Skip to main content
SpringerLink
Log in

Semi-cyclic holey group divisible designs with block size three

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

Abstract

In this paper we discuss the existence problem for a semi-cyclic holey group divisible design of type \((n,m^t)\) with block size \(3\), which is denoted by a \(3\)-SCHGDD of type \((n,m^t)\). When \(n=3\), a \(3\)-SCHGDD of type \((3,m^t)\) is equivalent to a \((3,mt;m)\)-cyclic holey difference matrix, denoted by a \((3,mt;m)\)-CHDM. It is shown that there is a \((3,mt;m)\)-CHDM if and only if \((t-1)m\equiv 0\ (\mathrm{mod}\ 2)\) and \(t\ge 3\) with the exception of \(m\equiv 0\ (\mathrm{mod}\ 2)\) and \(t=3\). When \(n\ge 4\), the case of \(t\) odd is considered. It is established that if \(t\equiv 1\ (\mathrm{mod}\ 2)\) and \(n\ge 4\), then there exists a \(3\)-SCHGDD of type \((n,m^t)\) if and only if \(t\ge 3\) and \((t-1)n(n-1)m\equiv 0\ (\mathrm{mod}\ 6)\) with some possible exceptions of \(n=6\) and \(8\). The main results in this paper have been used to construct optimal two-dimensional optical orthogonal codes with weight \(3\) and different auto- and cross-correlation constraints by the authors recently.

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. Abel R.J.R., Assaf A.: Modified group divisible designs with block size \(5\) and \(\lambda =1\). Discret. Math. 256, 1–22 (2002).

  2. Abel R.J.R., Assaf A.: Modified group divisible designs with block size \(5\) and even index. Discret. Math. 308, 3335–3351 (2008).

    Google Scholar 

  3. Abel R.J.R., Buratti M.: Difference families. In: Colbourn, C.J., Dinitz, J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 392–410. CRC Press, Boca Raton (2007).

  4. Abel R.J.R., Bennett F.E., Greig M.: PBD-closure. In: Colbourn, C.J., Dinitz, J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 247–255. CRC Press, Boca Raton (2007).

  5. Assaf A.: Modified group divisible designs. Ars Comb. 29, 13–20 (1990).

    Google Scholar 

  6. Assaf A.: An application of modified group divisible designs. J. Comb. Theory A 68, 152–168 (1994).

    Google Scholar 

  7. Assaf A.: Modified group divisible designs with block size 4 and \(\lambda \) > 1. Australas. J. Comb. 16, 229–238 (1997).

    Google Scholar 

  8. Assaf A., Wei R.: Modified group divisible designs with block size \(4\) and \(\lambda =1\). Discret. Math. 195, 15–25 (1999).

  9. Baker C.: Extended Skolem sequences. J. Comb. Des. 3, 363–379 (1995).

    Google Scholar 

  10. Bryant D., Chang Y., Rodger C.A., Wei R.: Two-dimensional balanced sampling plans excluding contiguous units. Commun. Stat. Theory Methods 31, 1441–1455 (2002).

    Google Scholar 

  11. Buratti M., Pasotti A.: Combinatorial designs and the theorem of Weil on multiplicative character sums. Finite Fields Appl. 15, 332–344 (2009).

    Google Scholar 

  12. Cao H., Wang L., Wei R.: The existence of HGDDs with block size four and its application to double frames. Discret. Math. 309, 945–949 (2009).

    Google Scholar 

  13. Chang Y., Ji L.: Optimal \((4up, 5, 1)\) optical orthogonal codes. J. Comb. Des. 12, 346–361 (2004).

  14. Chang Y., Miao Y.: Constructions for optimal optical orthogonal codes. Discret. Math. 261, 127–139 (2003).

    Google Scholar 

  15. Chen Z.: The existence of balanced difference families and perfect difference families. Master Degree Thesis, Guangxi Normal University (2008).

  16. Colbourn C.J.: Difference matrices. In: Colbourn, C.J., Dinitz, J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 411–419. CRC Press, Boca Raton (2007).

  17. Colbourn C.J., Dinitz J.H.: Generating sets in Steiner triple systems. Math. Slov. 50, 259–269 (2000).

    Google Scholar 

  18. Gallant R.P., Jiang Z., Ling A.C.H.: The spectrum of cyclic group divisible designs with block size three. J. Comb. Des. 7, 95–105 (1999).

    Google Scholar 

  19. Ge G., Wei R.: HGDDs with block size four. Discret. Math. 279, 267–276 (2004).

    Google Scholar 

  20. Ge G., Wang J., Wei R.: MGDD with block size \(4\) and its application to sampling designs. Discret. Math. 272, 277–283 (2003).

    Google Scholar 

  21. Ge G., Miao Y., Sun X.: Perfect difference families, perfect difference matrices, and related combinatorial structures. J. Comb. Des. 18, 415–449 (2010).

    Google Scholar 

  22. Ge G., Fuji-Hara R., Miao Y.: Further combinatorial constructions for optimal frequency-hopping sequences. J. Comb. Theory A 113, 1699–1718 (2006).

    Google Scholar 

  23. Ge G., Ling A.C.H., Miao Y.: A systematic construction for radar arrays. IEEE Trans. Inf. Theory 56, 410–414 (2008).

    Google Scholar 

  24. Linek V., Mor S.: On partitions of \(\{1,\ldots , 2m+1\}\setminus \{k\}\) into differences \(d,\ldots , d+m-1\): extended Langford sequences of large defect. J. Comb. Des. 12, 421–442 (2004).

  25. Ling A.C.H., Colbourn C.J.: Modified group divisible designs with block size four. Discret. Math. 219, 207–221 (2000).

    Google Scholar 

  26. Mathon R., Rosa A.: \(2-(v, k,{\lambda })\) designs of small orders. In: Colbourn C.J., Dinitz J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 25–58. CRC Press, Boca Raton (2007).

  27. Shalaby N.: Skolem and Langford sequences. In: Colbourn C.J., Dinitz J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 612–616. CRC Press, Boca Raton (2007).

  28. Stinson D.R.: A general construction for group divisible designs. Discret. Math. 33, 89–94 (1981).

    Google Scholar 

  29. Stinson D.R., Zhu L.: On the existence of MOLS with equal-sized holes. Aequ. Math. 33, 96–105 (1987).

    Google Scholar 

  30. Wang X., Chang Y.: The spectrum of \((gv, g,3,\lambda )\)-DF in \(Z_{gv}\). Sci. China A 52, 1004–1016 (2009).

  31. Wang X., Chang Y.: Further results on \((v,4,1)\)-perfect difference families. Discret. Math. 310, 1995–2006 (2010).

  32. Wang K., Wang J.: Semicyclic \(4\)-GDDs and related two-dimensional optical orthogonal codes. Des. Codes Cryptogr. 63, 305–319 (2012).

  33. Wang J., Yin J.: Two-dimensional optical orthogonal codes and semicyclic group divisible designs. IEEE Trans. Inf. Theory 56, 2177–2187 (2010).

    Google Scholar 

  34. Wang J., Shan X., Yin J.: On constructions for optimal two-dimentional optical orthogonal codes. Des. Codes Cryptogr. 54, 43–60 (2010).

    Google Scholar 

  35. Wang X., Chang Y., Feng T.: Optimal \(2\)-D \((n\times m,3,2,1)\)-optical orthogonal codes. IEEE Trans. Inf. Theory 59, 710–725 (2013).

  36. Wei R.: Group divisible designs wih equal-sized holes. Ars Combin. 35, 315–323 (1993).

    Google Scholar 

  37. Yin J.: A general construction for optimal cyclic packing designs. J. Comb. Theory A 97, 272–284 (2002).

    Google Scholar 

  38. Yin J.: Cyclic difference packing and covering arrays. Des. Codes Cryptogr. 37, 281–292 (2005).

    Google Scholar 

  39. Zhang J., Chang Y.: The spectrum of cyclic \(BSA(v,3,\lambda ; \alpha )\) with \(\alpha =2,3\). J. Comb. Des. 13, 313–335 (2005).

  40. Zhu L.: Some recent developments on BIBDs and related designs. Discret. Math. 123, 189–214 (1993).

    Google Scholar 

Download references

Acknowledgments

Supported by the Fundamental Research Funds for the Central Universities under Grants \(2011\)JBM\(298,\,2013\)JB\(Z005\) (T. Feng), \(2011\)JBZ\(012\) (Y. Chang), and the NSFC under Grant \(10901016\) (T. Feng), the NSFC under Grants \(11126123\) and \(11201252\) (X. Wang), the NSFC under Grant \(11271042\) (Y. Chang). The authors would like to thank the editor and the two anonymous referees for their helpful comments and valuable suggestions to improve the readability of this paper.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Tao Feng & Yanxun Chang

  2. Department of Mathematics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Xiaomiao Wang

Authors
  1. Tao Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaomiao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanxun Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Feng.

Additional information

Communicated by C. J. Colbourn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feng, T., Wang, X. & Chang, Y. Semi-cyclic holey group divisible designs with block size three. Des. Codes Cryptogr. 74, 301–324 (2015). https://doi.org/10.1007/s10623-013-9859-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-013-9859-7

Keywords

Mathematics Subject Classification

Search

Navigation