Skip to main content
SpringerLink
Log in

Egalitarian Steiner triple systems for data popularity

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

Abstract

For an ordering of the blocks of a design, the point sum of an element is the sum of the indices of blocks containing that element. Block labelling for popularity asks for the point sums to be as equal as possible. For Steiner systems of order v and strength t in general, the average point sum is \(O(v^{2t-1})\); under various restrictions on block partitions of the Steiner system, the difference between the largest and smallest point sums is shown to be \(O(v^{(t+1)/2}\log v)\). Indeed for Steiner triple systems, direct and recursive constructions are given to establish that systems exist with all point sums equal for more than two thirds of the admissible orders.

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

Fig. 1

Similar content being viewed by others

References

  1. Abel R.J.R., Lamken E.R., Wang J.: A few more Kirkman squares and doubly near resolvable BIBDs with block size 3. Discret. Math. 308(7), 1102–1123 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon N., Spencer J.H.: The Probabilistic Method. Wiley, Hoboken, NJ (2008).

    Book  MATH  Google Scholar 

  3. Anderson I.: Some 2-rotational and cyclic designs. J. Combin. Des. 4(4), 247–254 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  4. Anstice R.R.: On a problem in combinations. Camb. Dublin Math. J. 7, 279–292 (1852).

    Google Scholar 

  5. Anstice R.R.: On a problem in combinations (continued). Camb. Dublin Math. J. 8, 149–154 (1853).

    Google Scholar 

  6. Bermond J.C., Jean-Marie A., Mazauric D., Yu J.: Well-balanced designs for data placement. J. Combin. Des. 24(2), 55–76 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  7. Beth T., Jungnickel D., Lenz H.: Design Theory. Vol. I, Encyclopedia of Mathematics and Its Applications, vol. 69, 2nd edn Cambridge University Press, Cambridge (1999).

    MATH  Google Scholar 

  8. Bose R.C.: On the construction of balanced incomplete block designs. Ann. Eugen. 9(4), 353–399 (1939).

    Article  MathSciNet  Google Scholar 

  9. Buratti M., Ghinelli D.: On disjoint \((3t,3,1)\) cyclic difference families. J. Stat. Plann. Inference 140(7), 1918–1922 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  10. Buratti M., Zuanni F.: \(G\)-invariantly resolvable Steiner \(2\)-designs which are \(1\)-rotational over \(G\). Bull. Belg. Math. Soc. Simon Stevin 5(2–3), 221–235 (1998).

    MathSciNet  MATH  Google Scholar 

  11. Buratti M., Zuanni F.: Explicit constructions for 1-rotational Kirkman triple systems. Util. Math. 59, 27–30 (2001).

    MathSciNet  MATH  Google Scholar 

  12. Chee Y.M., Colbourn C.J., Dau H., Gabrys R., Ling A.C.H., Lusi D., Milenkovic O.: Access balancing in storage systems by labeling partial Steiner systems. Des. Codes Cryptogr. 88, 2361–2376 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  13. Chernoff H.: A note on an inequality involving the normal distribution. Ann. Probab. 9(3), 533–535 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  14. Chung F., Lu L.: Concentration inequalities and martingale inequalities: a survey. Internet Math. 3(1), 79–127 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  15. Cohen M.B., Colbourn C.J.: Optimal and pessimal orderings of Steiner triple systems in disk arrays. Theor. Comput. Sci. 297(1–3), 103–117 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  16. Colbourn C.J.: Separations of Steiner triple systems: some questions. Bull. ICA 6, 53–56 (1992).

    MathSciNet  MATH  Google Scholar 

  17. Colbourn C.J.: Popularity block labelling for Steiner systems. In: Seventeenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT2020), pp. 41–46 (2020)

  18. Colbourn C.J.: Egalitarian edge orderings of complete graphs. Graphs Combin. 37(4), 1405–1413 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  19. Colbourn C.J., Hoffman D.G., Rees R.: A new class of group divisible designs with block size three. J. Combin. Theory Ser. A 59(1), 73–89 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  20. Colbourn C.J., Horsley D., Wang C.: Trails of triples in partial triple systems. Des. Codes Cryptogr. 65(3), 199–212 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  21. Colbourn C.J., Lamken E.R., Ling A.C.H., Mills W.H.: The existence of Kirkman squares–doubly resolvable \((\nu,3,1)\)-BIBDs. Des. Codes Cryptogr. 26(1–3), 169–196 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  22. Colbourn C.J., Rosa A.: Triple Systems. Oxford Mathematical Monographs. Clarendon Press, Oxford (1999).

    Google Scholar 

  23. Dau H., Milenkovic O.: MaxMinSum Steiner systems for access balancing in distributed storage. SIAM J. Discret. Math. 32(3), 1644–1671 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  24. Dewar M., Stevens B.: Ordering Block Designs: Gray Codes, Universal Cycles and Configuration Orderings CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2012).

    Book  MATH  Google Scholar 

  25. Dinitz J.H., Rodney P.: Disjoint difference families with block size \(3\). Util. Math. 52, 153–160 (1997).

    MathSciNet  MATH  Google Scholar 

  26. Dinitz J.H., Shalaby N.: Block disjoint difference families for Steiner triple systems: \(v\equiv 3\,\text{ mod }\,6\). J. Stat. Plann. Inference 106(1–2), 77–86 (2002).

    Article  MATH  Google Scholar 

  27. Feng T., Horsley D., Wang X.: Novák’s conjecture on cyclic Steiner triple systems and its generalization (2020)

  28. Grannell M..J.., Griggs T..S.: Product constructions for cyclic block designs. I. Steiner quadruple systems. J. Combin. Theory Ser. A 36(1), 56–65 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  29. Guregová M., Rosa A.: Using the computer to investigate cyclic Steiner quadruple systems. Mat. Časopis Sloven. Akad. Vied 18, 229–239 (1968).

    MathSciNet  MATH  Google Scholar 

  30. Hanani H.: A note on Steiner triple systems. Math. Scand. 8(1), 154–156 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  31. Hartman A.: The existence of resolvable Steiner quadruple systems. J. Combin. Theory Ser. A 44(2), 182–206 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  32. Hedayat A.S., Sloane N.J.A., Stufken J.: Orthogonal Arrays. Springer, New York (1999).

    Book  MATH  Google Scholar 

  33. Ji L., Zhu L.: Resolvable Steiner quadruple systems for the last 23 orders. SIAM J. Discrete Math. 19(2), 420–430 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  34. Longyear J.Q.: Nested group divisible designs and small nested designs. J. Stat. Plann. Inference 13(1), 81–87 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  35. Lu J.X.: Construction methods for balanced incomplete block designs and resolvable balanced incomplete block designs (in Chinese). In: L. Wu, L. Zhu, Q. Kang (eds.) Collected Works of Jiaxi Lu, pp. 1–24. Inner Mongolia People’s Press, Hunhot, Mongolia (1990). Unpublished (1965).

  36. Lu X.N., Jimbo M.: Affine-invariant strictly cyclic Steiner quadruple systems. Des. Codes Cryptogr. 83(1), 33–69 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  37. Mathon R.A., Phelps K.T., Rosa A.: Small Steiner triple systems and their properties. Ars Combin. 15, 3–110 (1983).

    MathSciNet  MATH  Google Scholar 

  38. Meng Z.: Doubly resolvable Steiner quadruple systems and related designs. Des. Codes Cryptogr. 84(3), 325–343 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  39. Meng Z., Du B.: The last twenty orders of \((1,2)\)-resolvable Steiner quadruple systems. Discret. Math. 312(24), 3574–3584 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  40. Mood A.M.F., Graybill F.A., Boes D.C.: Introduction to the Theory of Statistics. McGraw-Hill, Chennai (1974).

    MATH  Google Scholar 

  41. Novák J.: A note on disjoint cyclic Steiner triple systems. In: Recent advances in graph theory (Proceedings Second Czechoslovak Symposium, Prague, 1974), pp. 439–440 (1975)

  42. Peltesohn R.: Eine Lösung der beiden Heffterschen Differenzenprobleme. Compos. Math. 6, 251–257 (1939).

    MathSciNet  MATH  Google Scholar 

  43. Phelps K.T., Rosa A.: Steiner triple systems with rotational automorphisms. Discret. Math. 33(1), 57–66 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  44. Ray-Chaudhuri D.K., Wilson R.M.: Solution of Kirkman’s schoolgirl problem. In: Combinatorics (Proceedings of Symposium Pure Mathematics, Vol. XIX, University California, Los Angeles, CA, 1968), pp. 187–203 (1971).

  45. Siemon H.: Cyclic Steiner quadruple systems and Köhler’s orbit graphs. Des. Codes Cryptogr. 1(2), 121–132 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  46. Skolem T.: Some remarks on the triple systems of Steiner. Math. Scand. 6, 273–280 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  47. Stewart B.M.: Supermagic complete graphs. Can. J. Math. 19, 427–438 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  48. Stinson D.R.: Combinatorial Designs. Constructions and Analysis. Springer, New York (2004).

    MATH  Google Scholar 

  49. Vanstone S.A., Stinson D.R., Schellenberg P.J., Rosa A., Rees R.S., Colbourn C.J., Carter M.W., Carter J.E.: Hanani triple systems. Israel J. Math. 83(3), 305–319 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  50. Wei H., Ge G., Colbourn C.J.: The existence of well-balanced triple systems. J. Combin. Des. 24(2), 77–100 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  51. Yu W., Zhang X., Ge G.: Optimal fraction repetition codes for access-balancing in distributed storage. IEEE Trans. Inf. Theory 67(3), 1630–1640 (2021).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Thanks to Yeow Meng Chee, Dylan Lusi, and Olgica Milenkovic for helpful discussions.

Author information

Authors and Affiliations

  1. School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, AZ, USA

    Charles J. Colbourn

Authors
  1. Charles J. Colbourn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles J. Colbourn.

Additional information

Communicated by D. Ghinelli.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work was supported by NSF Grant CCF 1816913.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Colbourn, C.J. Egalitarian Steiner triple systems for data popularity. Des. Codes Cryptogr. 89, 2373–2395 (2021). https://doi.org/10.1007/s10623-021-00925-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-021-00925-0

Keywords

Mathematics Subject Classification

Search

Navigation