A disjoint (v, k, k − 1) difference family in an additive group G is a partition of G{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.

On disjoint (v,k,k-1) difference families / Buratti, Marco. - In: DESIGNS, CODES AND CRYPTOGRAPHY. - ISSN 0925-1022. - 87:4(2019), pp. 745-755. [10.1007/s10623-018-0511-4]

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

Buratti, Marco
2019

Abstract

A disjoint (v, k, k − 1) difference family in an additive group G is a partition of G{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.
2019
Disjoint difference family; Ferrero pair; Frobenius group; Pisano period; Zero difference balanced function
01 Pubblicazione su rivista::01a Articolo in rivista
On disjoint (v,k,k-1) difference families / Buratti, Marco. - In: DESIGNS, CODES AND CRYPTOGRAPHY. - ISSN 0925-1022. - 87:4(2019), pp. 745-755. [10.1007/s10623-018-0511-4]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1654660
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