New cyclic difference sets with Singer parameters

The main result in this paper is a general construction of @f(m)/2 pairwise inequivalent cyclic difference sets with Singer parameters (v,k,@l)=(2^m-1,2^m^-^1,2^m^-^2) for anym>=3. The construction was conjectured by the second author at Oberwolfach in 1998. We also give a complete proof of related conjectures made by No, Chung and Yun and by No, Golomb, Gong, Lee and Gaal which produce another difference set for each m>=7 not a multiple of 3. Our proofs exploit Fourier analysis on the additive group of GF(2^m) and draw heavily on the theory of quadratic forms in characteristic 2. By-products of our results are a new class of bent functions and a new short proof of the exceptionality of the Muller-Cohen-Matthews polynomials. Furthermore, following the results of this paper, there are today no sporadic examples of difference sets with these parameters; i.e. every known such difference set belongs to a series given by a constructive theorem.

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