Skip to main content
SpringerLink
Log in

Code constructions and existence bounds for relative generalized Hamming weight

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

Abstract

The relative generalized Hamming weight (RGHW) of a linear code C and a subcode C 1 is an extension of generalized Hamming weight. The concept was firstly used to protect messages from an adversary in the wiretap channel of type II with illegitimate parties. It was also applied to the wiretap network II for secrecy control of network coding and to trellis-based decoding algorithms for complexity estimation. For RGHW, bounds and code constructions are two related issues. Upper bounds on RGHW show the possible optimality for the applications, and code constructions meeting upper bounds are for designing optimal schemes. In this article, we show indirect and direct code constructions for known upper bounds on RGHW. When upper bounds are not tight or constructions are hard to find, we provide two asymptotically equivalent existence bounds about good code pairs for designing suboptimal schemes. Particularly, most code pairs (C, C 1) are good when the length n of C is sufficiently large, the dimension k of C is proportional to n and other parameters are fixed. Moreover, the first existence bound yields an implicit lower bound on RGHW, and the asymptotic form of this existence bound generalizes the usual asymptotic Gilbert–Varshamov bound.

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.

Similar content being viewed by others

References

  1. Chor B., Goldreich O., Håstad J., Friedmann J., Rudich S., Smolensky R.: The bit extraction problem of t-resilient functions. In: 26th Annual IEEE Symposium on Foundations of Computer Science, Portland, pp. 397–407 (1985).

  2. Cohen G., Litsyn S., Zémor G.: Upper bounds on generalized distances. IEEE Trans. Inf. Theory 40(6), 2090–2092 (1994)

    Article  MATH  Google Scholar 

  3. Encheva S., Kløve T.: Codes satisfying the chain condition. IEEE Trans. Inf. Theory 40(1), 175–180 (1994)

    Article  MATH  Google Scholar 

  4. Fan Y., Liu H.: Equidistant linear codes and maximal projective codes. J. China Inst. Commun. 6(22), 48–52 (2001)

    Google Scholar 

  5. Forney G.D.: Dimension/length profiles and trellis complexity of linear block codes. IEEE Trans. Inf. Theory 40(6), 1741–1752 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Helleseth T., Kløve T., Levenshtein V.I., Ytrehus Ø.: Bounds on the minimum support weights. IEEE Trans. Inf. Theory 41(2), 432–440 (1995)

    Article  MATH  Google Scholar 

  7. Helleseth T., Kløve T., Mykkeltveit J.: The weight distribution of irreducible cyclic codes with block lengths n 1((q l−1)/N). Discret. Math. 18, 179–211 (1977)

    Article  MATH  Google Scholar 

  8. Helleseth T., Kløve T., Ytrehus Ø.: Generalized Hamming weights of linear codes. IEEE Trans. Inf. Theory 38(3), 133–1140 (1992)

    Article  Google Scholar 

  9. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  10. Hughes D.R., Piper F.C.: Design Theory. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  11. Kasami T., Takata T., Fujiwara T., Lin S.: On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes. IEEE Trans. Inf. Theory 39(1), 242–243 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kløve T.: The worst case propability of undetected error for linear codes on the local binomial channels. IEEE Trans. Inf. Theory 42(1), 172–196 (1996)

    Article  Google Scholar 

  13. Komura T., Oka M., Fujiwara T., Onoye T., Kasami T., Lin S.: VLSI architecture of a recursive maximum-likelihood decoding algorithm for a (64,35) subcode of the (64,42) Reed–Muller code. In: International Symposium on Information Theory and Its Applications, Victoria, pp. 396–407 (1996).

  14. Lin S., Kasami T., Fujiwara T., Fossorier M.: Trellises and Trellis-based Decoding Algorithms for Linear Block Codes. Kluwer Academic Press, Boston (1998)

    Book  MATH  Google Scholar 

  15. Lin S., Uehara G.T., Nakamura E., Chu W.P.: Circuit design approaches for implementation of a subtrellis IC for a Reed–Muller subcode. NASA Tech. Rep. 96–001 (1996).

  16. Liu Z., Chen W., Luo Y.: The relative generalized Hamming weight of linear q-ary codes and their subcodes. Des. Codes Cryptogr. 48, 111–123 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luo Y., Fu F., Mitrpant C., Han Vinck A.J.: Relative MDS pairs. Unpublished manuscript.

  18. Luo Y., Mitrpant C., Han Vinck A.J., Chen K.F.: Some new characters on the wire-tap channel of type II. IEEE Trans. Inf. Theory 51(3), 1222–1229 (2005)

    Article  MATH  Google Scholar 

  19. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1988)

    Google Scholar 

  20. Ozarow L.H., Wyner A.D.: Wire-tap channel II. Bell Labs Tech. J. 63(10), 2135–2157 (1984)

    MATH  Google Scholar 

  21. Tsfasman M.A., Vlǎdut S.G.: Geometric approach to higher weights. IEEE Trans. Inf. Theory 41(6), 1564–1588 (1995)

    Article  MATH  Google Scholar 

  22. Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inf. Theory 37(5), 1412–1418 (1991)

    Article  MATH  Google Scholar 

  23. Wolf J.K.: Efficient maximum likelihood decoding of linear block codes. IEEE Trans. Inf. Theory, IT- 24(1), 76–80 (1978)

    Article  MATH  Google Scholar 

  24. Zhang Z.: Wiretap networks II with partial information leakage. In: 4th International Conference on Communications and Networking in China, Xi’an, pp. 1–5 (2009).

  25. Zhang Z., Zhuang B.: An application of the relative network generalized Hamming weight to erroneous wiretap networks. In: 2009 IEEE Information Theory Workshop, Taormina, pp. 70–74 (2009).

  26. Zhuang Z., Luo Y., Dai B., Han Vinck A.J.: Relative dimension/length profile and its applications. Des. Codes Cryptogr. (in manuscript).

  27. Zhuang Z., Luo Y., Han Vinck A.J.: Bounds on relative generalized Hamming weight. Inf. Sci. (in manuscript).

  28. Zhuang Z., Luo Y., Han Vinck A.J., Dai B.: Some new bounds on relative generalized Hamming weight. In: 13th International Conference on Communication Technology, Jinan, pp. 971–974 (2011).

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Min Hang District, Shanghai, 200240, China

    Zhuojun Zhuang, Yuan Luo & Bin Dai

  2. The State Key Laboratory of Integrated Services Networks, Xi’an, China

    Zhuojun Zhuang & Bin Dai

  3. National Mobile Communications Research Laboratory, Southeast University, Nanjing, China

    Yuan Luo

Authors
  1. Zhuojun Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bin Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Luo.

Additional information

Communicated by I. Shparlinski.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhuang, Z., Luo, Y. & Dai, B. Code constructions and existence bounds for relative generalized Hamming weight. Des. Codes Cryptogr. 69, 275–297 (2013). https://doi.org/10.1007/s10623-012-9657-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-012-9657-7

Keywords

Mathematics Subject Classification

Search

Navigation