New cyclic difference sets with Singer parameters

Abstract

The main result in this paper is a general construction of φ( m)/2 pairwise inequivalent cyclic difference sets with Singer parameters ( v, k, λ)=(2 m −1,2 m−1 ,2 m−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(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 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.