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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 745–755 | Cite as

On disjoint \((v,k,k-1)\) difference families

  • Marco BurattiEmail author
Article
  • 107 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

A disjoint \((v,k,k-1)\) difference family in an additive group G is a partition of \(G{\setminus }\{0\}\) into sets of size k whose lists of differences cover, altogether, every non-zero element of G exactly \(k-1\) times. The main purpose of this paper is to get the literature on this topic in order, since some authors seem to be unaware of each other’s work. We show, for instance, that a couple of heavy constructions recently presented as new, had been given in several equivalent forms over the last forty years. We also show that they can be quickly derived from a general nearring theory result which probably passed unnoticed by design theorists and that we restate and reprove, more simply, in terms of differences. This result can be exploited to get many infinite classes of disjoint \((v,k,k-1)\) difference families; here, as an example, we present an infinite class coming from the Fibonacci sequence. Finally, we will prove that if all prime factors of v are congruent to 1 modulo k, then there exists a disjoint \((v,k,k-1)\) difference family in every group, even non-abelian, of order v.

Keywords

Disjoint difference family Zero difference balanced function Frobenius group Ferrero pair Pisano period 

Mathematics Subject Classification

05B10 

Notes

Acknowledgements

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly

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