Skip to main content
SpringerLink
Log in

Undetected error probability of q-ary constant weight codes

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

Abstract

In this paper, we introduce a new combinatorial invariant called q-binomial moment for q-ary constant weight codes. We derive a lower bound on the q-binomial moments and introduce a new combinatorial structure called generalized (s, t)-designs which could achieve the lower bounds. Moreover, we employ the q-binomial moments to study the undetected error probability of q-ary constant weight codes. A lower bound on the undetected error probability for q-ary constant weight codes is obtained. This lower bound extends and unifies the related results of Abdel-Ghaffar for q-ary codes and Xia-Fu-Ling for binary constant weight codes. Finally, some q-ary constant weight codes which achieve the lower bounds are found.

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. Abdel-Ghaffar K.A.S. (1997). A lower bound on the undetected error probability and strictly optimal codes. IEEE Trans. Inform. Theory 43, 1489–1502

    Article  MathSciNet  MATH  Google Scholar 

  2. Ashikhmin A., Barg A. (1999). Binomial moments of the distance distribution: bounds and applications. IEEE Trans. Inform. Theory 45, 438–452

    Article  MathSciNet  MATH  Google Scholar 

  3. Barg A., Ashikhmin A. (1999). Binomial moments of the distance distribution and the probability of undetected error. Desi. Codes and Cryptogr. 16, 103–116

    Article  MathSciNet  MATH  Google Scholar 

  4. Blaum M., Bruck J. (2000). Coding for tolerance and detection of skew in parallel asynchronous communications. IEEE Trans. Inform. Theory 46, 2329–2335

    Article  MathSciNet  MATH  Google Scholar 

  5. Chung F.R.K., Salehi J.A., Wei V.K. (1989). Optical orthogonal codes: Design, analysis, and applications. IEEE Trans. Inform. Theory 35, 595–604

    Article  MathSciNet  MATH  Google Scholar 

  6. Dodunekova R. (2003). Extended binomial moments of a linear code and undetected error probability. Probl. Inform. Transmission 39, 255–265

    Article  MathSciNet  MATH  Google Scholar 

  7. Etzion T. (1997). Optimal constant weight codes over Z k and generalized designs. Discrete Math. 169, 55–82

    Article  MathSciNet  MATH  Google Scholar 

  8. Fu F.-W., Xia S.-T. (1998). Binary constant weight codes for error detection. IEEE Trans. Inform. Theory 44, 1294–1299

    Article  MathSciNet  MATH  Google Scholar 

  9. Fu F.-W., Kløve T., Wei V.K. (2003). On the undetected error probability for binary codes. IEEE Trans. Inform. Theory 49, 382–390

    Article  MathSciNet  MATH  Google Scholar 

  10. Fu F.-W., Kløve T., Xia S.-T.: On the undetected error probability of m-out-of-n codes on the binary symmetric channel. In: Buchmann J., Høholdt T., Stichtenoth H., Tapia-Recillas H. (eds.) Coding Theory, Cryptography, and Related Areas, pp. 102–110, Springer (2000).

  11. Fu F.-W., Kløve T., Xia S.-T. (2000). The undetected error probability threshold of m-out-of-n codes. IEEE Trans. Inform. Theory 46, 1597–1599

    Article  MathSciNet  MATH  Google Scholar 

  12. Hanani H. (1963). On some tactical configurations. Canad. J. Math. 15, 702–722

    MathSciNet  MATH  Google Scholar 

  13. Kløve T., Korzhik V. (1995). Error Detecting Codes: General Theory and Their Application in Feedback Communication Systems. Kluwer Acad. Press, Boston

    Google Scholar 

  14. MacWilliams F.J., Sloane N.J.A. (1981). The Theory of Error-Correcting Codes. North-Holland, Amsterdam

    Google Scholar 

  15. Mills W.H.: On the covering of triples by quadruple. In: Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Algorithms, pp. 573–581 (1974).

  16. Tallini L.G., Bose B. (1998). Design of balanced and constant weight codes for VLSI systems. IEEE Trans. Comput. 47, 556–572

    Article  MathSciNet  Google Scholar 

  17. Tarnanen H., Aaltonen M., Goethals J-.M. (1985). On the nonbinary Johnson scheme. Eur. J. Combin. 6, 279–285

    MathSciNet  MATH  Google Scholar 

  18. Wang X.M., Yang Y.X. (1994). On the undetected error probability of nonlinear binary constant weight codes. IEEE Trans. Commun. 42, 2390–2393

    Article  MATH  Google Scholar 

  19. Xia S.-T., Fu F.-W., Jiang Y., Ling S. (2005). The probability of undetected error for binary constant weight codes. IEEE Trans. Inform. Theory 51, 3364–3373

    Article  MathSciNet  Google Scholar 

  20. Xia S.-T., Fu F.-W., Ling S. (2006). A lower bound on the probability of undetected error for binary constant weight codes. IEEE Trans. Inform. Theory 52, 4235–4243

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Graduate School at Shenzhen, Tsinghua University, Shenzhen Guangdong, 518055, P.R. China

    Shu-Tao Xia

  2. National Mobile Communications Research Laboratory, Southeast University, Nanjing Jiangsu, P.R. China

    Shu-Tao Xia

  3. Chern Institute of Mathematics and KLPMC, Nankai University, Tianjin, 300071, P.R. China

    Fang-Wei Fu

Authors
  1. Shu-Tao Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fang-Wei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shu-Tao Xia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xia, ST., Fu, FW. Undetected error probability of q-ary constant weight codes. Des. Codes Cryptogr. 48, 125–140 (2008). https://doi.org/10.1007/s10623-007-9130-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-007-9130-1

Keywords

AMS Classifications

Search

Navigation