New cyclic difference sets with Singer parameters

AbstractThe main result in this paper is a general construction of φ(m)/2 pairwise inequivalent cyclic difference sets with Singer parameters (v,k,λ)=(2m−1,2m−1,2m−2) for any m⩾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(2m) 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 Müller–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