≥2 and q ∈ {3, 4}. The purpose of this paper is to characterize all {2v α+1+2v β+1, 2v α+2v β;t, q}-minihypers for any integers t,q,α and β such that q ∈ {3, 4}, 0≤α<β<t and βα+1 using several results in Hamada and Helleseth [12], [13], [14], [16], [17].

Keywords

  • Discrete Math
  • Linear Code
  • Noncollinear Point
  • Optimal Linear Code
  • Nonsingular Diagonal Matrix

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Partially supported by Grant-in-aid for Scientific Research of the Ministry of Education, Science and Culture under Contract Numbers 403-4005-02640182.

Partially supported by the Scandinavia Japan Sasakawa Foundation. Partially supported by the Norwegian Research Council for Science and the Humanities

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.A. Bruen and R. Silverman, Arcs and blocking sets II, Europ. J. Combin. 8 (1987), 351–356.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. J.H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop. 4 (1960), 532–542.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. N. Hamada, Characterization resp. nonexistence of certain q-ary linear codes attaining the Griesmer bound, Bull. Osaka Women's Univ. 22 (1985), 1–47.

    MathSciNet  Google Scholar 

  4. N. Hamada, Characterization of min·hypers in a finite projective geometry and its applications to error-correcting codes, Bull. Osaka Women's Univ. 24 (1987), 1–24.

    MathSciNet  Google Scholar 

  5. N. Hamada, Characterization of {12, 2; 2, 4}-min·hypers in a finite projective geometry PG(2, 4), Bull. Osaka Women's Univ. 24 (1987), 25–31.

    Google Scholar 

  6. N. Hamada, Characterization of {(q+1)+2, 1; t, q}-min·hypers and {2(q+1)+2, 2; 2, q}-min·hypers in a finite projective geometry, Graphs and Combin. 5 (1989), 63–81.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. N. Hamada, A characterization of some (n, k, d; q)-codes meeting the Griesmer bound using a minihyper in a finite projective geometry, to appear in Discrete Math., In Chapter 4 in “Combinatorial Aspect of Design Experiments”.

    Google Scholar 

  8. N. Hamada and M. Deza, A characterization of some (n, k, d; q)-codes meeting the Griesmer bound for given integers k≥3, q≥5 and d=q k−1q αq γ(0≤gaβ<γ<k−1 or 0≤α<βγ<k−1). In: First Sino-Franco Conference on Combinatorics, Algorithms, and Coding Theory, Bull. Inst. Math. Academia Sinica 16 (1988), 321–338.

    MathSciNet  Google Scholar 

  9. N. Hamada and M. Deza, Characterization of {2(q+1)+2, 2; t, q}-min·hyper in PG(t, q) (t≥3, q≥5) and its applications to error-correcting codes, Discrete Math. 71 (1988), 219–231.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. N. Hamada and M. Deza, A characterization of {v μ+1+ε,v μ;t, q}-min·hyper and its applications to error-correcting codes and factorial designs, J. Statist. Plann. Inference 22 (1989), 323–336.

    CrossRef  MathSciNet  Google Scholar 

  11. N. Hamada and M. Deza, A characterization of {2v α+1+2v β+1,2v α +2v β; t,q}-minhypers in PG(t, q) (t≥2, q≥5 and 0≤α<β<t) and its applications to error-correcting codes, Discrete Math. 91 (1991), xxx–xxx.

    MathSciNet  Google Scholar 

  12. N. Hamada and T. Helleseth, A characterization of some {3v 2, 3v 1; t, q}-minihypers and some {2v 2+v γ+1,2v 1+v γ; t, q}-minihypers (q=3 or 4, 2,≤γ<t) and its applications to error-correcting codes, Bull. Osaka Women's Univ. 27 (1990), 49–107.

    Google Scholar 

  13. N. Hamada and T. Helleseth, A characterization of some minihypers in a finite projective geometry PG(t, 4), Europ. J. Combin. 11 (1990) 541–548.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. N. Hamada and T. Helleseth, A characterization of some linear codes over GF(4) meeting the Griesmer bound, to appear in Mathematica Japonica 37 (1992).

    Google Scholar 

  15. N. Hamada and T. Helleseth, A characterization of some {3v μ+13v μ; k−1, q}-minihypers and some (n, k, q k−1−3q μ; q)-codes (k≥3, q≥5, 1≤μ<k−1) meeting the Griesmer bound, submitted for publication.

    Google Scholar 

  16. N. Hamada and T. Helleseth, A characterization of some {2v α+1+v γ+1,2v α+v γ; k−1, 3}-minihypers and some (n, k, 3k−1−2 · 3α−3γ; 3)-codes (k≥3, 0≤α<γ<k−1) meeting the Griesmer bound, to appear in Discrete Math.

    Google Scholar 

  17. N. Hamada and T. Helleseth, A characterization of some minihypers in PG(t, 3) and some ternary codes meeting the Griesmer bound, submitted for publication.

    Google Scholar 

  18. N. Hamada, T. Helleseth and Ø. Ytrehus, Characterization of {2(q+1)+2, 2; t, q}-minihypers in PG(t, q) (t≥3, q ∈ {3, 4}}), to appear in Discrete Math.

    Google Scholar 

  19. N. Hamada and F. Tamari, On a geometrical method of construction of maximal t-linearly independent sets, J. Combin. Theory 25 (A) (1978), 14–28.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. N. Hamada and F. Tamari, Construction of optimal linear codes using flats and spreads in a finite projective geometry, Europ. J. Combin. 3 (1982), 129–141.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. T. Helleseth, A characterization of codes meeting the Griesmer bound, Inform. and Control 50 (1981), 128–159.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. R. Hill, Caps and codes, Discrete Math. 22 (1978), 111–137.

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. F. Tamari, A note on the construction of optimal linear codes, J. Statist. Plann. Inference 5 (1981), 405–411.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. F. Tamari, On linear codes which attain the Solomon-Stiffler bound, Discrete Math. 49 (1984), 179–191.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Hamada, N., Helleseth, T. (1992). On a characterization of some minihypers in PG(t,q) (q=3 or 4) and its applications to error-correcting codes. In: Stichtenoth, H., Tsfasman, M.A. (eds) Coding Theory and Algebraic Geometry. Lecture Notes in Mathematics, vol 1518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087992

Download citation

  • DOI: https://doi.org/10.1007/BFb0087992

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55651-0

  • Online ISBN: 978-3-540-47267-4

  • eBook Packages: Springer Book Archive