152 research outputs found
Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3,3^{2h+1})
Using a class of permutation polynomials of obtained from the
Ree-Tits symplectic spreads in , we construct a family of skew
Hadamard difference sets in the additive group of . With the help
of a computer, we show that these skew Hadamard difference sets are new when
and . We conjecture that they are always new when .
Furthermore, we present a variation of the classical construction of the twin
prime power difference sets, and show that inequivalent skew Hadamard
difference sets lead to inequivalent difference sets with twin prime power
parameters.Comment: 18 page
Cyclotomic Constructions of Skew Hadamard Difference Sets
We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
, where is a prime and a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
Formal duality
We provide an overview of formal duality with an emphasis on the authors contributions. Every formally dual set can be obtained from a primitive formally dual set or, more generally from an irreducible formally dual set.Using several methods, including even set theory and the field-descent method, it is possible to obtain examples of primitive/irreducible formally dual sets as well as non-existence results. A graph search algorithm can be used for further investigation. Overall, primitive formally dual sets seem rare in cyclic groups, but occasionally exist in finite abelian groups.In dieser Dissertation geben wir einen Überblick über formale Dualität. Jede formal duale Menge kann von einer primitiven, oder allgemeiner von einer irreduziblen, formal dualen Menge, konstruiert werden. Methoden wie die 'even set' Theorie oder die 'field-descent' Methode, können genutzt werden, um Beispiele für primitive/irreduzible formal duale Mengen sowie nicht-Existenz Resultate zu erhalten. Weiterhin kann ein Suchalgorithmus genutzt werden. Primitive formal duale Mengen scheinen in zyklischen Gruppen selten zu sein, kommen aber gelegentlich in endlichen abelschen Gruppen vor
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