Qualified difference sets : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
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Date
2009
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Massey University
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Abstract
Qualified difference sets are a class of combinatorial configuration. The sets are related
to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer.
Qualified difference sets consist of a set of residues modulo an integer v and they possess
attractive properties that suggest potential applications in areas such as image formation,
signal processing and aperture synthesis. This thesis outlines the theory behind qualified
difference sets and gives conditions for the existence and nonexistence of these sets in
various cases.
A special case of the qualified difference sets is the qualified residue difference sets.
These consist of the set of nth power residues of certain types of prime. Necessary and
sufficient conditions for the existence of qualified residue difference sets are derived and
the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified
residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18.
A generalisation of the qualified residue difference sets is introduced. These are the
qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class
is defined for an integer power n and the results of an exhaustive computer search are
presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were
discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and
12.
An explanation of how qualified difference sets may be implemented in physical
applications is given and potential applications are discussed.
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Keywords
Residue difference sets, Cyclotomic class