Advertisement

Partitioned difference families versus zero-difference balanced functions

  • Marco BurattiEmail author
  • Dieter Jungnickel
Article
  • 60 Downloads

Abstract

In a recent paper by the first author in this journal it was pointed out that the literature on zero-difference balanced functions is often repetitive and of little value. Indeed it was shown that some papers published in the last decade on this topic reproduced in a very convoluted way simple results on difference families which were known since the 90s or even earlier. In spite of this fact, unfortunately, a new paper of the same kind has recently appeared in this journal. Its main result was indeed already obtained by Furino in 1991 and here it will be shown that it is only a very special case of a much more general result by the first author. We take this opportunity to make a comparison between the equivalent notions of a partitioned difference family (PDF) and a zero-difference balanced function (ZDBF), explaining the reasons for which we prefer to adopt the terminology and notation of PDFs. Finally, “playing” with some known results on difference families, we produce a plethora of disjoint difference families with new parameters. Each of them can be viewed as a PDF with many blocks of size 1; therefore, even though the ZDBF community do not appear concerned about this, they are not so relevant from the design theory perspective. The main goal of this note is to explain the relationships between ZDBFs and the prior research, giving an example of how seemingly novel ZBDF results can be readily obtained from well known results on difference families.

Keywords

Disjoint difference family Partitioned difference family Relative difference family Zero-difference balanced function 

Mathematics Subject Classification

05B10 

Notes

Acknowledgements

The authors are very grateful to Chris Mitchell for reading and commenting on this note.

This work has been performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy.

References

  1. 1.
    Abel R.J.R., Buratti M.: Difference families. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 392–409. Chapman & Hall/CRC, Boca Raton, FL (2006).Google Scholar
  2. 2.
    Beth T., Jungnickel D., Lenz H.: Design Theory. Cambridge University Press, Cambridge (1999).CrossRefzbMATHGoogle Scholar
  3. 3.
    Buratti M.: On simple radical difference families. J. Comb. Des. 3, 161–168 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buratti M.: Recursive constructions for difference matrices and relative difference families. J. Comb. Des. 6, 165–182 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Buratti M.: Pairwise balanced designs from finite fields. Discret. Math. 208/209, 103–117 (1999).Google Scholar
  6. 6.
    Buratti M.: Old and new designs via difference multisets and strong difference families. J. Comb. Des. 7, 406–425 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buratti M.: Two new classes of difference families. J. Comb. Theory Ser. A 90, 353–355 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buratti M.: On disjoint \((v, k, k-1)\) difference families. Des. Codes Cryptogr. (2018).  https://doi.org/10.1007/s10623-018-0511-4.
  9. 9.
    Buratti M.: Hadamard partitioned difference families and their descendants. Cryptogr. Commun. (2018).  https://doi.org/10.1007/s12095-018-0308-3.
  10. 10.
    Buratti M., Ghinelli D.: On disjoint \((3t,3,1)\) cyclic difference families. J. Stat. Plann. Inference 140, 1918–1922 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Buratti M., Yan J., Wang C.: From a \(1\)-rotational RBIBD to a partitioned difference family. Electron. J. Comb. 17, \(\sharp \)R139 (2010).Google Scholar
  12. 12.
    Cai H., Zeng X., Helleseth T., Tang X., Yang Y.: A new construction of zero-difference balanced functions and its applications. IEEE Trans. Inf. Theory 59, 5008–5015 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Clay J.R.: Nearrings: Geneses and Applications. Oxford University Press, Oxford (1992).zbMATHGoogle Scholar
  14. 14.
    Ding C.: Optimal constant composition codes from zero-difference balanced functions. IEEE Trans. Inf. Theory 54, 5766–5770 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ding C., Yin J.: Combinatorial constructions of optimal constant composition codes. IEEE Trans. Inf. Theory 51, 3671–3674 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ding C., Wang Q., Xiong M.: Three new families of zero-difference balanced functions with applications. IEEE Trans. Inf. Theory 60, 2407–2413 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dinitz J.H., Rodney P.: Block disjoint difference families for Steiner triple systems. Util. Math. 52, 153–160 (1997).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dinitz J.H., Shalaby N.: Block disjoint difference families for Steiner triple systems: \(v\equiv 3\) (mod 6). J. Stat. Plan. Inference 106, 77–86 (2002).CrossRefzbMATHGoogle Scholar
  19. 19.
    Ferrero G.: Classificazione e costruzione degli stems \(P\)-singolari, Ist. Lomb. Acad. Sci. Lett. Rend. A 105, 597–613 (1968).zbMATHGoogle Scholar
  20. 20.
    Ferrero G.: Stems planari e BIB-disegni. Riv. Math. Univ. Parma 11, 79–96 (1970).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Furino S.: Difference families from rings. Discret. Math. 97, 177–190 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jungnickel D.: Composition theorems for difference families and regular planes. Discret. Math. 23, 151–158 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jungnickel D.: Transversal designs associated with Frobenius groups. J. Geom. 17, 140–154 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jungnickel D., Pott A., Smith K.W.: Difference sets. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 419–435. Chapman & Hall/CRC, Boca Raton, FL (2006).Google Scholar
  25. 25.
    Kaspers, C., Pott, A.: Solving isomorphism problems about \(2\)-designs from disjoint difference families. J. Comb. Des. (2019).  https://doi.org/10.1002/jcd.21648
  26. 26.
    Li S., Wei H., Ge G.: Generic constructions for partitioned difference families with applications: a unified combinatorial approach. Des. Codes Cryptogr. 82, 583–599 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Liu J., Jiang Y., Zheng Q., Lin D.: A new construction of zero-difference balanced functions and two applications. Des. Codes Cryptogr. (2019).  https://doi.org/10.1007/s10623-019-00616-x.
  28. 28.
    Momihara K.: Disjoint difference families from Galois rings. Electron. J. Comb. 24, 3–23 (2017).MathSciNetzbMATHGoogle Scholar
  29. 29.
    Netto E.: Zur Theorie der Tripelsysteme. Math. Ann. 42, 143–152 (1893).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wilson R.M.: Cyclotomic and difference families in elementary abelian groups. J. Number Theory 4, 17–47 (1972).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly
  2. 2.Mathematical InstituteUniversity of AugsburgAugsburgGermany

Personalised recommendations