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Partitioned difference families versus zero-difference balanced functions

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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.

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Notes

  1. This is a \((k\times |G_2|)\)-matrix with entries in \(G_2\) such that the differences of any two distinct rows give a permutation of \(G_2\).

  2. We mean the graph whose vertices are the blocks of \({\mathcal {F}}\) (repeated blocks have to be considered as distinct vertices) and where two blocks are adjacent if and only if they are not disjoint.

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Perugia, 06123, Perugia, Italy

    Marco Buratti

  2. Mathematical Institute, University of Augsburg, 86135, Augsburg, Germany

    Dieter Jungnickel

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  1. Marco Buratti
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  2. Dieter Jungnickel
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Correspondence to Marco Buratti.

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Communicated by M. Paterson.

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Buratti, M., Jungnickel, D. Partitioned difference families versus zero-difference balanced functions. Des. Codes Cryptogr. 87, 2461–2467 (2019). https://doi.org/10.1007/s10623-019-00632-x

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