Abstract
Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of difference families and the variations in priorities in constructions. We propose a definition of disjoint difference families that encompasses these variations and allows a comparison of the similarities and disparities. We then focus on two constructions of disjoint difference families arising from frequency hopping sequences and show that they are in fact the same. We conclude with a discussion of the notion of equivalence for frequency hopping sequences and for disjoint difference families.
Highlights
Difference sets and their generalisations to difference families arise from the study of designs and many other applications
In [28], Fuji-Hara et al stated “Often we are interested in properties of frequency hopping (FH) sequences, such as auto-correlation, randomness and generating method, which remain unchanged when passing from one FH sequence to another that is essentially the same
We have given a general definition of a disjoint difference family, and have seen a range of examples of applications in communications and information security for these difference families, with different applications placing different constraints on the associated properties and parameters
Summary
Difference sets and their generalisations to difference families arise from the study of designs and many other applications. The generalisation of difference sets to internal and This is one of several papers published in Designs, Codes and Cryptography comprising the 25th Anniversary Issue. B. Paterson external difference families arises from many applications in communications and information security. One particular class of internal difference family arises from frequency hopping (FH) sequences. In another paper by Fuji-Hara et al [28] various families of FH sequences were constructed using designs with particular automorphisms, and the question was raised there as to whether these constructions are the same as the LFSR constructions in [50]. The relationship between the equivalence of difference families and the equivalence of the designs and codes that arise from them has been much studied. We will focus on the notion of equivalence for frequency hopping sequences and for disjoint difference families
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