Skip to main content
SpringerLink
Log in

The characterization of binary constant weight codes meeting the bound of Fu and Shen

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

Abstract

Fu and Shen gave an upper bound on binary constant weight codes. In this paper, we present a new proof for the bound of Fu and Shen and characterize binary constant weight codes meeting this bound. It is shown that binary constant weight codes meet the bound of Fu and Shen if and only if they are generated from certain symmetric designs and quasi-symmetric designs in combinatorial design theory. In particular, it turns out that the existence of binary codes with even length meeting the Grey–Rankin bound is equivalent to the existence of certain binary constant weight codes meeting the bound of Fu and Shen. Furthermore, some examples are listed to illustrate these results. Finally, we obtain a new upper bound on binary constant weight codes which improves on the bound of Fu and Shen in certain case.

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. Bracken C (2006) New classes of self-complementary codes and quasi-symmetric designs. Des Codes Cryptogr 41(3):319–323

    Article  MathSciNet  MATH  Google Scholar 

  2. Bracken C, McGuire G, Ward HN (2006) New quasi-symmetric designs constructed using mutually orthogonal Latin squares and Hadamard matrices. Des Codes Cryptogr 41(2):195–198

    Article  MATH  MathSciNet  Google Scholar 

  3. Delsarte P (1973) An algebraic approach to the association schemes of coding theory. Philips Res Rep Suppl 10

  4. Fu F-W, Shen S-Y (2001) An upper bound for binary constant weight codes. J Stat Plan Infer 94(2):197–203

    Article  MATH  MathSciNet  Google Scholar 

  5. Fu F-W, Wei VK (2002) Self-complementary balanced codes and quasi-symmetric designs. Des Codes Cryptogr 27(3):271–279

    Article  MATH  MathSciNet  Google Scholar 

  6. Grey LD (1962) Some bounds for error-correcting codes. IEEE Trans Inf Theory 8:200–202

    Article  MathSciNet  Google Scholar 

  7. Jungnickel D, Tonchev VD (1991) Exponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound. Des Codes Cryptogr 1:247–253

    Article  MATH  MathSciNet  Google Scholar 

  8. Jungnickel D, Tonchev VD (1992) On symmetric and quasi-symmetric designs with the symmetric difference property and their codes. J Comb Theory Ser A 59:40–50

    Article  MATH  MathSciNet  Google Scholar 

  9. MacWilliams FJ, Sloane NJA (1981) The theory of error-correcting codes. North-Holland, Amsterdam

  10. McGuire G (1997) Quasi-symmetric designs and codes meeting the Grey-Rankin bound. J Comb Theory Ser A 78:280–291

    Article  MATH  MathSciNet  Google Scholar 

  11. Rankin RA (1955) The closest packing of spherical caps in n dimensions. Proc Glasgow Math Assoc 2:139–144

    Article  MATH  MathSciNet  Google Scholar 

  12. Rankin RA (1956) On the minimal points of positive definite quadratic forms. Mathematika 3:15–24

    Article  MathSciNet  MATH  Google Scholar 

  13. Shrikhande MS (1996) Quasi-symmetric designs. In: Colbourn CJ, Dinitz JH (eds) The CRC handbook of combinatorial designs Part IV, Section 37. CRC, New York

  14. Shrikhande MS, Sane SS (1991) Quasi-symmetric designs. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  15. Tonchev VD (1986) Quasi-symmetric designs and self-dual codes. European J Comb 7:67–73

    MATH  MathSciNet  Google Scholar 

  16. Tonchev VD (1993) Quasi-symmetric designs, codes, quadrics, and hyperplane sections. Geom Dedicata 48:295–308

    Article  MATH  MathSciNet  Google Scholar 

  17. Tonchev VD (1998) Codes and designs. In: Pless VS, Huffman WC (eds) Handbook of coding theory Chap. 15. North-Holland, Amsterdam

  18. Trung TV (1996) Symmetric designs. In: Colbourn CJ, Dinitz JH (eds) The CRC handbook of combinatorial designs Part I, Section 5. CRC, New York

Download references

Author information

Authors and Affiliations

  1. Chern Institute of Mathematics, Nankai University, Tianjin, 300071, People’s Republic of China

    Fang-Wei Fu

  2. Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, Singapore, 117508, Singapore

    Fang-Wei Fu

  3. Graduate School at Shenzhen, Tsinghua University, Shenzhen, Guangdong, 518055, People’s Republic of China

    Shu-Tao Xia

Authors
  1. Fang-Wei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shu-Tao Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fang-Wei Fu.

Additional information

This research is supported in part by the DSTA research grant R-394-000-025-422 and the National Natural Science Foundation of China under the Grant 60402031, and the NSFC-GDSF joint fund under the Grant U0675001

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fu, FW., Xia, ST. The characterization of binary constant weight codes meeting the bound of Fu and Shen. Des Codes Crypt 43, 9–20 (2007). https://doi.org/10.1007/s10623-007-9049-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-007-9049-6

Keywords

AMS Classifications

Search

Navigation