Advertisement

Designs, Codes and Cryptography

, Volume 76, Issue 3, pp 589–600 | Cite as

Completely reducible super-simple designs with block size five and index two

  • Hengjia Wei
  • Hui Zhang
  • Gennian Ge
Article

Abstract

Complete reducible super-simple (CRSS) designs are closely related to \(q\)-ary constant weight codes. A \((v,k,\lambda )\)-CRSS design is just an optimal \((v,2(k-1),k)_{\lambda +1}\) code. In this paper, we mainly investigate the existence of a \((v,5,2)\)-CRSS design and show that such a design exists if and only if \(v\equiv 1,5\pmod {20}\) and \(v\ge 21\), except possibly when \(v = 25\). Using this result, we determine the maximum size of an \((n,8,5)_3\) code for all \(n\equiv 0,1,4,5 \pmod {20}\) with the only length \(n=25\) unsettled. In addition, we also construct super-simple \((v,5,3)\)-BIBDs for \(v=45,65\).

Keywords

Completely reducible super-simple designs Constant weight codes Group divisible designs Super-simple designs 

Mathematics Subject Classification

Primary 05B05 94B25 

Notes

Acknowledgments

Research supported by the National Natural Science Foundation of China under Grant No. 61171198 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ13A010001.

References

  1. 1.
    Abel R.J.R., Bennett F.E.: Super-simple Steiner pentagon systems. Discret. Appl. Math. 156(5) 780–793 (2008).Google Scholar
  2. 2.
    Abel R.J.R., Colbourn C.J., Dinitz J.H.: Mutually orthogonal latin squares (MOLS). In: Colbourn C.J., Dinitz J.H., (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn., pp. 160–193. CRC Press, Boca Raton (2007).Google Scholar
  3. 3.
    Abel R.J.R., Ge G., Greig M., Ling A.C.H.: Further results on \((v,\{5, w^*\},1)\)-PBDs. Discret. Math. 309(8), 2323–2339 (2009).Google Scholar
  4. 4.
    Adams P., Bryant D.E., Khodkar A.: On the existence of super-simple designs with block size \(4\). Aequ. Math. 51(3), 230–246 (1996).Google Scholar
  5. 5.
    Blake-Wilson S., Phelps K.T.: Constant weight codes and group divisible designs. Des. Codes Cryptogr. 16(1), 11–27 (1999).Google Scholar
  6. 6.
    Bush K.A.: Orthogonal arrays of index unity. Ann. Math. Stat. 23, 426–434 (1952).Google Scholar
  7. 7.
    Caro Y., Yuster R.: Orthogonal decomposition and packing of complete graphs. J. Comb. Theory Ser. A 88(1), 93–111 (1999).Google Scholar
  8. 8.
    Cayley A.: On the triadic arrangements of seven and fifteen things. Lond. Edinb. Dublin Philos. Mag. J. Sci. 37(3), 50–53 (1850).Google Scholar
  9. 9.
    Chen K., Wei R.: Super-simple \((v,5,5)\) designs. Des. Codes Cryptogr. 39(2), 173–187 (2006).Google Scholar
  10. 10.
    Chen K., Wei R.: Super-simple \((v,5,4)\) designs. Discret. Appl. Math. 155(8), 904–913 (2007).Google Scholar
  11. 11.
    Chee Y.M., Ling S.: Constructions for \(q\)-ary constant-weight codes. IEEE Trans. Inf. Theory 53(1), 135–146 (2007).Google Scholar
  12. 12.
    Chee Y.M., Dau S.H., Ling A.C.H., Ling S.: The sizes of optimal \(q\)-ary codes of weight three and distance four: a complete solution. IEEE Trans. Inf. Theory 54(3), 1291–1295 (2008).Google Scholar
  13. 13.
    Chen K., Ge G., Zhu L.: Generalized Steiner triple systems with group size five. J. Comb. Des. 7(6), 441–452 (1999).Google Scholar
  14. 14.
    Chen K., Chen G., Li W., Wei R.: Super-simple balanced incomplete block designs with block size \(5\) and index \(3\). Discret. Appl. Math. 161(16–17), 2396–2404 (2013).Google Scholar
  15. 15.
    Colbourn C.J., Dinitz J.H. (eds.): The CRC Handbook of Combinatorial Designs. CRC Press Series on Discrete Mathematics and Its Applications, 2nd edn. CRC Press, Boca Raton (2007).Google Scholar
  16. 16.
    Etzion T.: Optimal constant weight codes over \(Z_k\) and generalized designs. Discret. Math. 169(1–3), 55–82 (1997).Google Scholar
  17. 17.
    Fu F.-W., Vinck A.J.H., Shen S.Y.: On the constructions of constant-weight codes. IEEE Trans. Inf. Theory 44(1), 328–333 (1998).Google Scholar
  18. 18.
    Fu F.-W., Kløve T., Luo Y., Wei V.K.: On the Svanström bound for ternary constant-weight codes. IEEE Trans. Inf. Theory 47(5), 2061–2064 (2001).Google Scholar
  19. 19.
    Ge G.: Generalized Steiner triple systems with group size \(g\equiv 1,5 (mod 6)\). Australas. J. Comb. 21, 37–47 (2000).Google Scholar
  20. 20.
    Ge G.: Further results on the existence of generalized Steiner triple systems with group size \(g\equiv 1,5 (mod 6)\). Australas. J. Comb. 25, 19–27 (2002).Google Scholar
  21. 21.
    Ge G.: Generalized Steiner triple systems with group size \(g\equiv 0, 3(mod 6)\). Acta Math. Appl. Sin. Engl. Ser. 18(4), 561–568 (2002).Google Scholar
  22. 22.
    Ge G.: Construction of optimal ternary constant weight codes via Bhaskar Rao designs. Discret. Math. 308(13), 2704–2708 (2008).Google Scholar
  23. 23.
    Gronau H.-D.O.F., Kreher D.L., Ling A.C.H.: Super-simple \((v,5,2)\)-designs, Discret. Appl. Math. 138(1–2), 65–77 (2004) (Optimal discrete structures and algorithms, ODSA 2000).Google Scholar
  24. 24.
    Hartmann S.: On simple and super-simple transversal designs. J. Comb. Des. 8, 311–320 (2000).Google Scholar
  25. 25.
    Hartmann S.: Superpure digraph designs. J. Comb. Des. 10(4), 239–255 (2002).Google Scholar
  26. 26.
    Lu J.X.: On large sets of disjoint Steiner triple systems I, II, and III. J. Comb. Theory Ser. A 34(2), 140–146, 147–155, 156–182 (1983).Google Scholar
  27. 27.
    Lu J.X.: On large sets of disjoint Steiner triple systems IV, V, and VI. J. Comb. Theory Ser. A 37(2), 136–163, 164–188, 189–192 (1984).Google Scholar
  28. 28.
    Östergård P.R.J., Svanström M.: Ternary constant weight codes. Electron. J. Comb. 9, no. 1, Research Paper 41, 23 pp (2002) (electronic).Google Scholar
  29. 29.
    Svanström M.: A lower bound for ternary constant weight codes. IEEE Trans. Inf. Theory 43(5), 1630–1632 (1997).Google Scholar
  30. 30.
    Svanström M.: Ternary codes with weight constraints. Ph.D dissertation, Linköpings Universitet, Linköping, Sweden (1999).Google Scholar
  31. 31.
    Svanström M., Östergård P.R.J. Bogdanova G.T.: Bounds and constructions for ternary constant-composition codes. IEEE Trans. Inf. Theory 48(1), 101–111 (2002).Google Scholar
  32. 32.
    Teirlinck L.: A completion of Lu’s determination of the spectrum of large sets of disjoint Steiner triple systems. J. Comb. Theory Ser. A 57, 302–305 (1991).Google Scholar
  33. 33.
    Wilson R.M.: Constructions and uses of pairwise balanced designs. Math. Centre Tracts 55, 18–41 (1974).Google Scholar
  34. 34.
    Zhang H., Ge G.: Optimal ternary constant-weight codes of weight four and distance six. IEEE Trans. Inf. Theory 56(5), 2188–2203 (2010).Google Scholar
  35. 35.
    Zhang H., Ge G.: Completely reducible super-simple designs with block size four and related super-simple packings. Des. Codes Cryptogr. 58(3), 321–346 (2011).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhou China
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijing China
  3. 3.Beijing Center for Mathematics and Information Interdisciplinary SciencesBeijing China

Personalised recommendations