On the Existence of Abelian Hadamard Difference Sets and a New Family of Difference Sets

Authors:
Yu Qing Chen
Affiliations:
Department of Mathematics, Ohio State University, Columbus, Ohio, 43210-1174, f1yuqchen@math.ohio-state.eduf1
Venue:
Finite Fields and Their Applications
Year:
1997

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Abstract

We present a construction of Hadamard difference sets in abelian groups of order 4p^4^n, whose Sylowp-subgroups are elementary. By a standard composition procedure, we can now conclude that (4h^2, 2h^2-h,h^2-h)-Hadamard difference sets exist forh= 2^@e^"^13^@e^"^2u^2, where @e"1, @e"2= 0 or 1 anduis a positive integer. We then generalize the construction of Hadamard difference sets to construct a family of (4q^2^n(q^2^n- 1)/(q^2-1),q^2^n-1[2(q^2^n- 1)/(q+ 1) + 1], (q^2^n-q^2^n-1)(q^2^n-1 + 1)/(q+ 1)-difference sets, whereqis an even power of an odd prime or any power of 3.