Skip to main content
SpringerLink
Log in

Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

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

Abstract

Let K q (n, w, t, d) be the minimum size of a code over Z q of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K q (n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2m + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.

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. Baker R.D.: Partitioning the planes of AG2m (2) into 2-designs. Discrete Math. 15(3), 205–211 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. de Caen D.: Extension of a theorem of Moon and Moser on complete subgraphs. Ars Comb. 16, 5–10 (1983)

    MATH  Google Scholar 

  3. Cohen G.D., Karpovsky M.G., Mattson H.F. Jr., Schatz J.R.: Covering radius—survey and recent results. IEEE Trans. Inform. Theory 31(3), 328–343 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Colbourn C.J., Dinitz J.H. (eds.): Handbook of Combinatorial Designs, 2nd edn. Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton (2007).

  5. Etzion T., Wei V., Zhang Z.: Bounds on the sizes of constant weight covering codes. Des. Codes Cryptogr. 5(3), 217–239 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fort M.K. Jr., Hedlund G.A.: Minimal coverings of pairs by triples. Pac. J. Math. 8, 709–719 (1958)

    MathSciNet  MATH  Google Scholar 

  7. Granville A., Hartman A.: Subdesigns in Steiner quadruple systems. J. Comb. Theory Ser. A 56(2), 239–270 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gray R.M., Davisson L.D.: Source coding theorems without the ergodic assumption. IEEE Trans. Inform. Theory IT-20, 502–516 (1974)

    Article  MathSciNet  Google Scholar 

  9. Hanani H.: On quadruple systems. Can. J. Math. 12, 145–157 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hartman A.: Tripling quadruple systems. Ars Comb. 10, 255–309 (1980)

    MathSciNet  MATH  Google Scholar 

  11. Hartman A.: A general recursive construction for quadruple systems. J. Comb. Theory Ser. A 33(2), 121–134 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hartman A., Mills W.H., Mullin R.C.: Covering triples by quadruples: an asymptotic solution. J. Comb. Theory Ser. A 41(1), 117–138 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heinrich K., Yin J.: On group divisible covering designs. Discrete Math. 202(1–3), 101–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Honkala I.S.: Modified bounds for covering codes. IEEE Trans. Inform. Theory 37(2), 351–365 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ji L.: An improvement on covering triples by quadruples. J. Comb. Des. 16(3), 231–243 (2008)

    Article  MATH  Google Scholar 

  16. Ji L.: An improvement on H design. J. Comb. Des. 17(1), 25–35 (2009)

    Article  MATH  Google Scholar 

  17. Kalbfleisch J.G., Stanton R.G.: Maximal and minimal coverings of (k − 1)-tuples by k-tuples. Pac. J. Math. 26, 131–140 (1968)

    MathSciNet  MATH  Google Scholar 

  18. Keranen M.S., Kreher D.L.: Transverse quadruple systems with five holes. J. Comb. Des. 15(4), 315–340 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lauinger K.A., Kreher D.L., Rees R., Stinson D.R.: Computing transverse t-designs. J. Comb. Math. Comb. Comput. 54, 33–56 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Mills W.H.: On the covering of triples by quadruples. In: Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing, Florida Atlantic University, Boca Raton, 1974, pp. 563–581. Congressus Numerantium, No. X. Utilitas Math., Winnipeg (1974).

  21. Mills W.H.: A covering of triples by quadruples. In: Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II, Baton Rouge, 1981, vol. 33, pp. 253–260 (1981).

  22. Mills W.H.: On the existence of H designs. In: Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing, Boca Raton 1990, vol. 79, pp. 129–141 (1990).

  23. Schönheim J.: On coverings. Pac. J. Math. 14, 1405–1411 (1964)

    MATH  Google Scholar 

  24. Swift J.D.: A generalized Steiner problem. Rend. Mat. 2(6), 563–569 (1969)

    MathSciNet  MATH  Google Scholar 

  25. Turán P.: On the theory of graphs. Colloquium Math. 3, 19–30 (1954)

    MathSciNet  MATH  Google Scholar 

  26. Wang J., Ji L.: A class of group divisible 3-designs and their applications. J. Comb. Des. 17(2), 136–146 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhuravlev A.A., Keranen M.S., Kreher D.L.: Small group divisible Steiner quadruple systems. Electron. J. Comb. 15(1), Research Paper 40, 14 pp (electronic) (2008).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, Zhejiang, People’s Republic of China

    Xiande Zhang, Hui Zhang & Gennian Ge

Authors
  1. Xiande Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gennian Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gennian Ge.

Additional information

Communicated by J. D. Key.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, X., Zhang, H. & Ge, G. Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four. Des. Codes Cryptogr. 62, 143–160 (2012). https://doi.org/10.1007/s10623-011-9499-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-011-9499-8

Keyword

Mathematics Subject Classification (2000)

Search

Navigation