Skip to main content
SpringerLink
Log in

On the intersection of binary linear codes

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

For a binary linear code, a new relation between the intersection and (2, 2)-separating property is addressed, and a relation between the intersection and the trellis complexity is also given. Using above relations, the authors will apply several classes of binary codes to secret sharing scheme and determine their trellis complexity and separating properties. The authors also present the properties of the intersection of certain kinds of two-weight binary codes. By using the concept of value function, the intersecting properties of general binary codes are described.

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. Ashikhmin A and Barg A, Minimal vectors in linear codes, IEEE Trans. Inform. Theory, 1998, 44(5): 2010–2017.

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlet C, Ding C, and Yuan J, Linear codes from highly perfect nonlinear functions and their secret sharing schemes, IEEE Trans. Inform. Theory, 2005, 51(6): 2089–2102.

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen G, Encheva S, Litsyn S, and Schaathun H G, Intersecting codes and separating codes, Discrete Applied Mathematics, 2003, 128(1): 75–83.

    Article  MathSciNet  MATH  Google Scholar 

  4. Cohen G, Encheva S, and Schaathun H G, More on (2,2)-separating systems, IEEE Trans. Inform. Theory, 2002, 48(9): 2606–2609.

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohen G and Zémor G, Intersecting codes and independent families, IEEE Trans. Inform. Theory, 1994, 40(6): 1872–1881.

    Article  MATH  Google Scholar 

  6. Forney G D, Coset codes II: Binary lattices and related codes, IEEE Trans. Inform. Theory, 1988, 34(5): 1152–1187.

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen W D and Kløve T, The weight hierarchies of q-ary codes of dimension 4, IEEE Trans. Inform. Theory, 1996, 42(6): 2265–2272.

    Article  MathSciNet  MATH  Google Scholar 

  8. Vardy A and Be’ery Y, Maximum-likelihood soft decision decoding of BCH codes, IEEE Trans. Inform. Theory, 1994, 40(2): 546–554.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ytrehus Ø, On the trellis complexity of certain binary linear block codes, IEEE Trans. Inform. Theory, 1995, 41(2): 559–560.

    Article  MathSciNet  MATH  Google Scholar 

  10. Li Z H, Xue T, and Lai H, Secret sharing schemes from binary linear codes, Inform. Sci., 2010, 180: 4412–4419.

    Article  MathSciNet  MATH  Google Scholar 

  11. Calderbank A R and Goethals J M, Three-weight codes and association schemes, Philips J. Res., 1984, 39: 143–152.

    MathSciNet  MATH  Google Scholar 

  12. Ding C, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 2009, 55(3): 955–960.

    Article  MathSciNet  Google Scholar 

  13. Lachaud G and Wolfmann J, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inform. Theory, 1990, 36(3): 686–692.

    Article  MathSciNet  MATH  Google Scholar 

  14. Schroof R, Families of curves and weight distribution of codes, B. Am. Math. Soc., 1995, 32(2): 171–183.

    Article  Google Scholar 

  15. Liu Z H and Wu X W, On relative constant-weight codes, Designs, Codes and Cryptography, 2015, 75(1): 127–144.

    Article  MathSciNet  MATH  Google Scholar 

  16. Helleseth T, Kløve T, and Ytrehus Ø, Generalized hamming weights of linear codes, IEEE Trans. Inform. Theory, 1992, 38(3): 1133–1140.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Huaihai Institute of Technology, Lianyungang, 222005, China

    Dajian Liao

  2. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Zihui Liu

Authors
  1. Dajian Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zihui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zihui Liu.

Additional information

This research was supported by the National Science Foundation of China under Grant Nos. 11171366 and 61170257.

This paper was recommended for publication by Editor DENG Yingpu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liao, D., Liu, Z. On the intersection of binary linear codes. J Syst Sci Complex 29, 814–824 (2016). https://doi.org/10.1007/s11424-015-4123-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-015-4123-z

Keywords

Search

Navigation